Lecture Notes

Math 524 – Linear Analysis on Manifolds Spring 2012 Pierre Albin

Contents Contents

i

1 Differential operators on manifolds

3

1.1

Smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.5

Pull-back bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.6

The cotangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7

Operations on vector bundles . . . . . . . . . . . . . . . . . . . . . . 12

1.8

Sections of bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.9

Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.10 Orientations and integration . . . . . . . . . . . . . . . . . . . . . . . 18 1.11 Exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.12 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.14 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 A brief introduction to Riemannian geometry

25

2.1

Riemannian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2

Parallel translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3

Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4

Associated bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 i

ii

CONTENTS 2.5

E-valued forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6

Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7

Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.8

Complete manifolds of bounded geometry

2.9

Riemannian volume and formal adjoint . . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . 42

2.10 Hodge star and Hodge Laplacian . . . . . . . . . . . . . . . . . . . . 44 2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.12 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Laplace-type and Dirac-type operators

53

3.1

Principal symbol of a differential operator . . . . . . . . . . . . . . . 53

3.2

Laplace-type operators . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3

Dirac-type operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4

Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5

Hermitian vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6.1

The Gauss-Bonnet operator . . . . . . . . . . . . . . . . . . . 68

3.6.2

The signature operator . . . . . . . . . . . . . . . . . . . . . . 70

3.6.3

The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . 71

3.6.4

Twisted Clifford modules . . . . . . . . . . . . . . . . . . . . . 73

3.6.5

The ∂ operator . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.7

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.8

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Pseudodifferential operators on Rm

79

4.1

Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3

4.2.1

Fundamental solutions . . . . . . . . . . . . . . . . . . . . . . 84

4.2.2

Principal values and finite parts . . . . . . . . . . . . . . . . . 85

Tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 86

CONTENTS

1

4.3.1

Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.2

Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4

Pseudodifferential operators on Rm . . . . . . . . . . . . . . . . . . . 92 4.4.1

Symbols and oscillatory integrals . . . . . . . . . . . . . . . . 92

4.4.2

Amplitudes and mapping properties . . . . . . . . . . . . . . . 98

4.5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.6

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5 Pseudodifferential operators on manifolds

109

5.1

Symbol calculus and parametrices . . . . . . . . . . . . . . . . . . . . 109

5.2

Hodge decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.3

The index of an elliptic operator . . . . . . . . . . . . . . . . . . . . . 118

5.4

Pseudodifferential operators acting on L2 . . . . . . . . . . . . . . . . 122

5.5

Sobolev spaces

5.6

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.7

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6 Spectrum of the Laplacian

135

6.1

Spectrum and resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2

The heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3

Weyl’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.4

Long-time behavior of the heat kernel . . . . . . . . . . . . . . . . . . 148

6.5

Determinant of the Laplacian . . . . . . . . . . . . . . . . . . . . . . 149

6.6

The Atiyah-Singer index theorem . . . . . . . . . . . . . . . . . . . . 151 6.6.1

Chern-Weil theory . . . . . . . . . . . . . . . . . . . . . . . . 151

6.6.2

The index theorem . . . . . . . . . . . . . . . . . . . . . . . . 155

6.7

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.8

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.9

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

2

CONTENTS

Lecture 1 Differential operators on manifolds 1.1

Smooth manifolds

The m-dimensional Euclidean space Rm , the m-dimensional sphere Sm and the mdimensional torus Tm = S1 × . . . × S1 are all examples of smooth manifolds, the context where our studies will take place. A general manifold is a space constructed from open sets in Rm by patching them together smoothly. Formally, M is a manifold if it is a paracompact1 Hausdorff topological space with an ‘atlas’, i.e., a covering by open sets {Vα } together with homeomorphisms φα : Uα −→ Vα for some open subsets {Uα } of Rm . The integer m is known as the dimension of M and the maps φα : Uα −→ Vα (or sometimes just the sets {Vα }) are known as coordinate charts. An atlas is smooth if whenever Vαβ := Vα ∩ Vβ 6= ∅ the map (1.1)

−1 ψαβ := φβ−1 ◦ φα : φ−1 α (Vαβ ) −→ φβ (Vαβ )

−1 m is a smooth diffeomorphism between the open sets φ−1 α (Vαβ ), φβ (Vαβ ) ⊆ R . (For us smooth will always mean C ∞ .) If the maps ψαβ are all C k -smooth, we say that the atlas is C k .

If φα : Uα −→ Vα is a coordinate chart, it is typical to write (y1 , . . . , ym ) = φ−1 α : Vα −→ Uα and refer to the yi as local coordinates of M on Vα . 1

A topological space is paracompact if every open cover has a subcover in which each point is covered finitely many times.

3

4

LECTURE 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

Two smooth atlases are compatible (or equivalent) if their union is a smooth atlas. A smooth structure on M is an equivalence class of smooth atlases or, equivalently, a maximal smooth atlas on M. A manifold M together with a smooth structure is a smooth manifold. To indicate that a manifold M is m-dimensional we will sometimes write M m or, more often, we will refer to M as an m-manifold. A more general notion than manifold is that of a manifold with boundary. The former is defined in the same way as a manifold, but the local model is m Rm − = {(x1 , . . . , xn ) ∈ R : x1 ≤ 0}.

If M is a smooth manifold with boundary, its boundary ∂M consists of those points in M that are in the image of ∂Rm − for some (and hence all) local coordinate charts. It is easy to check that ∂M is itself a smooth manifold (without boundary). The product of two manifolds with boundary is an example of a still more general notion, a manifold with corners. A map between two smooth manifolds (with or without boundary) F : M −→ M 0 is smooth if it is smooth in local coordinates. That is, whenever φα : Uα −→ Vα ⊆ M,

φ0γ : Uγ0 −→ Vγ0 ⊆ M 0 ,

are coordinate charts, the map (1.2)

(φ0γ )−1 ◦ F ◦ φα : Uα −→ Uγ0

is smooth (whenever it is defined). Notice that this definition is equivalent to making coordinate charts diffeomorphisms. The space of smooth maps between M and M 0 will be denoted C ∞ (M, M 0 ) and when M 0 = R, simply by C ∞ (M ). Notice that a map F ∈ C ∞ (M, M 0 ) naturally induces a pull-back map F ∗ : C ∞ (M 0 ) −→ C ∞ (M ),

F ∗ (h)(p) = h(F (p)) for all h ∈ C ∞ (M 0 ), p ∈ M.

(So we can think of C ∞ as a contravariant functor on smooth manifolds.) For us a manifold will mean a connected smooth manifold without boundary, unless otherwise specified. Similarly, unless we specify otherwise, a map between manifolds will be smooth.

1.2. PARTITIONS OF UNITY

5

A subset X ⊆ M is a submanifold of M of dimension k if among the m coordinates on M at each point of X there are k that restrict to give coordinates of X. That is, at each point p ∈ X there is a coordinate chart of M, φ : U −→ V with q ∈ V, whose restriction to U ∩ Rk × {0} maps into X, φ U ∩Rk ×{0} : U ∩ Rk × {0} −→ V ∩ X. These restricted charts then define a smooth k-manifold structure on X.

1.2

Partitions of unity

A manifold is defined by patching together many copies of Rm . It is often useful to take a construction on Rm and transfer it to a manifold by carrying it out on each patch and then adding the pieces together. For these constructions, it is very convenient that every open covering Vα of a smooth manifold carries a partition of unity. That is, a collection ωi of smooth functions ωi : M −→ R such that: i) At each p ∈ M only finitely many ωi (p) are non-zero ii) Each ωi is supported inside some Vα iii) The sum of the ωi is the constant function equal to one at each point p ∈ M. One remarkable fact that follows easily from the existence of partitions of unity is that any compact smooth manifold can be identified with a smooth submanifold of RN for some N. This is easy to show using a finite atlas with a corresponding partition of unity. Whitney’s embedding theorem shows that any smooth manifold M embeds into R2 dim M +1 and a compact smooth manifold into R2 dim M .

1.3

Tangent bundle

We know what a smooth function f : M −→ R is, what is its derivative? If γ is a smooth curve on M, that is, a smooth map γ : R −→ M, then f ◦ γ is a map from R to itself so its derivative is unambiguously defined. We can take advantage of this to define the derivative of f at a point p ∈ M in terms of the curves through that point.

6


Indeed, let us say that two curves γ1 and γ2 are equivalent at p if γ1 (0) = γ2 (0) = p and ∂ ∂ −1 (φ−1 ◦ γ2 ) (φ ◦ γ ) = 1 ∂t t=0 ∂t t=0 for some (and hence any) coordinate chart φ : U −→ V ⊆ M containing p. It is easy to see that this is an equivalence relation and we refer to an equivalence class as a tangent vector at p ∈ M. The tangent space to M at p is the set of all tangent vectors at p and is denoted Tp M. For the case of Euclidean space, the tangent space to Rm at any point is another copy of Rm , Tp Rm = Rm . The tangent space of the m-sphere is also easy to describe. By viewing Sm as the unit vectors in Rm+1 , curves on Sm are then also curves in Rm+1 so we have Tp Sm ⊆ Tp Rm+1 = Rm+1 , and a tangent vector at p will correspond to a curve on Sm precisely when it is orthogonal to p as a vector in Rm+1 , thus Tp Sm = {v ∈ Rm+1 : v · p = 0}. Clearly if γ1 is equivalent to γ2 at p then ∂ ∂ (f ◦ γ1 ) = (f ◦ γ2 ) ∂t t=0 ∂t t=0 for all f : M −→ R, so we can think of the derivative of f at p as a map from equivalence classes of curves through p to equivalence classes of curves through f (p), thus Dp f : Tp M −→ Tf (p) R. More generally, any smooth map F : M1 −→ M2 defines a map, known as its differential, Dp F : Tp M1 −→ TF (p) M2 by sending the equivalence class of a curve γ on M1 to the equivalence class of the curve F ◦γ on M2 . When the point p is understood, we will omit it from the notation. If the differential of a map is injective we say that the map is an immersion, if it is surjective we say that it is a submersion. If F : M1 −→ M2 is an injective immersion which is a homeomorphism between M1 and F (M1 ) then we say that F is an embedding. A connected subset X ⊆ M is a submanifold of M precisely when the inclusion map X ,→ M is an embedding.

1.3. TANGENT BUNDLE

7

If F is both an immersion and a submersion, i.e., if DF is always a linear isomorphism, we say that it is a local diffeomorphism. By the inverse function theorem, if F is a local diffeomorphism then every point p ∈ M1 has a neighborhood V ⊆ M1 such that F V is a diffeomorphism onto its image. A diffeomorphism is a bijective local diffeomorphism. The chain rule is the fact that commutative diagrams M1

F1

/

M2 !

F2 ◦F1

F2

M3

of smooth maps between manifolds induce, for each p ∈ M1 , commutative diagrams between the corresponding tangent spaces, Tp M1

DF1

/ TF (p) M2 1

D(F2 ◦F1 )

&

DF2

TF2 (F1 (p)) M3 i.e., that D(F2 ◦ F1 ) = DF2 ◦ DF1 . Taking differentials of coordinate charts of M, we end up with coordinate charts on [ TM = Tp M p∈M

showing that T M, the tangent bundle of M, is a smooth manifold of twice the dimension of M. For Rm the tangent bundle is simply the product T Rm = Rm × Rm where the first factor is thought of as a point in Rm and the second as a vector ‘based’ at that point; the tangent bundle of an open set in Rm has a very similar description, we merely restrict the first factor to that open set. (We can think of M 7→ T M together with F 7→ DF as a covariant functor on smooth manifolds.) If φ : U −→ V is a coordinate chart on M and p ∈ V ⊆ M, then Dp φ identifies Tp M with Rm . A different choice of coordinate chart gives a different identification of Tp M with Rm , but these identifications differ by an invertible linear map. This means that each Tp M is naturally a vector space of dimension m, which however does not have a natural basis. Now that we know the smooth structure of T M, note that the differential of a smooth map between manifolds is a smooth map between their tangent bundles.

8


There is also a natural smooth projection map π : T M −→ M that sends elements of Tp M to p and when M = Rm is simply the projection onto the first factor. It is typical, for any subset Z ⊆ M, to use the notations π −1 (Z) =: T M Z =: TZ M.

1.4

Vector bundles

A coordinate chart φ : U −→ V on a smooth m-manifold M induces a coordinate chart on T M and an identification Dφ−1

φ×id

π −1 (V) −−−−→ T U = U × Rm −−−−→ V × Rm

(1.3)

which restricts to each fiber π −1 (p) to a linear isomorphism. We say that a coordinate chart of M locally trivializes T M as a ‘vector bundle’. A vector bundle over a manifold M is a locally trivial collection of vector spaces, one for each point in M. Formally a rank k real vector bundle consists of a smooth manifold E together with a smooth surjective map π : E −→ M such that the fibers π −1 (p) form vector spaces over R and satisfying a local triviality condition: for any point p ∈ M and small enough neighborhood V of p there is a diffeomorphism Φ : V × Rk −→ π −1 (V) whose restriction to any q ∈ V is a linear isomorphism Φ q∈V : {q} × Rk −→ π −1 (q). We refer to the pair (V, Φ) as a local trivialization of E. We will often refer to a rank k real vector bundle as a Rk -bundle. The discussion below of Rk -vector bundles applies equally well to complex vector bundles, or Ck -vector bundles, which are the defined by systematically replacing Rk with Ck above. π

If X is a submanifold of M, and E −−→ M is a vector bundle, then π|X E X −−−→ X is a vector bundle.

1.5. PULL-BACK BUNDLES

9

A subset of a vector bundle is a subbundle if it is itself a bundle in a consistent fashion. That is, E1 is a subbundle of E2 if E1 and E2 are both bundles over M, E1 ⊆ E2 , and the projection map E1 −→ M is the restriction of the projection map E2 −→ M. A bundle morphism between two vector bundles π

π

E1 −−1→ M1 ,

E2 −−2→ M2

is a map F : E1 −→ E2 that sends fibers of π1 to fibers of π2 and acts linearly between fibers. Thus for any bundle morphism F there is a map F : M1 −→ M2 such that the diagram (1.4)

E1 π1

M1

/

F

F

/

E2

π2

M2

commutes; we say that F covers the map F . An invertible bundle morphism is called a bundle isomorphism. The trivial Rk -bundle over a manifold M is the manifold Rk × M with projection the map onto the second factor. We will use Rk to denote the trivial Rk -bundle. A bundle is trivial if it is isomorphic to Rk for some k. A manifold is said to be parallelizable if its tangent bundle is a trivial bundle, e.g., Rm is parallelizable. We will show later that Sm is not parallelizable when m is even. The circle S1 is parallelizable and hence (by exercise 1) so are all tori Tm . Some interesting facts which we will not show is that any three dimensional manifold is parallelizable and that the only parallelizable spheres are S1 , S3 and S7 . The fact that these sphere are parallelizable is easy to see by viewing them as the unit vectors in the complex numbers, the quaternions, and the octionians, respectively; the fact that these are the only parallelizable ones is not trivial.

1.5

Pull-back bundles

Any bundle morphism covers a map between manifolds (1.4). There is a sort of converse to this: given a map between manifolds f : M1 −→ M2 , and a Rk -bundle π E −−→ M2 the set f ∗ E = {(p, v) ⊆ M1 × E : f (p) = π(v)}

10


fits into the commutative diagram f ∗E

fb

/E

π b

M1

f

/

π

M2

where, for any (p, v) ∈ f ∗ E, po

π b

(p, v)

fb

/

v.

Moreover, if (U, Φ) is a local trivialization of E and V is a coordinate chart in M1 such that f (V) ⊆ U then f ∗ E ∩ (V × π −1 (U)) = {(p, (Φ f (p) )−1 (x)) : p ∈ V, x ∈ Rk } and we can use this to introduce a C ∞ manifold structure on f ∗ E such that f ∗ E ∩ π b (V × π −1 (U)) is diffeomorphic to V × Rk making f ∗ E −−→ M1 a Rk -bundle. The bundle f ∗ E is called the pull-back of E along f . The simplest example of a pull-back bundle is the restriction of a bundle. If π π X is a submanifold of M, and E −−→ M is a vector bundle, then E X −−−X−→ X is (isomorphic to) the pull-back of E along the inclusion map X ,→ M. We see directly from the definition that the pull-back bundle of a trivial bundle is again trivial, but note that the converse is not true. Given a smooth map f : M1 −→ M2 , pull-back along f defines a functor between bundles on Y and bundles on X. Indeed, if E1 −→ Y and E2 −→ Y are bundles over Y and F : E1 −→ E2 is a bundle map between them then f ∗ F : f ∗ E1 −→ f ∗ E2 ,

f ∗ F (p, v) = (p, F (p)v) for all (p, v) ∈ f ∗ E1

is a bundle map between f ∗ E1 and f ∗ E2 . Another useful relation between pull-backs and bundle maps is the following: π π If E1 −−1→ M1 and E2 −−2→ M2 are vector bundles and F : E1 −→ E2 is a bundle map so that F / E2 E1 π1

M1

F

/

π2

M2 ∗

then F is equivalent to a bundle map between E1 −→ M1 and F E2 −→ M1 , namely ∗

E1 3 v 7→ (π1 (v), F (π1 (v))v) ∈ F E2 .

1.6. THE COTANGENT BUNDLE

11

We will show later (Remark 3) that if f : M1 −→ M2 and g : M1 −→ M2 are homotopic smooth maps, i.e., if there is a smooth map H : [0, 1] × M1 −→ M2 s.t. H(0, ·) = f (·) and H(1, ·) = g(·) then f ∗ E ∼ = g ∗ E for any vector bundle E −→ M2 . In particular, this implies that all vector bundles over a contractible space are trivial.

1.6

The cotangent bundle

Every real vector space V has a dual vector space V ∗ = Hom(V, R) consisting of all the linear maps from V to R. Given a linear map between vector spaces T : V −→ W there is a dual map between the dual spaces T ∗ : W ∗ −→ V ∗ ,

T ∗ (ω)(v) = ω(T (v)) for all ω ∈ W ∗ , v ∈ V. π

By taking the dual maps of the local trivializations of a vector bundle E −−→ M, the manifold E ∗ = {(p, ω) : p ∈ M, ω ∈ (π −1 (p))∗ } with the obvious projection E ∗ −→ M is seen to be a vector bundle, the dual bundle to E. Every vector v ∈ Rk defines a dual vector v ∗ ∈ (Rk )∗ , v ∗ : Rk −→ R,

v ∗ (w) = v · w,

and it is easy to see that the dual vectors of a basis of Rk form a basis of (Rk )∗ so that in fact v 7→ v ∗ is a vector space isomorphism. In particular the dual bundle of an Rk -bundle is again a Rk -bundle. Given vector bundles E1 −→ M1 and E2 −→ M2 and F : E1 −→ E2 a bundle map, we get a dual bundle map F ∗ : E2∗ −→ E1∗ . Thus taking duals defines a contravariant functor on the category of vector bundles.

12


The dual of a bundle map F : E1 −→ E2 between two bundles Ei −→ M is a map between their duals F ∗ : E2∗ −→ E1∗ . Thus taking duals defines a contravariant functor on the category of vector bundles on M. It is easy to see that this functor commutes with pull-back along a smooth map. The dual bundle to the tangent bundle is the cotangent bundle T ∗ M −→ M. Remark 1. The assignation M 7→ T ∗ M is natural, but is not a functor in the same way that M 7→ T M is. Rather, a map F : M1 −→ M2 between manifolds induces a map DF : T M1 −→ T M2 between their tangent bundles which, as explained above, we can interpret as a map DF : T M1 −→ F ∗ (T M2 ) between bundles over M1 . Now we can take duals and we get a bundle map (DF )∗ : F ∗ (T M2∗ ) −→ T M1∗ . Thus we can view M 7→ T ∗ M as a functor but its target category should consist of vector bundles where a morphism between E1 −→ M1 and E2 −→ M2 consists of a map F : M1 −→ M2 and a bundle map Φ : F ∗ E2 −→ E1 and composition is (F1 , Φ1 ) ◦ (F2 , Φ2 ) = (F1 ◦ F2 , Φ1 ◦ (F1∗ Φ2 )). This is called the category of star bundles in [Kolar-Michor-Slovak].

1.7

Operations on vector bundles

We constructed the vector bundle E ∗ −→ M from the vector bundle E −→ M by applying the vector space operation V 7→ V ∗ fiber-by-fiber. This works more generally whenever we have a smooth functor of vector spaces. Let Vect denote the category of finite dimensional vector spaces with linear isomorphisms, and let T be a (covariant) functor on Vect. We say that T is smooth if for all V and W in Vect the map Hom(V, W ) −→ Hom(T (V ), T (W )) is smooth. π

Let E −−→ M be a vector bundle over M, define G Tp (E) = T (π −1 (p)), T (E) = Tp (E) p∈M

1.7. OPERATIONS ON VECTOR BUNDLES

13

and let T (π) : T (E) −→ M be the map that sends Tp (E) to p. We claim that T (π)

T (E) −−−−→ M is a smooth vector bundle. Applying T fibrewise to a local trivialization of E, Φ : V × Rk −→ π −1 (V), we obtain a map T (Φ) : V × T (Rk ) −→ T (π)−1 (V), and we can use these maps as coordinate charts on T (E) to obtain the structure of a smooth manifold. Indeed, we only need to see that if two putative charts overlap then the map T (Φ1 )−1 T (Φ2 )

(V1 ∩ V2 ) × T (Rk ) −−−−−−−−−−→ (V1 ∩ V2 ) × T (Rk ) is smooth, and this follows from the smoothness of T . Thus T (E) is a smooth manifold, T (π) is a smooth function, and T (π)

T (E) −−−−→ M satisfies the local triviality condition, so we have a new vector bundle over M. The same procedure works for contravariant functors (meaning that Hom(V, W ) is sent to Hom(T (W ), T (V ))) as well as functors with several variables. Therefore, for instance, we can define for vector bundles E −→ M and F −→ M the vector bundles: i) Hom(E, F ) −→ M with fiber over p, Hom(Ep , Fp ). ii) E ⊗ F −→ M with fiber over p, Ep ⊗ Fp . iii) S 2 E −→ M whose fiber over p is the space of all symmetric bilinear transformations from Ep × Ep to R. iv) S k E −→ M whose fiber over p is the space of all symmetric k-linear transformations from Epk to R. v) Λk E −→ M whose fiber over p is the space of all antisymmetric k-linear transformations from Epk to R. We also obtain natural isomorphisms of vector bundles from isomorphisms of vector spaces. Thus, for instance, for vector bundles E, F, and G over M, we have2 2

Recall that R is the trivial R-bundle, R × M −→ M.

14

LECTURE 1. DIFFERENTIAL OPERATORS ON MANIFOLDS i) E ⊕ F ∼ =F ⊕E ii) E ⊗ R ∼ =E

iii) E ⊗ F ∼ =F ⊗E iv) E ⊗ (F ⊕ G) ∼ = (E ⊗ F ) ⊕ (E ⊗ G) v) Hom(E, F ) = E ∗ ⊗ F (so E ∗ ∼ = Hom(E, R)) M vi) Λk (E ⊕ F ) ∼ (Λi E ⊗ Λj F ) = i+j=k

vii) (Λk E)∗ = Λk (E ∗ )

1.8

Sections of bundles π

A section of a vector bundle E −−→ M is a smooth map s : M −→ E such that >E s

M

π id

/

M

commutes. We will denote the space of smooth sections of E by C ∞ (M ; E). This forms a vector space over R and, as such, it is infinite-dimensional (unless we are in a ‘trivial’ situation where (dim M )(rankE) = 0). However it is also a module over C ∞ (M ) (with respect to pointwise multiplication) and here it is finitely generated. For instance, the sections of a trivial bundle form a free C ∞ (M ) module, C ∞ (M ; Rn ) = (C ∞ (M ))n . We also introduce notation for functions and sections with compact support, Cc∞ (M ),

Cc∞ (M ; E)

and point out that the latter is a module over the former. One consequence of the module structure is that we can use a partition of unity on M to decompose a section into a sum of sections supported on trivializations of E. A section of the tangent bundle is known as a vector field. Thus a vector field on M is a smooth function that assigns to each point p ∈ M a tangent vector in Tp M. A vector field V ∈ C ∞ (M ; T M ) naturally acts on a smooth function by differentiation V : C ∞ (M ) −→ C ∞ (M ),

(V f )(p) = Dp f (V (p)) for all f ∈ C ∞ (M ), p ∈ M.

1.8. SECTIONS OF BUNDLES

15

If Vα is a coordinate chart on M with local coordinates y1 , . . . , ym then we can write V

Vα

=

m X

ai (y1 , . . . , ym )

i=1

∂ . ∂yi

Indeed, this is just the representation in the induced coordinates of T M with the identification (1.3). This action of V on C ∞ (M ) is linear and satisfies V (f h) = V (f )h + f V (h), for all f, h ∈ C ∞ (M ), i.e., it is a derivation of C ∞ (M ). In fact, it is possible to identify vector fields on M with the derivations of C ∞ (M ). (see e.g., [Lee]) As maps on C ∞ (M ), we can compose vector fields (V W )(f ) = V (W (f )) but the result is no longer a vector field. However, if we form the commutator of two vector fields, [V, W ]f = V (W (f )) − W (V (f )) then the result is again a vector field! In fact it is easy to check that this is a derivation on C ∞ (M ). The commutator of vector fields satisfies the ‘Jacobi identity’ [[U, V ], W ] + [[V, W ], U ] + [[W, U ], V ] = 0 and hence the commutator, combined with the R-vector space structure, gives C ∞ (M, T M ) the structure of a Lie algebra. Remark 2. Whereas a smooth map F : M1 −→ M2 induces a map DF : T M1 −→ T M2 , it is not always possible to ‘push-forward’ a vector field. We say that two vector fields V ∈ C ∞ (M1 ; T M1 ), and W ∈ C ∞ (M2 ; T M2 ) are F -related if the diagram TM O 1

DF

TM O 2 W

V

M1

/

F

/

M2

commutes, i.e., if DF ◦ V = W ◦ F. If F is a diffeomorphism, then we can push-forward a vector field: the vector fields V ∈ C ∞ (M1 ; T M1 ) and F∗ V = DF ◦ V ◦ F −1 ∈ C ∞ (M2 ; T M2 ) are always F -related. Moreover, F∗ [V1 , V2 ] = [F∗ V1 , F∗ V2 ] for all V1 , V2 ∈ C ∞ (M1 ; T M1 ), so M 7→ C ∞ (M ; T M ) is a functor from smooth manifolds with diffeomorphisms to Lie algebras and isomorphisms.

16


1.9

Differential forms

Sections of the cotangent bundle of M, T ∗ M, are known as differential 1-forms. More generally, sections of the k-th exterior power of the cotangent bundle, Λk T ∗ M, are known as differential k-forms. The typical notation for this space is Ωk (M ) = C ∞ (M ; Λk T ∗ M ),

Ω∗ (M ) =

dim MM

Ωk (M )

k=0

where by convention Ω0 (M ) = C ∞ (M ). A differential form in Ωk (M ) is said to have degree k. A differential k-form ω ∈ Ωk (M ) can be evaluated on k-tuples of vector fields, ω : C ∞ (M ; T M )k −→ R, it is linear in each variable and has the property that ω(Vσ(1) , . . . , Vσ(k) ) = sign(σ)ω(V1 , . . . , Vk ) for any permutation σ of {1, . . . , k}. Differential forms of all degrees Ω∗ (M ) form a graded algebra with respect to the wedge product Ωk (M ) × Ω` (M ) 3 (ω, η) 7→ ω ∧ η ∈ Ωk+` (M ) 1 X sign(σ)ω(Vσ(1) , . . . , Vσ(k) )η(Vσ(k+1) , . . . , Vσ(k+`) ). ω ∧ η(V1 , . . . , Vk+` ) = k!`! σ This product is graded commutative (so sometimes commutative and sometimes anti-commutative) ω ∈ Ωk (M ),

η ∈ Ω` (M ) =⇒ ω ∧ η = (−1)k` η ∧ ω.

In particular if ω ∈ Ωk (M ) with k odd, then ω ∧ ω = 0. Given a differential k-form, ω, we will write the operator ‘wedge product with ω, by eω : Ω∗ (M ) −→ Ω∗+k (M ),

eω (η) = ω ∧ η, for all η ∈ Ω∗ (M ).

We say that a k-form is elementary if it is the wedge product of k 1-forms. Elementary k-forms are easy to evaluate (ω1 ∧ · · · ∧ ωk )(V1 , . . . , Vk ) = det(ωi (Vj )).

1.9. DIFFERENTIAL FORMS

17

Let p be a point in a local coordinate chart V on M with coordinates y1 , . . . , ym . A basis of Tp∗ M dual to the basis {∂y1 , . . . , ∂ym } of Tp M is given by {dy 1 , . . . , dy m } where dy i (∂yj ) = δji . A basis for the space of k-forms in Tp∗ M is given by {dy i1 ∧ · · · ∧ dy ik : i1 < · · · < ik }. Any k-form supported in V can be written as a sum of elementary k-forms. Using a partition of unity we can write any k-form on M as a sum of elementary k-forms. Notice that anticommutativity of the wedge product implies that all elementary kforms, and hence all k-forms, of degree greater than the dimension of the manifold vanish. There is also an interior product of a vector field and a differential form, which is simply the contraction of the differential form by the vector field. For every W ∈ C ∞ (M ; T M ), the interior product is a map iW : Ω∗ (M ) −→ Ω∗−1 (M ) (iW ω)(V1 , . . . , Vk−1 ) = ω(W, V1 , . . . , Vk−1 ), for all ω ∈ Ωk , {Vi } ⊆ C ∞ (M ; T M ). We will also use the notation iW ω = W y ω. Notice that antisymmetry of differential forms induces anticommutativity of the interior product in that iV iW = −iW iV for any V, W ∈ C ∞ (M ; T M ). In particular, (iV )2 = 0. If F : M1 −→ M2 is a smooth map between manifolds, we can pull-back differential forms from M2 to M1 , F ∗ : Ωk (M2 ) −→ Ωk (M1 ), F ∗ ω(V1 , . . . , Vk ) = ω(DF (V1 ), . . . , DF (Vk )) for all ω ∈ Ωk (M2 ), Vi ∈ C ∞ (M ; T M1 ). Pull-back respects the wedge product, F ∗ (ω ∧ η) = F ∗ ω ∧ F ∗ η,

(1.5) and the interior product,

F ∗ (DF (W ) y ω) = W y F ∗ (ω). In the particular case of Rm , the pull-back of an m-form by a linear map T is (1.6)

T ∗ (dx1 ∧ · · · ∧ dxm ) = (det T )(dx1 ∧ · · · ∧ dxm ),

directly from the definition of the determinant.

18


1.10

Orientations and integration

If W is an m-dimensional vector space, and {u1 , . . . , um }, {v1 , . . . , vm } are bases of W, we will say that they represent the same orientation if the linear map that sends ui to vi has positive determinant. Equivalently, since Λm W is a one-dimensional vector space, we know that u1 ∧ · · · ∧ um and v1 ∧ · · · vm differ by multiplication by a non-zero real number and we say that they represent the same orientation if this number is positive. A representative ordered basis or non-zero m-form is said to be an orientation of W. The canonical orientation of Euclidean space Rm corresponds to the canonical ordered basis, {e1 , . . . , em }. An Rk -vector bundle E −→ M is orientable if we can find a smooth orientation of its fibers, i.e., if there is a nowhere vanishing section of Λk E −→ M. Notice that the latter is an R-bundle over M, so the existence of a nowhere vanishing section is equivalent to its being trivial. Thus E −→ M is orientable if and only if Λk E −→ M is a trivial bundle. An orientation is a choice of trivializing section, and a manifold together with an orientation is said to be oriented. Note that, since Λk (E ∗ ) = (Λk E)∗ , E is orientable if and only if E ∗ is orientable. An m-manifold M is orientable if and only if its tangent bundle is orientable. Equivalently, a manifold is orientable when its cotangent bundle is orientable, so when there is a nowhere vanishing m-form. A choice of nowhere vanishing m-form is called an orientation of M or also a volume form on M. A coordinate chart φ : U −→ V is oriented if it pulls-back the orientation of M to the canonical orientation of Rm . A manifold is orientable precisely when it has an atlas of oriented coordinate charts. If M is oriented, there is a unique linear map the integral Z : Cc∞ (M ; Λm M ) −→ R which is invariant under diffeomorphims and coincides in local coordinates with the Lebesgue integral. That is, if ω ∈ Ωm M is supported in a oriented coordinate chart φ : U −→ V then φ∗ ω ∈ Ωm (Rm ) and so we can take its Lebesgue integral. The assertion is that the result is independent of the choice of oriented coordinate chart, and the proof follows from (1.6) and the usual change of variable formula for the Lebesgue integral. By using a partition of unity, integration of any m-form reduces to integration of m-forms supported in oriented coordinate charts. It is convenient to formally extend the integral to a functional on forms of all degrees, by setting it equal to zero on all forms of less than maximal degree. What happens when a manifold is not orientable? At each point p ∈ M, the set of equivalence classes of orientations of T M, Op , has two elements, say o1 (p) and

1.11. EXTERIOR DERIVATIVE

19

o2 (p). We define the orientable double cover of M to be the set Or(M ) = {(p, oi (p)) : p ∈ M, oi (p) ∈ Op }. f is a smooth manifold. The manifold structure on M easily extends to show that M Indeed, given a coordinate chart φ : U −→ V of M we set, for each q ∈ U, e = (φ(q), [φ−1 (e1 ∧ · · · ∧ em )]) ∈ Or(M ) φ(q) f. The mapping π : M f −→ M that sends (p, oi (p)) and we get a coordinate chart on M to p is smooth and surjective. Moreover if V is a coordinate neighborhood in M then π −1 (V) = V1 ∪ V2 is the union of two disjoint open sets in Or(M ) and π restricted to each Vi is a diffeomorphism onto V, hence the term ‘double cover’. It is easy to see that Or(M ) is always orientable (at each point (p, oi (p)) choose the orientation corresponding to oi (p)). If M is connected and not orientable then Or(M ) is connected and orientable, and we can often pass to Or(M ) if we need an orientation. If M is orientable, then the same construction of Or(M ) yields a disconnected manifold consisting of two copies of M. Orientations are one place where the theory of complex vector bundles differs from that of general real vector bundles. Complex vector bundles are always orientable. Indeed, first note that a C-vector space W of complex dimension k has a canonical orientation as a R-vector space of real dimension 2k. We just take any basis of W over C, say b = {v1 , . . . , vk }, and then take bb = {v1 , iv1 , v2 , iv2 , . . . , vk , ivk } as a basis for W over R. The claim is that if b0 = {w1 , . . . , wk } is any other basis of W over C, then bb0 = {w1 , iw1 , . . . , wk , iwk } gives us the same orientation for W as a Rb from bb to bb0 has positive determinant. vector space, i.e., the change of basis matrix B The proof is just to note that, if B is the complex change of basis matrix from b to b0 then b = (det B)(det B) > 0. det B As the orientation is canonical, every Ck -vector bundle is orientable as a R2k -vector bundle.

1.11

Exterior derivative

If f is a smooth function on M, its exterior derivative is a 1-form df ∈ C ∞ (M ; T ∗ M ) defined by df (V ) = V (f ), for all V ∈ C ∞ (M ; T M ).

20


If ω ∈ C ∞ (M ; T ∗ M ), its exterior derivative is a 2-form dω ∈ C ∞ (M ; Λ2 T ∗ M ) defined by dω(V, W ) = V (ω(W )) − W (ω(V )) − ω([V, W ]), for all V, W ∈ C ∞ (M ; T M ). In the same way, if ω ∈ Ωk (M ), we define its exterior derivative to be the (k+1)-form dω ∈ Ωk+1 (M ) defined by dω(V0 , . . . , Vk ) =

p X

(−1)i Vi (ω(V0 , . . . , Vbi , . . . , Vp ))

i=0

X + (−1)i+j ω([Vi , Vj ], V0 , . . . , Vbi , . . . , Vbj , . . . , Vp ) i 0 such that: For all p ∈ U and tangent vector v ∈ Tp M with |v|g < ε there is a unique geodesic γ(p,v) : (−2ε2 , 2ε2 ) −→ M, satisfying the initial conditions (2.10)

γ(p,v) (0) = p,

0 γ(p,v) (0) = v.

Furthermore, the geodesics depend smoothly on the initial conditions. The solution to the system (2.9) is unique up to its domain of definition. Since (2.9) is non-linear the geodesic may not be defined on all of R, but rather there is a maximal interval around zero, say (a, b), in which this solution is defined. With this domain, the solution is unique, so we say that for each (p, v) ∈ M there is a unique maximally extends geodesic satisfying (2.10). We will use γ(p,v) to refer to this maximally extended geodesic. Remark 4. If M is a submanifold of RN with the induced metric then a curve in M is a geodesic if and only if its acceleration as a curve in RN is normal to M. In particular, if the intersection of M with a subspace of RN that is everywhere orthogonal to M is a curve, then any constant speed parametrization of that curve must be a geodesic. The same is true replacing RN with a Riemannian manifold, X. If M ⊆ X is a Riemannian submanifold and all of the geodesics of M are also geodesics of X, we say that M is a totally geodesic submanifold. By uniqueness we have the following homogeneity property γ(p,λv) (t) = γ(p,v) (λt) for all λ ∈ R for which one of the two sides is defined. Thus by making ε1 smaller in the lemma, we can take, e.g., ε2 = 1. At each point p ∈ M we can define the exponential map by expp : U ⊆ Tp M −→ M,

expp (v) = γ(p,v) (1)

where U is the subset of Tp M of vectors v for which γ(p,v) (1) is defined. We know that U contains a neighborhood of the origin and that expp is smooth on U. Let {y1 , . . . , ym , ∂y1 , . . . , ∂ym } be local coordinates for T M over a coordinate chart V of p. Consider the smooth function F : U −→ M × M, defined by F (p, v) = (p, expp (v)).

2.7. GEODESICS

41

Notice that, if {y1 , . . . , ym , z1 , . . . , zm } are the corresponding coordinates on V ×V ⊆ M × M, then DF : T U −→ T M × T M satisfies DF (∂yi ) = ∂yi + ∂zi , DF (∂(∂yi ) ) = ∂zi Id Id and hence has the form . In particular, we can apply the implicit function 0 Id theorem to F to conclude as follows: Lemma 5. For each p ∈ M there is a neighborhood U of M and a number ε > 0 such that: a) Any two points p, q ∈ U are joined by a unique geodesic in M of length less than ε. b) This geodesic depends smoothly on the two points. c) For each q ∈ U the map expq maps the open ε-ball in Tq M diffeomorphically onto an open set Uq containing U. Since expp is a diffeomorphism from Bε (0) ⊆ Tp M into M, it defines a coordinate chart on M. These coordinates are known as normal coordinates. At any point p ∈ M, one can show that if v ∈ Tp M is small enough, the distance (2.4) between p and exp(v) is |v|g . In particular d(p, q) = 0 =⇒ p = q and hence (M, d) is a metric space. A very nice property of a normal coordinate system is that near the origin the coefficients of the metric agree to second order with the identity matrix up to second order. More precisely2 : Proposition 6. Let (M, g) be a Riemannian manifold, p ∈ M, and let {y1 , . . . , ym } be normal coordinates centered at p. Then we have gij (0) = δij , Γkij (0) = 0, 2 1 ∂ gj` ∂ 2 gjk ∂ 2 gik ∂ 2 gi` − + − Rijk` (0) = 2 ∂yi ∂yk ∂yi ∂y` ∂yj ∂y` ∂yj ∂yk and the Taylor expansion of the functions gij at the origin has the form 1X gij (y) ∼ δij − Rikj` yk y` + O(|y|3 ). 3 k,` 2

`le Vergne See, e.g., Proposition 1.28 of Nicole Berline, Ezra Getzler, Miche Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. x+363 pp. ISBN: 3-540-20062-2

42 LECTURE 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

2.8

Complete manifolds of bounded geometry

At each point p in a Riemannian radius, the injectivity radius at p, injM (p), is the largest radius R such that expp : BR (0) ⊆ Tp M −→ M is a diffeomorphism. The injectivity radius of M is defined by injM = inf{injM (p) : p ∈ M }. By Lemma 5, a compact manifold has a positive injectivity radius. We say that a manifold is complete if there is a point p ∈ M for which expp is defined on all of Tp M (though it need not be injective on all of Tp M ). The nomenclature comes from the Hopf-Rinow theorem, which states that the following assertions are equivalent: a) The metric space (M, d), with d defined by (2.4), is complete. b) Any closed and bounded subset of M is compact. c) There is a point p ∈ M for which expp is defined on all of Tp M. d) At any point p ∈ M, expp is defined on all of Tp M. Moreover in this case one can show that any two points p, q ∈ M can be joined by a geodesic γ with d(p, q) = `(γ). A Riemannian manifold (M, g) is called a manifold with bounded geometry if: 1) M is complete and connected 2) injM > 0 3) For every k ∈ N0 , there is a constant Ck such that |∇k R|g ≤ Ck . Every compact Riemannian manifold has bounded geometry. Many noncompact manifolds, such as Rm and Hm , have bounded geometry. It is often the case that constructions that one can do on Rm can be carried out also on manifolds of bounded geometry but that one runs into trouble trying to carry them out on manifolds that do not have bounded geometry.

2.9

Riemannian volume and formal adjoint

On Rm with the Euclidean metric and the canonical orientation, there is a natural choice of volume form dvolRm = dy1 ∧ · · · ∧ dym .

2.9. RIEMANNIAN VOLUME AND FORMAL ADJOINT

43

This assigns unit volume to the unit square. The volume of a parallelotope generated by m-vectors z1 , . . . , zm is equal to the determinant of the change of basis matrix from y to z, which we can express as the ‘Gram determinant’ q det(zi · xj ) . It turns out that a Riemannian metric g on an orientable manifold M also determines a natural volume form, dvolg , determined by the property that, at any p ∈ M and any local coordinates in which gij (p) = δij we have dvolg (p) = dvolRm . In arbitrary oriented local coordinates, x1 , . . . , xm , dvol(p) should be the volume of the parallelotope generated by ∂xi and hence q dvolg (p) = det(gij (p)) dx1 ∧ · · · ∧ dxm . The volume form is naturally associated to the metric in that, e.g., dvolf ∗ g = f ∗ dvolg and one can check that the volume form is covariantly constant with respect to the Levi-Civita connection. Given a metric gE on a bundle E −→ M and a Riemannian metric on M, we obtain an inner product on sections of E by setting: h·, ·iE : C ∞ (M ; E) × C ∞ (M ; E) −→ R, Z hσ, τ iE = gE (σ, τ ) dvolg , for all σ, τ ∈ C ∞ (M ; E). M

This is a non-degenerate, positive definite inner product on C ∞ (M ; E) because gE is a non-degenerate positive definite inner product on each Ep . Using this inner product, we can define the Lp -norms in the usual way Z 1/p ∞ p k·kLp (M ;E) : C (M ; E) −→ R, kskLp (M ;E) = |s|gE dvolg M

and then define the Lp space of sections of E, Lp (M ; E), by taking the k·kLp (M ;E) closure of the smooth functions with finite Lp (M ; E)-norm. These are Banach spaces, and we will mostly work with L2 (M ; E), which is also a Hilbert space. If we have a Riemannian manifold (M, g) and two vector bundles E −→ M, F −→ M with bundle metrics, we say that two linear operators T : C ∞ (M ; E) −→ C ∞ (M ; F ),

T ∗ : C ∞ (M ; F ) −→ C ∞ (M ; E)

44 LECTURE 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY are formal adjoints if they satisfy hT σ, τ iF = hσ, T ∗ τ iE ,

for all σ ∈ Cc∞ (M ; E), τ ∈ Cc∞ (M ; F ).

(We use the word formal because we are not working with a complete vector space like L2 or Lp but just with smooth functions, this avoids some technicalities.) As a simple example, the formal adjoint of a scalar differential operator on R is easy to compute ∗ X X ak (x)∂xk u(x) = (−1)k ∂xk (ak (x)u(x)), for all u ∈ C ∞ (R). The formal adjoint of any differential operator is similarly computed using integration by parts in local coordinates (the compact supports allow us not to worry about picking up ‘boundary terms’). We say that a linear operator is formally self-adjoint if it coincides with its formal adjoint. For instance, if T and T ∗ are formal adjoints then T T ∗ and T ∗ T are formally self-adjoint. Moreover, these operators are positive semidefinite since on compactly supported sections hT ∗ T s, si = hT s, T si = |T s|2 . In the next section we will compute the formal adjoints of d and ∇E , which will lead to natural Laplace-type operators.

2.10

Hodge star and Hodge Laplacian

We have already looked at the space of differential forms on M and seen that the smooth structure on M induces a lot of structure on Ω∗ (M ), such as the exterior product, interior product and exterior derivative. Now we suppose that (M, g) is a Riemannian manifold and we will see that there is still more structure on Ω∗ (M ). First, we have already noted above that a metric on M induces metrics on associated vector bundles, such as Λ∗ T ∗ M. It is easy to describe this metric in terms of the metric on T ∗ M when dealing with elementary k-forms, namely gΛk T ∗ M (ω 1 ∧ · · · ∧ ω k , θ1 ∧ · · · θk )p = det gT ∗ M (ω i , θj )p , for all {ω i , θj } ⊆ Tp∗ M. On any oriented Riemannian m-manifold, we define the Hodge star operator ∗ : Ωk (M ) −→ Ωm−k (M )

2.10. HODGE STAR AND HODGE LAPLACIAN

45

by demanding, for all ω, η ∈ Ωk (M ), ω ∧ ∗η = gΛk T ∗ M (ω, η) dvolg . Thus, for example, in R4 with Euclidean metric and canonical orientation, with y1 , . . . , y4 an orthonormal basis, ∗1 = dy 1 ∧ dy 2 ∧ dy 3 ∧ dy 4 ∗dy 1 = dy 2 ∧ dy 3 ∧ dy 4 , ∗dy 2 = −dy 1 ∧ dy 3 ∧ dy 4 , ∗dy 1 ∧ dy 2 = dy 3 ∧ dy 4 , ∗dy 2 ∧ dy 4 = −dy 1 ∧ dy 3 ∗dy 1 ∧ dy 2 ∧ dy 3 = dy 4 , ∗dy 1 ∧ dy 2 ∧ dy 3 ∧ dy 4 = 1 and so on. Note that ∗ is symmetric in that ω ∧ ∗η = gΛk T ∗ M (ω, η) dvolg = gΛk T ∗ M (η, ω) dvolg = η ∧ ∗ω. The Hodge star is almost an involution, ω ∈ Ωk (M ) =⇒ ∗(∗ω) = (−1)k(m−k) ω. The Hodge star acts on differential forms and we have Z hω, ηiΛ∗ T ∗ M = ω ∧ ∗η. M

This lets us get a formula for the formal adjoint of the exterior derivative, (M ), and η ∈ Ωkc (M ), d∗ = δ. Namely, whenever ω ∈ Ωk−1 c Z

Z

(dω) ∧ ∗η = d(ω ∧ ∗η) − (−1)k−1 ω ∧ d ∗ η M M Z Z k k = (−1) ω ∧ d ∗ η = (−1) ω ∧ (−1)(m−k+1)(k−1) ∗2 d ∗ η

hdω, ηiΛk T ∗ M =

M

M

= hω, (−1)m(k+1)+1 ∗ d ∗ ηiΛk−1 T ∗ M and so, when acting on Ωk (M ), δ = d∗ = (−1)m(k+1)+1 ∗ d∗ = (−1)k ∗−1 d ∗ . A form is called coclosed if it is in the null space of δ. (Functions are thus automatically coclosed.) In particular on one-forms we have ω ∈ Ω1 (M ) =⇒ δω = − ∗ d ∗ ω,

46 LECTURE 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY and, if ω ∈ Ω1c (M ), then Stokes’ theorem (on manifolds with boundary) says that Z Z Z Z δ(ω) dvolg = − ∗d ∗ ω dvolg = − d(∗ω) = − ∗ω. M

M

M

∂M

Hence on manifolds without boundary, Z δ(ω) dvolg = 0. M

The Hodge Laplacian on k-forms is the differential operator ∆k : Ωk (M ) −→ Ωk (M ),

∆k = dδ + δd.

Note that this operator is formally self-adjoint. Elements of the null space of the Hodge Laplacian are called harmonic differential forms, and we will use the notation Hk (M ) = {ω ∈ Ωk (M ) : ∆k ω = 0} for the vector space of harmonic k-forms. A form that is closed and coclosed is automatically harmonic. Conversely, if ω ∈ Ωkc (M ) is compactly supported and harmonic, then 0 = h∆ω, ωiΛk T ∗ M = hdω, dωiΛk+1 T ∗ M + hδω, δωiΛk−1 T ∗ M and so ω is both closed and coclosed. In particular, if M is compact, we have a natural map Hk (M ) −→ HkdR (M ) since each harmonic form is closed. We will show later that this map is an isomorphism by showing that every de Rham cohomology class on a compact manifold contains a unique harmonic form. Since d2 = 0 and δ 2 = 0, we can also write the Hodge Laplacian as ∆k = (d + δ)2 . Also, notice that ∗∆k = ∆m−k ∗, d∆k = dδd = ∆k+1 d, δ∆k = δdδ = ∆k−1 δ and hence ω is harmonic implies dω and δω harmonic, and is equivalent to ∗ω harmonic. In particular, Hk (M ) = Hm−k (M ),

2.11. EXERCISES

47

which we call Poincaré duality for harmonic forms. Let us describe the Laplacian on functions ∆ = ∆0 in a little more detail. Since δ vanishes on functions, we have ∆ = δd = − ∗ d ∗ d. A closed function is necessarily a constant, so on compact manifolds all harmonic functions are constant. In local coordinates, {y1 , . . . , ym }, the Laplacian is given by ∂ p 1 ij ∂f ∆f = − p = − g ij ∂i ∂j − g ij Γkij ∂k f. det gij g ∂yi det gij ∂yj On Euclidean space our Laplacian is ∆=−

m X ∂ 2f i=1

∂yi2

which is, up to a sign, the usual Laplacian on Rm . With our sign convention, the Laplacian is sometimes called the ‘geometer’s Laplacian’ while the opposite sign convention yields the ‘analyst’s Laplacian’. The Laplacian on functions on a Riemannian manifold is sometimes called the Laplace-Beltrami operator to emphasize that there is a Riemannian metric involved.

2.11

Exercises

Exercise 1. Show that if (M1 .g1 ) and (M2 , g2 ) are Riemannian manifolds, then so is (M1 × M2 , g1 + g2 ). Exercise 2. It is tempting to think that the interior product and the exterior derivative will define a covariant derivative on T ∗ M without having to use a metric, say by e V ω = V y dω, for all ω ∈ C ∞ (M ; T ∗ M ), V ∈ C ∞ (M ; T M ). ∇ Show that this is not a covariant derivative on T ∗ M. Show that the exterior derivative does define a connection on the trivial bundle R × M −→ M. Exercise 3. Show that parallel transport determines the connection.

48 LECTURE 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY Exercise 4. Consider the 2-dimensional sphere of radius r centered at the origin of R3 , S2r = {(y1 , y2 , y3 ) ∈ R3 : |y| = r}. The Euclidean metric on R3 induces the ‘round’ metric on S2r . Compute the sectional curvatures of this metric. Hint: Use the spherical coordinates (θ, ψ) that parametrize the sphere via φ(θ, φ) = (r cos θ cos ψ, r cos θ sin φ, r sin θ). Notice that any curve on S2 can be parametrized γ(t) = (θ(t), ψ(t)) and the length of γ 0 (t) is p r (θ0 )2 + (ψ 0 )2 cos2 θ . Hence the round metric in these coordinates is gS2r = r2 dθ2 + r2 cos2 θdψ 2 . Exercise 5. One model of hyperbolic space of dimension m, (Hm , gHm ) is the space m Rm + = {(x, y1 , . . . , ym−1 ) ∈ R : x > 0}

with the metric

dx2 + |dy|2 g = . x2 Compute the Christoffel symbols and the sectional curvatures of Hm . Hm

Exercise 6. (Riemann’s formula) Let K be a constant. Show that a neighborhood of the origin in Rm with coordinates {y1 , . . . , ym } endowed with the metric |dy|2 1+

K 2 2 |y| 4

has constant curvature K. Exercise 7. Use stereographic projection to write the metric on the sphere as a particular case of Riemann’s formula. Exercise 8. Consider the flat torus Tm = Rm /Zm with the metric it inherits from Rm . Compute the curvature and geodesics of Tm . Exercise 9. If γ(t) is a geodesic on a Riemannian manifold show that γ 0 (t) has constant length.

2.11. EXERCISES

49

Exercise 10. a) Determine the geodesics on the sphere. (Hint: Remark 4.) b) All of the points of the sphere have the same injectivity radius, what is it? c) Consider the geodesic triangle on the sphere S2 with a vertex at the ‘north pole’ and other vertices on the ‘equator’, suppose that at the north pole the edge form an angle θ. Describe the parallel translation map along this triangle. Hint: Because the sides are geodesics, any vector parallel along the sides forms a constant angle with the tangent vector of the edge. Exercise 11. Give an example of two curves on a Riemannian manifold with the same endpoints and with different parallel transport maps. Exercise 12. a) Show that any differential k-form α on M × [0, 1] has a ‘standard decomposition’ α = αt (s) + ds ∧ αn (s) where ds is the unique positively oriented unit 1-form on [0, 1] and, for each s ∈ [0, 1], αt (s) is a differential k-form on M and αn (s) is a differential (k − 1)-form on M. What is the standard decomposition of dα in terms of αt and αn ? b) For each s ∈ [0, 1] let is : M −→ M × [0, 1],

is (p) = (p, s).

We can think of the pull-back of a differential form by is as the ‘restriction of a form on M × [0, 1] to M × {s}’. Consider the map π∗ that takes k-forms on M × [0, 1] to (k − 1)-forms on M given by Z 1 π∗ (α) = αn (s) ds. 0

Show that for any differential k-form α on M × [0, 1] i∗1 α − i∗0 α = dπ∗ α + π∗ (dα). c) Show that whenever [0, 1] 3 t 7→ ωt is a smooth family of closed k-forms on a manifold M, the de Rham cohomology class of ωt is independent of t. Exercise 13. (Poincaré’s Lemma) A manifold M is (smoothly) contractible if there is a point p0 ∈ M, and a smooth map H : M × [0, 1] −→ M such that H(p, 1) = p,

H(p, 0) = p0 for all p ∈ M.

50 LECTURE 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY H is known as a ‘contraction’ of M. a) Show that Rn is contractible and any manifold is locally contractible (i.e, every point has a contractible neighborhood). b) Let M be a contractible smooth n-dimensional manifold, and let ω be a closed differential k-form on M (i.e., dω = 0). Prove that there is a (k − 1)-form η on M such that dη = ω. (Hint: Let H be a contraction of M, and let α = H ∗ ω. Then ω = (H ◦ i1 )∗ ω = i∗1 α and i∗0 α = 0. Check that η = π∗ α works.) Exercise 14. Let M be an n-dimensional manifold with boundary ∂M. A (smooth) retraction of M onto its boundary is a smooth map φ : M −→ ∂M such that φ(ζ) = ζ, whenever ζ ∈ ∂M. Prove that if M is compact and oriented, then there do not exist any retractions of M onto its boundary. R (Hint: Let ω be an (n − 1)-form on ∂M with ∂M ω > 0 (how do we know there is such an ω?), notice that dω = 0 (why?). Assume that φ is a retraction of M onto its boundary, and apply Stokes’ theorem to φ∗ ω to get a contradiction.) If φ : M −→ ∂M is a continuous retraction, one can approximate it by a smooth retraction, so there are in fact no continuous retractions from M to ∂M. Exercise 15. (Brouwer fixed point theorem) Let B be the closed unit ball in Rn , show that any continuous map F : B −→ B has a fixed point (i.e., there is a point ζ ∈ B such that F (ζ) = ζ). (Hint: If F : B −→ B is a smooth map without any fixed points, let φ(ζ) be the point on ∂B where the ray starting at F (ζ) and passing through ζ hits the boundary. Show that φ is continuous and then apply the continuous version of the previous exercise.) Exercise 16. Let Vi be a local orthonormal frame for T M with dual coframe θi . Show that X X iVi ∇Vi . d= e θ i ∇ Vi , δ = −

2.12

Bibliography

My favorite book for Riemannian geometry is the eponymous book by Gallot, Hulin, and Lafontaine. Some other very good books are Riemannian geometry and geometric analysis by Jost, and Riemannian geometry by Petersen.

2.12. BIBLIOGRAPHY

51

Perhaps the best place to read about connections is the second volume of Spivak, A comprehensive introduction to differential geometry which includes translations of articles by Gauss and Riemann. Spivak does a great job of presenting the various approaches to connections and curvature. In each approach, he explains how to view the curvature as the obstruction to flatness. An excellent concise introduction to Riemannian geometry, which we follow when discussing geodesics, can be found in Milnor, Morse theory. Another excellent treatment is in volume one of Taylor, Partial differential equations.

52 LECTURE 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

Lecture 3 Laplace-type and Dirac-type operators 3.1

Principal symbol of a differential operator

Let M be a smooth manifold, E −→ M, F −→ M real vector bundles over M, and D a differential operator of order ` acting between sections of E and sections of F, D ∈ Diff ` (M ; E, F ). In any coordinate chart V of M that trivializes both E and F, D is given by X D= aα (y)∂yα |α|≤`

where {y1 , . . . , ym } are the coordinates on V and aα ∈ C ∞ (V; Hom(E, F ) V ). This expression for D changes quite a bit if we choose a different set of coordinates. However it turns out we can make invariant sense of the ‘top order part’ of D, X aα (y)∂yα , |α|=`

as a constant coefficient differential operator on each fiber of the tangent space of M. It is more common to take the dual approach and interpret this as a function on the cotangent bundle of M as follows: Let {y1 , . . . , ym , dy 1 , . . . , dy m } be the induced coordinates on the cotangent bundle of M, π : T ∗ M −→ M, over V and define the principal symbol of D at the point ξ = ξj dy j of the cotangent bundle, to be X σ` (D)(y, ξ) = i` aα (y)ξ α ∈ C ∞ (T ∗ M ; π ∗ Hom(E, F )). |α|=`

That is, we just ignore all the terms in the expression for the differential operator 53

54

LECTURE 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

of order less than ` and replace differentiation with a polynomial in the cotangent variables. Why the factor of i? A key fact that we will exploit later is that the Fourier transform on Rn interchanges differentiation by yj with multiplication by iξj , Z Z −iy·ξ e ∂yj f (y) dy = e−iy·ξ (iξj )f (y) dy. Rn

Rn

Of course here one should be working with complex valued functions, and soon we will be. For the moment, just think of the power of i as a formal factor. Another way of defining the principal symbol, which has the advantage of being obviously independent of coordinates, is: At each ξ ∈ Tp∗ M, choose a function f such that df (p) = ξ and then set (3.1) σ` (D)(p, ξ) = lim t−` e−itf ◦ D ◦ eitf (p) ∈ Hom(Ep , Fp ). t→∞

By the Leibnitz rule, e−itf ◦ D ◦ e−itf is a polynomial in t of order ` and the top order term at p is of the form X X αm (it)` aα (p)(∂yα11 f )(p) · · · (∂yαmm f )(p) = (it)` aα (p)ξ1α1 · · · ξm |α|=`

|α|=`

which shows that (3.1) is independent of the choice of f and hence well-defined. Notice that the principal symbol of a differential operator is not just a smooth function on T ∗ M, but in fact a homogeneous polynomial of order ` on each fiber of T ∗ M, we will denote these functions by P` (T ∗ M ; π ∗ Hom(E, F )) ⊆ C ∞ (T ∗ M ; π ∗ Hom(E, F )). For an operator of order one or two, it is easy to carry out the differentiations in (3.1) and see that σ1 (D)(p, ξ) =

1 [D, f ] , i

1 σ2 (D)(p, ξ) = − [[D, f ] , f ] 2

for any f such that df (p) = ξ. A similar formula holds for higher order operators. The principal symbol of order ` vanishes if and only if the operator is actually of order ` − 1, so we have a short exact sequence σ

0 −→ Diff `−1 (M ; E, F ) −→ Diff ` (M ; E, F ) −−`→ P` (T ∗ M ; π ∗ Hom(E, F )) −→ 0. One very important property of the principal symbol map is that it is a homomorphism. That is, if D1 : C ∞ (M ; E1 ) −→ C ∞ (M ; E2 ),

D2 : C ∞ (M ; E2 ) −→ C ∞ (M ; E3 )

3.1. PRINCIPAL SYMBOL OF A DIFFERENTIAL OPERATOR

55

are differential operators of orders `1 and `2 respectively, then D2 ◦ D1 : C ∞ (M ; E1 ) −→ C ∞ (M ; E3 ) is a differential operator with symbol σ`1 +`2 (D2 ◦ D1 )(p, ξ) = σ`2 (D2 )(p, ξ) ◦ σ`1 (D1 )(p, ξ), as follows directly from (3.1). This is especially useful for scalar operators Ei = R since then multiplication in P`1 +`2 (T ∗ M ) is commutative, so we always have σ`1 (D1 )(p, ξ) ◦ σ`2 (D2 )(p, ξ) = σ`2 (D2 )(p, ξ) ◦ σ`1 (D1 )(p, ξ), even though D1 ◦ D2 is generally different from D2 ◦ D1 . It is also true that the symbol map respects adjoints in that, if D and D∗ are formal adjoints, then σ` (D∗ ) = σ` (D)∗ where on the right we take the the adjoint in Hom, so in local coordinates just the matrix transpose. We can use (3.1) to find the symbols of d and δ by noting that [d, f ] ω = d(f ω) − f dω = edf ω, [δ, f ] ω = δ(f ω) − f δω = −idf ] ω hence

σ1 (d)(p, ξ) = −ieξ : Λk Tp∗ M −→ Λk+1 Tp∗ M σ1 (δ)(p, ξ) = iiξ] : Λk Tp∗ M −→ Λk−1 Tp∗ M.

Then the symbol of the Hodge Laplacian is (3.2)

σ2 (∆k )(p, ξ) = (σ1 (d)σ1 (δ) + σ1 (δ)σ1 (d)) (p, ξ) = eξ iξ] + iξ] eξ = |ξ|2g

i.e., it is given by multiplying vectors in Λk Tp∗ M by the scalar |ξ|2g . A differential operator is called elliptic if σ` (D)(p, ξ) is invertible at all ξ 6= 0. Hence ∆k is an elliptic differential operator, d and δ are not elliptic, and d ± δ are elliptic. The main aim of this course is to understand elliptic operators. One particularly interesting property of an elliptic operator D on a compact manifold is that its ‘index’ ind D = dim ker D − dim ker D∗ is well-defined (in that both terms in the difference are finite) and remarkably stable, as we will show below. Before doing that, however, we will start by constructing examples of elliptic operators.

56


3.2

Laplace-type operators

A second-order differential operator on a Rk -bundle E −→ M over a Riemannian manifold (M, g) is of Laplace-type if its principal symbol at ξ ∈ Tp∗ M is scalar multiplication by |ξ|2g . Thus by (3.2) the Hodge Laplacian ∆k is, for every k, a Laplace-type operator. A general Laplace-type operator H can be written in any local coordinates as H = −g ij ∂yi ∂yj + ak ∂yk + b, with ak , b ∈ C ∞ (V; Hom(E, E)). If E −→ M is a vector bundle with connection ∇E , then we can form the composition ∇E

∇T

∗ M ⊗E

C ∞ (M ; E) −−−→ C ∞ (M ; T ∗ M ⊗ E) −−−−−−−→ C ∞ (M ; T ∗ M ⊗ T ∗ M ⊗ E) and then take the trace over the two factors of T ∗ M to define the connection Laplacian, ∆∇ = − Tr(∇T

∗ M ⊗E

∇E ) : C ∞ (M ; E) −→ C ∞ (M ; E).

(This is also known as the Bochner Laplacian.) Explicitly, we have ∗ E E (∇T M ⊗E ∇E s)(V, W ) = ∇E ∇ − ∇ V W ∇V W s and so in local coordinates we have ij k E E ∇ ij E ∆ s = − g ∇∂yi ∇∂yj − g Γij ∇∂y s. k

Thus this is a Laplace-type operator. Directly from the formula in local coordinates we see that the Laplacian on functions is the connection Laplacian for the Levi-Civita connection ∆∇ = ∆. It turns out that any Laplace-type operator on a bundle E −→ M is equal to a connection Laplacian plus a zero-th order differential operator1 . Proposition 7. Let (M, g) be a Riemannian manifold, E −→ M a vector bundle, and H a Laplace-type operator on sections of E. There exists a unique connection ∇E on E and a unique section Q ∈ C ∞ (M ; Hom(E)) such that H = ∆∇ + Q. 1

For the proof of this proposition see, e.g., Proposition 2.5 of Nicole Berline, Ezra Get`le Vergne Heat kernels and Dirac operators. Corrected reprint of the 1992 original. zler, Miche Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. x+363 pp. ISBN: 3-540-20062-2

3.3. DIRAC-TYPE OPERATORS

3.3

57

Dirac-type operators

The physicist P.A.M. Dirac constructed first-order differential operators whose squares were Laplace-type operators (albeit in ‘Lorentzian signature’) for the purpose of extending quantum mechanics to the relativistic setting. After these operators were rediscovered by Atiyah and Singer, they were christened Dirac operators and later generalized to Dirac-type operators. Let E −→ M and F −→ M be Rk -vector bundles over a Riemannian manifold (M, g). A first order differential operator D ∈ Diff 1 (M ; E, F ) is a Dirac-type operator if D∗ D and DD∗ are Laplace-type operators. If E = F and D = D∗ , we say that D is a symmetric Dirac-type operator. If D ∈ Diff 1 (M ; E, F ) is a Dirac-type operator, then 0 D∗ e D= ∈ Diff 1 (M ; E ⊕ F ) D 0 is a symmetric Dirac-type operator. We have seen one example of a symmetric Dirac-type operator so far, the de Rham operator: 1

∗

∗

d + δ ∈ Diff (M ; Ω (M )), where Ω (M ) =

m M

Ωk (M ) = C ∞ (M ; Λ∗ T ∗ M )

k=0

Indeed, we know that (d + δ)∗ = d + δ and (d + δ)2 = ∆, the Hodge Laplacian on forms of all degrees. We can use the symbol of an arbitrary symmetric Dirac operator D to uncover some algebraic structure on E. For every p ∈ M, and ξ ∈ Tp M, let θ(p, ξ) = σ1 (D)(p, ξ) : Ep −→ Ep , and, when the point p is fixed, let θ(ξ) = θ(p, ξ). Since D is symmetric, we have θ(ξ) = θ(ξ)∗ and since it is a Dirac-type operator, we have θ(ξ)2 = g(ξ, ξ) Id . We can ‘polarize’ this identity by applying it to ξ + η ∈ Tp M ∗ , θ(ξ + η)2 = θ(ξ)2 + θ(ξ)θ(η) + θ(η)θ(ξ) + θ(η)2 = g(ξ + η, ξ + η) Id = (g(ξ, ξ) + 2g(ξ, η) + g(η, η)) Id

58


and hence θ(ξ)θ(η) + θ(η)θ(ξ) = 2g(ξ, η) Id . More generally, suppose we have two vector spaces V and W, a quadratic form q on V, and a linear map cl : V −→ End(W ) such that2 (3.3)

cl (v) cl (v) = q(v) Id .

We call cl Clifford multiplication by elements of V. As above, we can polarize (3.3). Define q(v, w) = 21 (q(v + w) − q(v) − q(w)) and apply (3.3) to v + w to obtain (3.4)

cl (v) cl (w) + cl (w) cl (v) = 2q(v, w) Id .

The map cl canonically extends to the Clifford algebra of (V, q), which is defined as follows. First let O V = R ⊕ V ⊕ (V ⊗ V ) ⊕ (V ⊗ V ⊗ V ) ⊕ . . . N and note that the map cl extends to a map clb : V −→ End(W ). This extension vanishes on elements of the form v ⊗ w + w ⊗ v − 2q(v, w) Id and hence on the ideal generated by these elements n O o Icl (V ) = span κ ⊗ (v ⊗ w + w ⊗ v − 2q(v, w) Id) ⊗ χ : κ, χ ∈ V . The quotient is the Clifford algebra of (V, q), O Cl(V, q) = V /Icl (V ), it has an algebra structured induced by the tensor product in

N

V which satisfies

v · w + w · v = 2q(v, w) whenever v, w ∈ V. By construction, Cl(V, q) has the universal property that any linear map cl : V −→ End(W ) satisfying (3.3) extends to a unique algebra homomorphism cl : Cl(V, q) −→ End(W ), with cl (1) = Id . 2

Many authors use a different convention, namely cl (v) cl (v) = −q(v) Id .

3.4. CLIFFORD ALGEBRAS

59

This universal property makes the Clifford algebra construction functorial. Indeed, if (V, q) and (V 0 , q 0 ) are vector spaces with quadratic forms and T : V −→ V 0 is a linear map such that T ∗ q 0 = q then there is an induced algebra homomorphism Te : Cl(V, q) −→ Cl(V 0 , q 0 ). Uniqueness of Te shows that T] ◦ S = Te ◦ Se for any quadratic-form preserving linear map S. One advantage of functoriality is that we can extend this construction to vector bundles: Given a vector bundle F −→ M and a bundle metric gF there is a vector bundle Cl(F, gF ) −→ M whose fiber over p ∈ M is Cl(Fp , gF (p)). In particular, coming back to our first order Dirac-type operator D ∈ C ∞ (M ; E), we can conclude that its principal symbol σ1 (D) induces an algebra bundle morphism (3.5)

cl : Cl(T ∗ M, g) −→ End(E) with cl (1) = Id .

A bundle E −→ M together with a map (3.5) is called a Clifford module. (It should really be called a bundle of Clifford modules, but the abbreviation is common.) In the case of the de Rham operator we have σ1 (d + δ)(p, ξ) = −i(eξ − iξ] ) : Λ∗ Tp∗ M −→ Λ∗ Tp∗ M, and an induced algebra homomorphism (3.6)

3.4

cl : Cl(T ∗ M, g) −→ End(Λ∗ Tp∗ M ).

Clifford algebras

Let us consider the structure of the algebra Cl(V, q) associated to a quadratic form q on V. If e1 , . . . , ek is a basis for V then any element of Cl(V, q) can be written as a polynomial in the ei . In fact, using (3.4) we can choose the polynomial so that no exponent larger than one is used, and so that the ei occur in increasing order; thus any ω ∈ Cl(V, q) can be written in the form X (3.7) ω= aα eα1 1 · eα2 2 · · · eαmm αj ∈{0,1}

60


with real coefficients aα . This induces a filtration on Cl(V, q), Cl(k) (V, q) = {ω as in (3.7), with |α| ≤ k}, which is consistent with the algebraic structure, Cl(k) (V, q) · Cl(`) (V, q) ⊆ Cl(k+`) (V, q). The map (3.6), which there was induced by the symbol of the de Rham operator, always exists, M : Cl(V, q) −→ End(Λ∗ V ). There is a distinguished element 1 ∈ Λ∗ V, and evaluation at one induces f : Cl(V, q) −→ Λ∗ V, M

f(ω) = M (ω)(1). M

f(v) = v. Moreover, Clearly if we think of v ∈ V as an element of Cl(V, q), we have M f is comparing the anti-commutation rules in Cl(V, q) and Λ∗ V, we can show that M ∗ an isomorphism of vector spaces. In fact, Λ V is the graded algebra associated to the filtered algebra Cl(V, q), i.e., Cl(k) (V, q)/ Cl(k−1) (V, q) ∼ = Λk V. If we set q = 0, then we recognize Cl(V, 0) = Λ∗ V as algebras. In general the algebras Cl(V, q) and Λ∗ V only coincide as vector spaces. Let us identify a few of the low-dimensional Clifford algebras. From the vector space isomorphism Cl(V, q) ∼ = Λ∗ V we know that dim Cl(V, q) is a power of two. Let us agree that R itself, with its usual algebra structure, is the unique one-dimensional Clifford algebra. In dimension two, let us identify Cl(R, gR ) and Cl(R, −gR ). Let e1 be a unit vector in R, so that both of these Clifford algebras are generated by 1, e1 . In the second case, where the quadratic form is minus the usual Euclidean norm, we have e21 = −1, and it is easy to see that Cl(R, −gR ) ∼ =C as R-algebras. In the first case, where the quadratic form is the usual Euclidean norm, the elements v = 21 (1 + e1 ) and w = 12 (1 − e1 ) satisfy v · w = 0, and we can identify

v · v = v,

w·w =w

Cl(R, gR ) ∼ =R⊕R

3.4. CLIFFORD ALGEBRAS

61

as R-algebras, where the summands consist of multiples of v and w, respectively. Next consider Cl(R2 , ±gR2 ). Let e1 , e2 be an orthonormal basis of (R2 , gR2 ). Using the universal property, we can define an isomorphism f between Cl(R2 , ±gR2 ) and a four dimensional algebra A by specifying the images of e1 and e2 and verifying that f (ei ) · f (ei ) = ±1, i ∈ {1, 2}. For Cl(R2 , gR2 ) the map 0 1 e1 → 7 , 1 0

e2 7→

1 0 0 −1

extends linearly to an algebra isomorphism Cl(R2 , gR2 ) ∼ = End(R2 ). Similarly, we can identify Cl(R2 , −gR2 ) with the quaternions by extending the map e1 7→ i,

e2 7→ j

(so, e.g., e1 · e2 maps to k). Thus (real) Clifford algebras generalize both the complex numbers and the quaternions. It is possible to compute all of the Clifford algebras of finite dimensional vector spaces3 and an incredible fact is that they recur (up to ‘suspension’) with periodicity eight! It is useful to consider the complexified Clifford algebra Cl(V, q) = C ⊗ Cl(V, q) as these complex algebras have a simpler structure than Cl(V, q). In particular, we next show that they repeat with periodicity two. Let us restrict attention to V = Rm and q = gRm , and abbreviate Cl(Rm , gRm ) = Cl(m),

Cl(Rm , gRm ) = Cl(m).

Proposition 8. There are isomorphisms of complex algebras Cl(1) ∼ = C2 , Cl(2) ∼ = End(C2 ) and Cl(m + 2) ∼ = Cl(m) ⊗ Cl(2) and so

Cl(2k) ∼ = End(C2 ), k

3

k k Cl(2k + 1) ∼ = End(C2 ) ⊕ End(C2 ).

See for instance, Lawson, H. Blaine, Jr. and Michelsohn, Marie-Louise Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989.

62


Proof. The first two identities are just the complexifications of Cl(R, gR ) and Cl(R2 , gR2 ) computed above. To prove periodicity, embed Rm+2 into Cl(m) ⊗ Cl(2) by picking an orthonormal basis {e1 , . . . , em+2 } and taking ej 7→ icl (ej ) ⊗ cl (em+1 )cl (em+2 ) for 1 ≤ j ≤ m, ej 7→ 1 ⊗ cl (ej ) for j > m. Then the universal property of Cl(m + 2) leads to the isomorphism above. Given a vector space with quadratic form (V, q), let e1 , . . . , em be an oriented orthonormal basis and define the chirality operator m

Γ = id 2 e e1 · · · em ∈ Cl(V, q). This is an involution (i.e., Γ2 = Id), and is independent of the choice of oriented orthonormal basis used in its definition. Moreover, Γ · v = −v · Γ if dim V even,

Γ · v = v · Γ if dim V odd.

From the explicit description of Cl(m), we can draw some immediate consequences about the representations of this algebra4 . Indeed if m is even, say m = 2k, k then Cl(2k) = End(C2 ) and up to isomorphism there is a unique irreducible Cl(2k)/ and it has dimension 2k , module, ∆, / Cl(2k) = End(∆). k

k

If m is odd, say m = 2k + 1, then Cl(2k + 1) = End(C2 ) ⊕ End(C2 ) and up to / ±, isomorphism there are two irreducible Cl(2k + 1) modules ∆ / + ) ⊗ End(∆ / −) Cl(2k + 1) = End(∆ each of dimension 2k distinguished by Γ ∆/ = ±1. ±

Any finite dimensional complex representation of Cl(2k) must be a direct sum / or equivalently of the form of copies of ∆, / ⊗ W0 W =∆ 4

For more details see, e.g., Roe, John Elliptic operators, topology and asymptotic methods. Second edition. Pitman Research Notes in Mathematics Series, 395. Longman, Harlow, 1998.

3.5. HERMITIAN VECTOR BUNDLES

63

for some auxiliary ‘twisting’ vector space W 0 . We can recover W 0 from W via / W) W 0 = HomCl(2k) (∆, and in the same fashion we have End(W ) = Cl(2k) ⊗ End(W 0 ) = Cl(2k) ⊗ EndCl(2k) (W ). / ±. One can similarly decompose representations of Cl(2k + 1) in terms ∆ / follows directly from the proposition, it can be useAlthough Cl(2k) = End(∆) / So let us describe how to construct ∆ / ful to have a more concrete representative of ∆. m from (R , gRm ) or indeed any m-dimensional vector space V with a positive-definite quadratic form, q. Let e1 , . . . , em be an oriented orthonormal basis of V and consider the subspaces of V ⊗ C, 1 K = span{ηj = √ (e2j−1 − ie2j ) : 2 1 K = span{η j = √ (e2j−1 + ie2j ) : 2

j ∈ {1, . . . , m}} j ∈ {1, . . . , m}}.

We can use the fact that V ⊗ C = K ⊕ K to define a natural homomorphism cl : V ⊗ C −→ End(Λ∗ K) √ √ cl (k) = 2 (ek ), cl k = 2 (ik ) for k ∈ K. This satisfies the Clifford relations and so extends to a homomorphism of Cl(V, q) into End(Λ∗ K). Comparing dimensions, we see this is in fact an isomorphism so we / = Λ∗ K. We refer to this representation as the spin representation. can take ∆ Below, we will use the consequence that the metric q on V induces a Hermitian / compatible with Clifford multiplication. metric on ∆

3.5

Hermitian vector bundles

Recall that on a complex vector space V a Hermitian inner product is a map h·, ·i : V × V −→ C that is linear in the first entry and satisfies hv, wi = hw, vi, for all v, w ∈ V. It is positive definite if hv, vi ≥ 0, and moreover hv, vi = 0 implies v = 0.

64


A Hermitian metric hE on a complex vector bundle E −→ M is a smooth Hermitian inner product on the fibers of E. Compatible connections are defined in the usual way, namely, a connection ∇E on E is compatible with the Hermitian metric hE if dhE (σ, τ ) = hE (∇E σ, τ ) + hE (σ, ∇E τ ), for all σ, τ ∈ C ∞ (M ; E). One natural source of complex bundles is the complexification of real bundles. If F −→ M is a real bundle of rank k then F ⊗ C is a complex bundle of rank k. We can think of F ⊗ C as F ⊕ F together with a bundle map J :F ⊕F (u, v)

/

F ⊕F

/ (−v, u)

satisfying J 2 = − Id . (Such a map is called an almost complex structure.) A bundle metric gF on F −→ M induces a Hermitian metric hF on F ⊗ C by hF ((u1 , v1 ), (u2 , v2 )) = gF (u1 , u2 ) + gF (v1 , v2 ) + i (gf (u2 , v1 ) − gF (u1 , v2 )) thus the ‘real part’ of hF is gF . Similarly a connection on F, ∇F , induces a connection on F ⊗ C, namely the induced connection on F ⊕ F. If ∇F is compatible with gF , then the induced connection on F ⊗ C is compatible with hF . A Riemannian metric on a manifold induces a natural Hermitian inner product on C (M ; C) or more generally on sections of a complex vector bundle E endowed with a Hermitian bundle metric, hE . In the former case, Z hf, hi = f (p)h(p) dvolg for all f, h ∈ C ∞ (M ; C) ∞

M

and in the latter, Z

hE (σ, τ ) dvolg for all σ, τ ∈ C ∞ (M ; E)

hσ, τ iE = M

(though the integrals need not converge if M is not compact). A complex vector bundle E over a Riemannian manifold (M, g) is a complex Clifford module if there is a homomorphism, known as Clifford multiplication, cl : Cl(T ∗ M, g) −→ End(E). Clifford multiplication is compatible with the Hermitian metric hE if hE (cl (θ) σ, τ ) = −hE (σ, cl (θ) τ ), for all θ ∈ C ∞ (M ; T ∗ M ), σ, τ ∈ C ∞ (M ; E).


65

Clifford multiplication is compatible with a connection ∇E if [∇E , cl (θ)] = cl (∇θ) , where on the right hand side we are using the Levi-Civita connection. A connection compatible with Clifford multiplication is known as a Clifford connection. Let us define a Dirac bundle to be a complex Clifford module, E −→ M, with a Hermitian metric hE and a metric connection ∇E , both of which are compatible with Clifford multiplication. Any Dirac bundle has a generalized Dirac operator ðE ∈ Diff 1 (M ; E) given by ∇E

icl

ðE : C ∞ (M ; E) −−−→ C ∞ (M ; T ∗ M ⊗ E) −−−→ C ∞ (M ; E). Note that this is indeed a Dirac-type operator, since [ðE , f ] = [icl ◦ ∇E , f ] = icl (df ) shows that σ1 (ðE )(p, ξ) = cl (ξ) ,

σ2 (ð2E )(p, ξ) = |ξ|2g .

It is also a self-adjoint operator thanks to the compatibility of the metric with the connection and Clifford multiplication. We know that ð2E is a Dirac-type operator and hence can be written as a connection Laplacian up to a zero-th order term. There is an explicit formula for this term which is often very useful. First given two vector fields V, W ∈ C ∞ (M ; T M ), the curvature R(V, W ) can be thought of as a two-form e W )(X, Y ) = g(R(V, W )Y, X) for all X, Y ∈ C ∞ (M ; T M ) R(V, e W) . and so we have an action of R(V, W ) on E by cl R(V,

e W ) ∈ Ω2 (M ), R(V,

Next note that for any complex Clifford module, E, (for simplicity over an even dimensional manifold) and a contractible open set U ⊆ M we have / E U ) / ⊗ HomCl(T ∗ M ) (∆, E U ∼ =∆ / denotes the trivial bundle ∆ / × U −→ U. Correspondingly we have where ∆ End(E) = Cl(T ∗ M ) ⊗ EndCl(T ∗ M ) (E) locally. However both sides of this equality define bundles over M and the natural map from the right hand side to the left hand side is locally an isomorphism so these are in fact globally isomorphic bundles.

66


Proposition 9. The curvature RE ∈ Ω2 (M ; End(E)) of a Clifford connection ∇E on E −→ M decomposes under the isomorphism End(E) = Cl(T ∗ M ) ⊗ EndCl(T ∗ M ) (E) into a sum E 1 e bE (V, W ), R (V, W ) = 4 cl R(V, W ) + R e W ) ∈ Cl(T ∗ M ), R bE (V, W ) ∈ EndCl(T ∗ M ) (E) cl R(V, for any V, W ∈ C ∞ (M ; T M ). bE ∈ Ω2 (M ; EndCl(T ∗ M ) (E)) the twisting curvature of the Clifford We call R E connection ∇ . Proof. From the compatibility of the connection with Clifford multiplication, and the definition of the curvature in terms of the connection, it is easy to see that for any V, W ∈ C ∞ (M ; T M ), ω ∈ C ∞ (M ; T ∗ M ), and s ∈ C ∞ (M ; E), RE (V, W )cl (ω) s = cl (ω) RE (V, W )s + cl (R(V, W )ω) s where R(V, W )ω denotes the action of R(V, W ) as a map from one-forms to oneforms. We can write the last term as a commutator 1 1 e e cl (R(V, W )ω) s = 4 cl R(V, W ) cl (ω) s − 4 cl (ω) cl R(V, W ) s since we have, in any local coordinates θi for T ∗ M with dual coordinates ∂i , e W ) cl (θp ) − cl (θp ) cl R(V, e W) cl R(V, = g(R(V, W )∂i , ∂j )cl θi · θj · θp − g(R(V, W )∂i , ∂j )cl θp · θi · θj = −2g(R(V, W )∂p , ∂j )cl θj + 2g(R(V, W )∂i , ∂p )cl θi = 4g(R(V, W )∂i , ∂p )cl θi . So we conclude that RE − 41 cl (R) commutes with Clifford multiplication by one-forms, and hence it commutes with all Clifford multiplication. With this preliminary out of the way we can now compare ð2E and ∆∇ . The proof is through a direct computation in local coordinates5 . 5

`le Vergne See Theorem 3.52 of Nicole Berline, Ezra Getzler, Miche Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. x+363 pp. ISBN: 3-540-20062-2


67

Theorem 10 (Lichnerowicz formula). Let E −→ M be a Dirac bundle, ∆∇ the 0 Bochner Laplacian of the Clifford connection ∇E , RE ∈ Ω2 (M ; EndCl(T ∗ M ) (E)) the twisting curvature of ∇E and scalM the scalar curvature of M, then bE + 1 scalM ð2E = ∆∇ + cl R 4 where we define (in local coordinates ∂i for T M and dual coordinates θi ) 0 X bE = bE (∂i , ∂j )cl θi · θj . cl R R i 0 =⇒ ker ðE = {0} =⇒ ind ð+ = 0. cl R E 4 bE + 1 scalM > 0. So the vanishing of the index of is an obstruction to having cl R 4 This is most interesting when the former is a topological invariant. ð+ E

3.6

Examples

3.6.1

The Gauss-Bonnet operator

Our usual example is the de Rham operator ðdR = d + δ : Ω∗ (M ) −→ Ω∗ (M ), although this is a real generalized Dirac operator, so our conventions will have to be slightly different from those above. As we mentioned above, in this case the Clifford module is the bundle of forms of all degrees, Λ∗ T ∗ M −→ M. Clifford multiplication by a 1-form ω ∈ C ∞ (M ; T ∗ M ) is given by cl (ω) = (eω − iω] ), this is compatible with both the Riemannian metric on Λ∗ T ∗ M and the Levi-Civita connection, and we have cl ◦ ∇ = d + δ. Thus Λ∗ T ∗ M is a (real) Dirac bundle with generalized Dirac operator the de Rham operator. This Dirac bundle is Z2 -graded. Let us define M M Λeven T ∗ M = Λk T ∗ M, Λodd T ∗ M = Λk T ∗ M k even

kodd

3.6. EXAMPLES

69

and correspondingly Ωeven (M ) and Ωodd (M ). Notice that Clifford multiplication exchanges these subbundles and that the covariant derivative with respect to a vector field preserves these subbundles, so even (M ) −→ Ωodd (M ), ðeven dR : Ω

odd ðodd (M ) −→ Ωeven (M ). dR : Ω

The operator ðeven dR is the Gauss-Bonnet operator. A form in the null space of d + δ is necessarily in the null space of the Hodge Laplacian ∆ = (d + δ)2 and so is a harmonic form. Conversely any compactly supported harmonic form is in the null space of d + δ. So on compact manifolds, ker(d + δ) = H∗ (M ),

even ind ðeven (M ) − dim Hodd (M ). dR = dim H

The latter is the Euler characteristic of the Hodge cohomology of M, χHodge (M ), an important topological invariant of M. Notice that on odd-dimensional compact manifolds, the Hodge star is an isomorphism between Heven (M ) and Hodd (M ), so the Euler characteristic of the Hodge cohomology vanishes. The Lichnerowicz formula in this case is known as the Weitzenböck formula, and simplifies to ∆k = ∇∗ ∇ −

X

g(R(ei , ej )ek , e` )eθk ie` eθi iθj

ijk`

where ek is a local orthonormal frame with dual coframe θk . In particular on functions we get ∆0 = ∇∗ ∇ and on one-forms we get Bochner’s formula f ∆1 = ∇∗ ∇ + Ric f is the endomorphism of T ∗ M associated to the Ricci curvature via the where Ric metric. In particular, if the Ricci curvature is bounded below by a positive constant then ∆1 does not have null space, i.e., Ric > 0 =⇒ H1 (M ) = 0.

70


3.6.2

The signature operator

Now consider the complexification of the differential forms on M, Λ∗C T ∗ M = Λ∗ T ∗ M ⊗ C and the corresponding space of sections, Ω∗C (M ). Clifford multiplication is as before, but now we can allow complex coefficients, cl (ω) = 1i (eω − iω] ), this is compatible with both the Riemannian metric on Λ∗C T ∗ M (extended to a Hermitian metric) and the Levi-Civita connection, and we have icl ◦ ∇ = d + δ. Since Λ∗C T ∗ M is a complex Clifford module, we have an involution given by Clifford multiplication by the chirality operator Γ. One can check that, up to a factor of i, Γ coincides with the Hodge star, (M ), cl (Γ) : ΩkC (M ) −→ Ωm−k C

1

cl (Γ) = ik(k−1)+ 2 dim M ∗ .

If M is even dimensional, this induces a Z2 grading of Λ∗C T ∗ M as a Dirac bundle, Λ∗C T ∗ M = Λ∗+ T ∗ M ⊕ Λ∗− T ∗ M. The de Rham operator splits as d+δ =

ð− sign 0

0

ð+ sign

and we refer to ð+ sign as the Hirzebruch signature operator. The index of this operator + − ind ð+ sign = dim ker ðsign − dim ker ðsign

is again an important topological invariant of M. It is always zero if dim M is not a multiple of four, and otherwise it is equal to the signature of the quadratic form 1

/C

H 2 dim M (M ) u It is called the signature of M, sign(M ).

/

R M

u∧u

3.6. EXAMPLES

3.6.3

71

The Dirac operator

If E −→ M is a complex Clifford module over manifold M of even dimension, say m = 2k, then Clifford multiplication cl : Cl(T ∗ M, g) −→ End(E) is, for each p ∈ M, a finite dimensional complex representation of Cl(2k) on Ep . We / can decompose Ep in terms of the unique irreducible representation of Cl(Tp∗ M ), ∆, / ⊗ HomCl(2k) (∆, / Ep ). Ep = ∆ / as Clifford modules for all p ∈ M we say that E is a If it happens that Ep = ∆ C spin bundle over M. If a spinC bundle satisfies the symmetry condition HomCl(T ∗ M ) (E, E) is trivial we say that E is a spin bundle over M. We will generally denote a spin bundle over M by / −→ M. S Remark 5. In the interest of time we have avoided talking about principal bundles and structure groups and hence also spin structures and spinC structures. In brief, the structure group of a Rk -vector bundle is GLk (R) because the transition maps between trivializations are valued in GLk (R). If we choose a metric on the bundle then we can restrict attention to trivializations where the transition maps are valued in O(n), and if we choose an orientation we can use oriented frames and reduce the structure group to SO(n). It turns out that a double cover of SO(n) is a Lie group called the spin group Spin(n). (We can identify Spin(n) with the multiplicative group of invertible elements in Cl(even) (Rn , gRn ).) If the structure group of the tangent bundle of a manifold M can be ‘lifted’ to Spin(n) we say that M admits a spin structure, and we can combine a spin structure with the spin representation to obtain a spin bundle. The group SpinC (n) is the subgroup of Cl(Rk , gRk ) generated by Spin(n) together with the unit complex numbers. If the structure group of the tangent bundle of a manifold M can be ‘lifted’ to SpinC (n) we say that M admits a spinC structure, and we can combine a spinC structure with the spin representation to obtain a spinC bundle. For details we refer to any of the books in the bibliography of this lecture. There are topological obstructions to the existence of a spin bundle over a manifold M and the number of spin bundles when they do exist is also determined topologically6 . For example, a genus g Riemann surface admits 22g distinct spin 6

For a thorough discussion see Lawson, H. Blaine, Jr. and Michelsohn, Marie-Louise Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989.

72


bundles. All spheres in dimension greater than two admit spin bundles. All compact orientable manifolds of dimension three or less admit spin bundles. The complex projective space CPk admits a spin bundle if and only if k is odd. Additionally, any (almost) complex manifold has a spinC bundle. The Riemannian metric and Levi-Civita connection induce the structure of a Dirac bundle on any spin bundle. Indeed, we saw above that the metric on a 2k-dimensional vector space V with positive quadratic form induces a metric on the / so it suffices to describe the induced compatible connection. In a corresponding ∆ local coordinate chart U that trivializes T ∗ M, if we choose an orthonormal frame θ1 , . . . , θm for T M U , the Levi-Civita connection is determined by an m × m matrix of one-forms ω defined through ∇θi = ωij θj . (The fact that the connection respects the metric is reflected in ωij = −ωji .) We can / use ω to define a one-form valued in End(S ) U , namely ω e=

1X ωij cl θi · θj 2 i 2 m−1 ) |x| Indeed, note that on |x| > 0, ∆Qm = 0, so we can write, for any φ ∈ Cc∞ (Rm ), Z Qm ∆φ dx hφ, ∆Qm i = h∆φ, Qm i = lim ε→0 |x|>ε Z Z = lim Qm ∆φ − φ∆Qm dx = − lim Qm ∂r φ − φ∂r Qm dS ε→0

ε→0

|x|>ε

|x|=ε

Z  log ε∂r φ − φε−1   − lim dS  ε→0 2π |x|=ε Z = ε2−m ∂r φ − φ(2 − m)ε1−m   dS − lim ε→0 |x|=ε (2 − m) Vol(Sm−1 )

if m = 2 if m > 2

= φ(0) where in the fourth equality we have used Green’s formula. If we identify R2 with C by identifying (x, y) with z = x + iy then the operator ∂ = 12 (∂x + i∂y ) on complex valued functions has as null space the space of holomorphic functions. 1 as a fundamental A similar application of Green’s theorem allows us to identify πz solution for ∂.

4.2. EXAMPLES

4.2.2

85

Principal values and finite parts

Let us see a few more examples of distributions. The function x 7→ xα is locally integrable on R if α > −1. For α = −1 we define the (Cauchy) principal value of 1 , PV( x1 ) ∈ C −∞ (R) by x Z ∞ Z φ(x) − φ(−x) φ(x) 1 hφ, PV( x )i = lim dx = dx ε→0 |x|>ε x x 0 Z ∞Z 1 φ0 (tx) dt dx, for all φ ∈ Cc∞ (R). = −1

0

This is indeed an element of C −∞ (R), since whenever supp φ ⊆ [−C, C] we have Z ∞ Z 1 0 ≤ 2C sup |φ0 |, φ (tx) dt dx 0

−1

so PV( x1 ) is a distribution of order at most one (in fact of order one). Notice that x PV( x1 ) = 1 since, for any φ ∈ Cc∞ (R), Z Z φ(x)x 1 1 hφ, x PV( x )i = hxφ, PV( x )i = lim dx = φ(x) dx = hφ, 1i. ε→0 |x|>ε x It is then easy to see that T ∈ C −∞ (R),

xT = 1 =⇒ T = PV( x1 ) + Cδ0 , C ∈ C.

so in this sense PV( x1 ) is a good replacement for

1 x

as a distribution.

What about higher powers of x−1 ? Simply taking the principal value will not work for x−2 since we have Z Z φ(x) φ(x) + φ(−x) 2φ(0) dx = dx = + ( bounded ) 2 x2 ε |x|>ε x x>ε so instead we define the (Hadamard) finite part of x−2 , FP( x12 ) by Z Z ∞ φ(x) − φ(0) φ(x) − 2φ(0) + φ(−x) 1 hφ, FP( x2 )i = lim dx = dx ε→0 |x|>ε x2 x2 0 Z ∞ Z 1 Z 1 00 = s φ (stx) dt ds dx, for all φ ∈ Cc∞ (R). 0

0

−1

As before, this is clearly a distribution of order at most two and we have T ∈ C −∞ (R),

x2 T = 1 =⇒ T = FP( x12 ) + C0 δ0 + C1 ∂x δ0 ,

C0 , C1 ∈ C.

LECTURE 4. PSEUDODIFFERENTIAL OPERATORS ON RM

86

We can continue this and define FP( x1n ) for any n ∈ N, n ≥ 2, by P Z 1 (k) k φ − n−2 k=0 k! φ (0)x 1 dx. hφ, FP( xn )i = lim ε→0 |x|>ε xn This defines a distribution of order at most n and moreover T ∈C

−∞

(R),

n

x T = 1 =⇒ T =

FP( x1n )

+

n−1 X

Ci ∂x(k) δ0 ,

{Ci } ⊆ C.

k=0

One can similarly define finite parts of non-integer powers of x and also define finite parts in higher dimensions.

4.3

Tempered distributions

On Rm , the class of Schwartz functions S(Rm ) sits in-between the smooth functions and the smooth functions with compact support Cc∞ (Rm ) ⊆ S(Rm ) ⊆ C ∞ (Rm ) n o S(Rm ) = f ∈ C ∞ (Rm ) : pα,β (f ) = sup |xα ∂ β f | < ∞, for all α, β Rm

Correspondingly the dual spaces satisfy Cc−∞ (Rm ) ⊆ S 0 (Rm ) ⊆ C −∞ (Rm ) S 0 (Rm ) = T ∈ C −∞ (Rm ) : pα,β (f ) → 0 for all α, β =⇒ T (fj ) → 0 . The dual space of S, S 0 (Rm ), is known as the space of tempered distributions. Notice that these inclusions are strict. Any Lp function will be in S 0 (Rm ) but will only be in Cc−∞ (Rm ) if it has compact support. On the other hand, the function 2 e|x| defines a distribution in C −∞ (Rm ) but not a tempered distribution.

4.3.1

Convolution

The convolution of two functions in S(Rm ) is Z Z (f ∗ h)(x) = f (x − y)h(y) dy = Rm

h(x − y)f (y) dy = (h ∗ f )

Rm

and defines a bilinear map S(Rm ) × S(Rm ) −→ S(Rm ). The commutativity is particularly clear from the expression Z Z hφ, f ∗ hi = f (x)h(y)φ(x + y) dx dy,

4.3. TEMPERED DISTRIBUTIONS

87

which we can also write as Z Z Z Z hφ, f ∗ hi = h(y) f (x)φ(x + y) dxdy = h(y) f (z − y)φ(z) dzdy = hfˇ ∗ φ, hi, where fˇ(x) = f (−x). This allows us to use duality to define convolution of a tempered distribution with a Schwartz function ∗ : S(Rm ) × S 0 (Rm ) −→ S 0 (Rm ) hφ, f ∗ Si = hfˇ ∗ φ, Si for all f, φ ∈ S(Rm ) and similarly ∗ : Cc∞ (Rm ) × C −∞ (Rm ) −→ C −∞ (Rm ) hφ, f ∗ Si = hfˇ ∗ φ, Si for all f, φ ∈ C ∞ (Rm ). c

For instance, notice that we have hφ, f ∗ δ0 i = hfˇ ∗ φ, δ0 i =

Z f (x)φ(x) dx = hφ, f i Rm

and so f ∗ δ0 = f for all f ∈ S(Rm ). If P is any constant coefficient differential operator, and f, h ∈ S(Rm ) then P (f ∗ h) = P (f ) ∗ h = f ∗ P (h) and so by duality we also have P (f ∗ S) = P (f ) ∗ S = f ∗ P (S), for all f ∈ S(Rm ), S ∈ S 0 (Rm ). Putting these last two observations together, suppose that S ∈ S 0 (Rm ) is a fundamental solution of the constant coefficient differential operator P, so that P (S) = δ0 , then we have P (f ∗ S) = f ∗ P (S) = f ∗ δ0 = f, for all f ∈ S(Rm ). Thus a fundamental solution for P is tantamount to a right inverse for P, even though P is generally not invertible. In particular, with Qm as above and m > 2 for simplicity,

(4.1)

u ∈ S 0 (Rm ), f ∈ S(Rm ), and ∆u = f =⇒ u = f ∗ Qm ∈ C ∞ (Rm ) Z Z f (y) 1 dy u(x) = Qm (x − y)f (y) dy = m−1 (2 − m) Vol(S ) Rm |x − y|m−2 Rm


88

Remark 6. We can also use convolution to prove that test functions are dense in distributions, be they Cc−∞ , S 0 , or C −∞ . Indeed, first say that a sequence φn ∈ Cc∞ (Rm ) is an approximate identity if Z φn (x) dx = 1, φn (x) = 0 if |x| > n1 , φn ≥ 0, Rd

and note that, as distributions, φn → δ0 . Let χn ∈ Cc∞ (Rm ) approximate the constant function one, and notice that, for any distribution S, the sequence Sn ∈ Cc∞ (Rm ),

Sn (x) = χn (x)(φn ∗ S)(x)

converges as distributions to S.

4.3.2

Fourier transform

Recall that Fourier transform of a function f ∈ L1 (Rm ) is defined by F : L1 (Rm ) −→ L∞ (Rm ) Z 1 b F(f )(ξ) = f (ξ) = f (x)e−ix·ξ dx (2π)n/2 Rm and satisfies for f ∈ S(Rm ), F(∂xj f )(ξ) = iξj F(f )(ξ),

∂ξj F(f )(ξ) = −iF(xj f )(ξ).

To simplify powers of i, let us define Ds = 1i ∂s so that these identities become F(Dxj f )(ξ) = ξj F(f )(ξ),

Dξj F(f )(ξ) = −F(xj f )(ξ),

and indeed for any multi-indices α, β, ξ α Dξβ F(f )(ξ) = (−1)|β| F(Dxα xβ f )(ξ).

(4.2)

In particular we have the very nice result that m F : S(Rm x ) −→ S(Rξ ).

The formal adjoint of F, F ∗ , is defined by hF(f ), hiRm = hf, F ∗ (h)iRm , for all f, h ∈ Cc∞ (Rm ) x ξ and since we have Z

hF(f ), hiRm ξ

1 = F(f )(ξ)h(ξ) dξ = (2π)m/2 Rm

Z Rm ×Rm

e−ix·ξ f (x)h(ξ) dξdx

4.3. TEMPERED DISTRIBUTIONS we conclude that

89

1 F (h)(x) = (2π)m/2 ∗

Z

h(ξ)eix·ξ dξ.

Rm

Using the fact Dj∗ = Dj we find xα Dxβ F ∗ (φ)(x) = (−1)|α| F ∗ (Dξα ξ β φ)(x)

(4.3) and hence

m F ∗ : S(Rm ξ ) −→ S(Rx ).

Proposition 11 (Fourier inversion formula). As a map on S(Rm ), the Fourier transform F is invertible with inverse F ∗ . Proof. We want to show that, for any f ∈ S(Rm ) we have F ∗ (F(f )) = f, i.e., Z Z ix·ξ −iy·ξ e e f (y) dy dξ = (2π)m f (x) but note that the double integral does not converge absolutely and hence we can not change the order of integration. We get around this by introducing a function ψ ∈ S with ψ(0) = 1, letting ψε (x) = ψ(εx) and then using Lebesgue’s dominated convergence theorem to see that F ∗ (h) = lim F ∗ (ψε h). ε→0

Hence the integral we need to compute is given by Z Z Z Z −iy·ξ ix·ξ e f (y) dy dξ = lim ei(x−y)·ξ ψ(εξ)f (y) dξ dy lim e ψ(εξ) ε→0 ε→0 Z Z y−x dy m/2 m/2 b b lim f (x + εη)ψ(η) dη = (2π) lim f (y)ψ( ε ) εm = (2π) ε→0 ε→0 Z m/2 b = (2π) f (x) ψ(η) dη. 2 b If we take ψ(x) = e−|x| /2 , then ψ(η) = ψ(η) and ∗ follows. One can prove FF = Id similarly.

R

b ψ(η) dη = (2π)m/2 so the result

We can extend the Fourier transform and its inverse to tempered distributions in the usual way F, F ∗ : S 0 (Rm ) −→ S 0 (Rm ) hφ, F(T )i = hF ∗ (φ), T i, hφ, F ∗ (T )i = hF(φ), T i for all φ ∈ S(Rm )

90


and these are again mutually inverse and satisfy (4.2), (4.3). In particular, we can compute that 1 hφ, F(δ0 )i = hF (φ), δ0 i = F (φ)(0) = (2π)m/2 ∗

∗

Z

φ(x) dx = hφ, (2π)−m/2 i

Rm

so that F(δ0 ) = (2π)−m/2 . Then, using (4.2) and (4.3), we can deduce that F(Dα δ0 ) = (−1)|α| (2π)−m/2 xα ,

F ∗ (1) = (2π)m/2 δ0 ,

F ∗ (xα ) = (−1)|α| (2π)m/2 Dα δ0 .

Another consequence of Proposition 11 is that, for any φ ∈ S(Rm ), b 2 2 = hF(φ), F(φ)i = hF ∗ F(φ), φi = hφ, φi = kφk2 2 kφk L L thus F has a unique extension to an invertible map F : L2 (Rm ) −→ L2 (Rm ) and this map is an isometry – a fact known as Plancherel’s theorem. The Fourier transform is a very powerful tool for understanding differential operators. Note that if P ∈ Diff k (Rm ), X P = aα (x)Dα , |α|≤k

and the coefficients aα (x) are bounded with bounded derivatives, then for any φ ∈ S(Rm ) (or even S 0 (Rm )) we have P φ = F ∗ (p(x, ξ)Fφ) X with p(x, ξ) = aα (x)ξ α . |α|≤k

The polynomial p(x, ξ) is called the total symbol of P and will be denoted σtot (P ). One consequence which can seem odd at first is that a differential operator acts by integration! Indeed, we have Z Z 1 ∗ (4.4) (P φ)(x) = F (σtot (P )Fφ)(x) = ei(x−y)·ξ p(x, ξ)φ(y) dy dξ m (2π) although note that this integral is not absolutely convergent. Notice the commonality between (4.4) and (4.1). In each case we have a map of the form Z (4.5) A(f )(x) = KA (x, y)f (y) dy Rm

where f and KA , the integral kernel of A, may have low regularity. In fact, this describes a very general class of maps.

4.3. TEMPERED DISTRIBUTIONS

91

Theorem 12 (Schwartz kernel theorem). Let M and M 0 be manifolds, E −→ M and E 0 −→ M 0 vector bundles, and let HOM(E, E 0 ) −→ M × M 0 be the bundle whose fiber at (ζ, ζ 0 ) ∈ M × M 0 is Hom(Eζ , Eζ0 0 ). If A : Cc∞ (M ; E) −→ C −∞ (M 0 ; E 0 ) is a bounded linear mapping, there exists KA ∈ C −∞ (M × M 0 ; Hom(E, E 0 )) known as the Schwartz kernel of A or the integral kernel of A such that hψ, A(φ)i = hψ φ, KA i where ψ φ ∈ Cc∞ (M × M 0 ),

(ψ φ)(ζ, ζ 0 ) = ψ(ζ)φ(ζ 0 ).

We will not prove this theorem6 and we will not actually use it directly. Rather we view as an inspiration to study operators by describing their Schwartz kernels. In fact, it is common to purposely confuse an operator and its Schwartz kernel. Note that if the integral kernel is given by a function, say KA ∈ L1loc (M × M 0 ), then A is given by (4.5). A simple example is the kernel of the identity map KId ∈ C −∞ (M × M ),

KId = δdiagM

and we abuse notation by writing Z f (ζ) = δdiagM f (ζ 0 ) dvol(ζ 0 ). M m

Note that on R we can also write its integral kernel in terms of the Fourier transform Z 1 ∗ ei(x−y)·ξ dξ. (4.6) Id = F F =⇒ KId = (2π)m Rm Similarly we can write the kernel of a differential operator P ∈ Diff ∗ (M ) as KP ∈ C −∞ (M × M ),

KP = P ∗ δdiagM

or, on Rm , (4.7) 6

For a proof, see ...

1 KP = (2π)m

Z

ei(x−y)·ξ p(x, ξ) dξ.


92

Pseudodifferential operators on Rm

4.4 4.4.1

Symbols and oscillatory integrals

In both (4.6) and (4.7) we have integral kernels of the form Z 1 ei(x−y)·ξ p(x, ξ) dξ (2π)m where p(x, ξ) is a polynomial in ξ of degree k and the integral makes sense as a distribution. The way we define pseudodifferential operators is to take a similar integral, but replace p(x, ξ) with a more general function. Let U ⊆ Rm be a (regular7 ) open set and r ∈ R a real number. A function a ∈ C ∞ (U × Rm ) is a symbol of order r on U, a ∈ S r (U) if for all multi-indices α, β there is a constant Cα,β such that p sup |Dxα Dξβ a(x, ξ)| ≤ Cα,β ( 1 + |ξ|2 )r−|β| . x∈U

Notice that the symbol condition is really only about the growth of a and its derivatives as |ξ| → ∞. So we always have 0

S r (U) ⊆ S r (U; Rm ) if r < r0 and, if a is smooth in (x, ξ) and compactly supported (or Schwartz) in ξ, then \ a∈ S r (U) = S −∞ (U; Rm ). r∈R

For example, if r ∈ N then a polynomial of degree r is an example of a symbol of order r. Also, for arbitrary r ∈ R, we can take a function a(x, θ) defined for |θ| = 1, extend it to (x, ξ) with |ξ| > 1 by ξ ) for |ξ| > 1 a(r, ξ) = |ξ|r a(x, |ξ|

and choose any smooth extension of a for |ξ| < 1 and we obtain a symbol of order r. It is easy to see that pointwise multiplication defines a map S r1 (U; Rm ) × S r2 (U; Rm ) −→ S r1 +r2 (U; Rm ) 7

Recall that an open set is regular if it is equal to the interior of its closure.

4.4. PSEUDODIFFERENTIAL OPERATORS ON RM

93

and that differentiation defines a map Dxα Dξβ : S r (U) −→ S r−|β| (U; Rm ). Given a symbol a ∈ S r (U), we will make sense of the oscillatory integral Z (4.8) ei(x−y)·ξ a(x, ξ) dξ as a distribution. If r < −m then the integral converges to a continuous function (equal to Fξ∗ (a)(x, x − y)). Similarly if a ∈ S −∞ (U) then the distribution is actually a smooth function. In this case, the action of this distribution on φ ∈ Cc∞ (U) is Z 1 φ 7→ ei(x−y)·ξ a(x, ξ)φ(x) dξ dx. (2π)m Notice that (1 + ξ · Dx )ei(x−y)·ξ = (1 + |ξ|2 )ei(x−y)·ξ so for a ∈ S r with r < −m and for any ` ∈ N we have Z Z i(x−y)·ξ e a(x, ξ)φ(x) dξ dx = (1 + |ξ|2 )−` (1 + ξ · Dx )` ei(x−y)·ξ a(x, ξ)φ(x) dξ dx Z = ei(x−y)·ξ (1 − ξ · Dx )` ((1 + |ξ|2 )−` a(x, ξ)φ(x)) dξ dx Moreover we have (1 − ξ · Dx )` ((1 + |ξ|2 )−` a(x, ξ)φ(x)) =

X

aα (x, ξ)Dxα φ(x)

|α|≤`

with each aα ∈ S r−` (U). So for arbitrary r ∈ R and a ∈ S r (U), we define the distribution (4.8) on Cc∞ (U) by Z 1 ei(x−y)·ξ (1 − ξ · Dx )` ((1 + |ξ|2 )−` a(x, ξ)φ(x)) dξ dx φ 7→ (2π)m where ` is any non-negative integer larger than r + m. We will continue to denote this distribution by (4.8). Alternately, we can interpret (4.8) as Z 1 φ 7→ lim ei(x−y)·ξ ψ(εξ)a(x, ξ)φ(x) dξ dx ε→0 (2π)m


94

where ψ ∈ Cc∞ (Rm ) is equal to one in a neighborhood of the origin. Then we can carry out the integration by parts from the previous paragraph before taking the limit. For any open set U ⊆ Rm , and s ∈ R we define pseudodifferential operators on U of order r in terms of their integral kernels Z 1 i(x−y)·ξ r r −∞ e a(x, ξ) dξ, for some a ∈ S (U) . Ψ (U) = K ∈ C (U × U) : K = (2π)m The symbol a is known as the total symbol of the pseudodifferential operator. Given a ∈ S r (U), there are several notations for the corresponding pseudodifferential operator, e.g., Op(a) and a(x, D). Let us determine the singular support of K ∈ Ψr (U). The useful observation that (x − y)α ei(x−y)·ξ = Dξα ei(x−y)·ξ

(4.9) shows that

1 (x − y) K = (2π)m

Z

(x − y)α ei(x−y)·ξ a(x, ξ) dξ Z Z 1 (−1)|α| α i(x−y)·ξ = (Dξ e )a(x, ξ) dξ = ei(x−y)·ξ (Dξα a(x, ξ)) dξ (2π)m (2π)m α

is an absolutely convergent integral whenever a ∈ S r (U) for r < |α|−m and moreover taking further derivatives shows that (x − y)α K ∈ C j (U × U) if r − |α| < −m − j so we can conclude that sing supp K ⊆ diagU ×U . As with symbols, pseudodifferential operators are filtered by degree 0

Ψr (Rm ) ⊆ Ψr (Rm ) if r < r0 and a special rôle is played by the operators of order −∞, \ Ψ−∞ (Rm ) = Ψr (Rm ). r∈R

Each a(x, D) ∈ Ψs (U) defines a continuous linear operator a(x, D) : Cc∞ (U) −→ C −∞ (U) Z Z 1 (a(x, D)φ)(x) = ei(x−y)·ξ a(x, ξ)φ(y) dy dξ = F ∗ (a(x, ξ)F(φ)). (2π)m


95

Proposition 13. For a ∈ S r (U), the operator a(x, D) is a continuous map a(x, D) : Cc∞ (U) −→ C ∞ (U).

(4.10)

For a ∈ S r (Rm ), the operator a(x, D) also extends to a linear operator a(x, D) : S(Rm ) −→ S(Rm ) b is Schwartz Proof. If a ∈ S r (U) and φ ∈ Cc∞ (U) then, for every x, a(x, ξ)φ(ξ) function of ξ, hence the integral Z b dξ a(x, D)(φ)(x) = eix·ξ a(x, ξ)φ(ξ) is absolutely convergent, and one can differentiate under the integral sign and the derivatives will also be given by absolutely convergent integrals. Similarly, when a ∈ S r (Rm ), Z Z α α α ix·ξ α b b x a(x, D)(φ) = (−1) (Dξ e )a(x, ξ)φ(ξ) dx = (−1) eix·ξ (Dξα a(x, ξ)φ(ξ)) dx is absolutely convergent for every α. The total symbol of a differential operator is a polynomial so it has an expansion p(x, ξ) =

k X X

pβ (x)ξ β =

j=0 |β|=j

k X

pj (x, ξ)

j=0

where each pj (x, ξ) is homogeneous pj (x, λξ) = λj pj (x, ξ) and a symbol of order j. It turns out we can ‘asymptotically sum’ any sequence of symbols of decreasing order. Proposition 14 (Asymptotic completeness). Let U be an open subset of Rm , {rj } an unbounded decreasing sequence of real numbers and aj ∈ S rj (U). There exists a ∈ S r0 (U) such that, for all N ∈ N, (4.11)

a−

N −1 X

aj ∈ S rN (U)

j=0

which we denote a∼

X

aj .

96


This is analogous to Borel’s lemma to the effect that any sequence of real numbers can occur as the coefficients of the Taylor expansion of a smooth function at the origin. Proof. Let us write U as the union of an increasing sequence of compact sets Kj ⊆ U, [

Ki = U, and Kj ⊆ Kj+1 for all i ∈ N.

j∈N

Pick χ ∈ C ∞ (Rm , [0, 1]) vanishing on B1 (0) and equal to one outside B2 (0), pick a sequence converging to zero, (εj ) ⊆ R+ , and set a(x, ξ) =

∞ X

χ(εj ξ)aj (x, ξ).

j=0

Note that on any ball {|ξ| < R} only finitely many terms of the sequence are nonzero, so the sum is a well-defined smooth function. We will see that by choosing εj decaying sufficiently quickly, we can arrange (5.2). To see that a(x, ξ) ∈ S r0 (U) we need to show that, for each α and β there is a constant Cα,β such that p |Dxα Dξβ χ(εj ξ)a(x, ξ)| ≤ Cα,β ( 1 + |ξ|2 )r0 −|β| . Let us estimate the individual terms in the sum X |β|−|µ| |Dxα Dξβ χ(εj ξ)aj (x, ξ)| ≤ |Dxα Dξγ aj (x, ξ)|εj |(Dβ−µ χ)(εj ξ)|. γ+µ=β

The term with γ = β is supported in |ξ| ≤ ε−1 and the other terms are supported j −1 −1 in εj ≤ |ξ| ≤ 2εj . We can bound the former term by q p p rj −r0 Cα,β (aj )( 1 + |ξ|2 )rj −|β| ≤ Cα,β (aj )( 1 + |ξ|2 )r0 −|β| ( 1 + ε−2 j ) p r −r ≤ Cα,β (aj )εj 0 j ( 1 + |ξ|2 )r0 −|β| and by similar reasoning we can bound the latter terms by p r −r Cα,γ (aj )kDβ−µ χkL∞ εj 0 j ( 1 + |ξ|2 )r0 −|β| . So altogether we have r −rj

|Dxα Dξβ χ(εj ξ)aj (x, ξ)| ≤ Cα,β (χaj )εj 0

p ( 1 + |ξ|2 )r0 −|β| .


97

for some constant Cα,β (χaj ) independent of εj . Given L ∈ N we can choose a sequence εj (L) so that r −rj

|α| + |β| ≤ L =⇒ Cα,β (χaj )εj 0

(L) ≤ j −2

and hence p |α| + |β| ≤ L =⇒ |Dxα Dξβ χ(εj (L)ξ)aj (x, ξ)| ≤ j −2 ( 1 + |ξ|2 )r0 −|β| and |Dxα Dξβ a(x, ξ)| satisfies the bounds we need for |α| + |β| ≤ L. By diagonalization (taking εj = εj (j)) we can arrange the convergence of p ( 1 + |ξ|2 )−(r0 −|β|) |Dxα Dξβ a(x, ξ)| for all α, β. The same argument applies to show that X χ(εj ξ)aj (x, ξ) ∈ S rN (U) j≥N

and hence that a ∼

P

aj .

It is easy to see that if a∼

X

aj

then {aj } determine a up to an element of S −∞ (U). Indeed, if b ∼ a − b = (a −

N −1 X

aj ) − (b −

j=0

N −1 X

P

aj then

aj ) ∈ S rN (U)

j=0

for all N, hence a − b ∈ S −∞ (U). There is a further subset of symbols that imitates polynomials a little closer. We say that a symbol a ∈ S r (U) is a classical symbol, a ∈ Sclr (U), if for each j ∈ N there is a symbol ar−j ∈ S r−j (U) homogeneous in ξ for |ξ| ≥ 1, i.e., ar−j (x, λξ) = λr−j ar−j (x, ξ) whenever |ξ|, |λξ| ≥ 1, and we have a∼

∞ X

ar−j .

j=1

We refer to ar as the principal symbol of a(x, D).


98

4.4.2

Amplitudes and mapping properties

In order to prove that pseudodifferential operators are closed under composition and adjoints, it is convenient to allow the symbol to depend on both x and y. Remarkably, this does not lead to a larger class of operators. Let U ⊆ Rm be an open set, we say that a function b ∈ C ∞ (U × U × Rm ) is an amplitude of order r ∈ R, b ∈ S r (U × U; Rm ) if for any multi-indices α, β, γ, there is a constant Cα,β,γ such that (4.12)

sup |Dxα Dyβ Dξγ b(x, y, ξ)| ≤ Cα,β,γ (1 + |ξ|)r−|γ| .

x,y∈U

Thus an amplitude is a symbol where there are twice as many ‘space’ variables (x, y) as there are ‘cotangent’ variables ξ. An amplitude defines a distribution Z 1 ei(x−y)·ξ b(x, y, ξ)φ(y) dξ dy. φ 7→ m (2π) For fixed x this is a well-defined oscillatory integral in (y, ξ) by the discussion above. Since the estimates are uniform in x, this defines a continuous function in x. We will denote the set of distributions with amplitudes of order r by e r (U). Ψ e r (U), and we will soon show that Ψ e r (U) = Ψr (U). We clearly have Ψr (U) ⊆ Ψ We start by showing this is true for operators in \ e −∞ (U) = e r (U). Ψ Ψ r∈R

Proposition 15. The following are equivalent: i) K ∈ Ψ−∞ (U), R i(x−y)·ξ 1 e a(x, ξ) dξ for some a ∈ S −∞ (U), ii) K = (2π) m iii) K ∈ C ∞ (U × U) and for every N ∈ N and multi indices α, β, (4.13)

sup |(1 + |x − y|)N Dxα Dyβ K(x, y)| < ∞.

x,y∈U

e −∞ (U). iv) K ∈ Ψ


99

Proof. Clearly, (ii) implies (i) and (i) implies (iv). e −∞ (U) means that, Let us check that (iv) implies (iii). By definition, K ∈ Ψ for each r ∈ R, there is an amplitude br ∈ S r (U × U; Rm ) such that Z 1 K(x, y) = ei(x−y)·ξ br (x, y, ξ) dξ. (2π)m Let us assume that r −m so that the integral converges absolutely and we can use (4.9) and integrate by parts to find Z 1 |α| α β γ (x − y) Dx Dy K(x, y) = (−1) ei(x−y)·ξ (Dx + iξ)β (Dy − iξ)γ br (x, y, ξ) dξ (2π)m which converges absolutely and uniformly in x, y provided |β| + |γ| + r < |α| − m. Thus (x − y)α Dxβ Dyγ K(x, y) is uniformly bounded. Finally we show that (iii) implies (ii). If K satisfies (iii) then the function L(x, z) = K(x, x − z) satisfies sup |(1 + |z|)N Dxα Dzβ L(x, z)| < ∞ for all N, α, β. Thus L(x, z) is a Schwartz function in z with bounded derivatives in x. We can write Z 1 K(x, y) = L(x, x − y) = ei(x−y)·ξ (Fz L)(x, ξ) dξ (2π)m/2 and Fz L ∈ S −∞ (U) so we have (ii). At the moment, the mapping property (4.10) is not good enough to allow us to compose operators. We will need an extra condition on the support of the Schwartz kernels of pseudodifferential operators to guarantee that they preserve Cc∞ (U). Let us say that a subset R ⊆ Rm × Rm is a proper support if the left and right projections Rm × Rm −→ Rm , restricted to R, are proper functions8 . An example of a proper support is a compact set, a better example is a set R satisfying R ⊆ {(x, y, ξ) : d(x, y) ≤ ε} for any fixed ε > 0. We will say that a distribution K ∈ C −∞ (Rm × Rm ) is properly supported if supp K is a proper support. We will say that an amplitude b(x, y, ξ) ∈ C ∞ (Rm × Rm ; Rm ) is properly supported if suppx,y b = closure of {(x, y) : b(x, y, ξ) 6= 0 for some ξ} is a proper support. 8

Recall that any continuous function maps compact sets to compact sets, while a function is called proper if the inverse image of a compact set is always a compact set.


100

Theorem 16. Let b(x, y, ξ) ∈ C ∞ (U × U; Rm ) be a properly supported amplitude of order r. The operator (4.14)

B : Cc∞ (U) −→ C −∞ (U) Z Z 1 Bφ(x) = ei(x−y)·ξ b(x, y, ξ)φ(y) dy dξ (2π)m Rm U

in fact maps B : Cc∞ (U) −→ Cc∞ (U) and induces maps B : C ∞ (U) −→ C ∞ (U), B : C −∞ (U) −→ C −∞ (U), and B : Cc−∞ (U) −→ Cc−∞ (U). The formal adjoint of B is of the same form B ∗ : Cc∞ (U) −→ C −∞ (U) Z Z 1 ∗ ei(x−y)·ξ b(y, x, ξ)φ(y) dy dξ. B φ(x) = (2π)m Rm U Moreover, B ∈ Ψr (U) and its total symbol has an asymptotic expansion (4.15)

a(x, ξ) ∼

X i|α| α

α!

(Dξα Dyα b(x, y, ξ)) x=y

where for a multi-index α = (α1 , . . . , αm ), α! denotes α1 ! · · · αm !. Proof. First note that the description of the adjoint follows immediately from Z Z Z 1 hBφ, f i = ei(x−y)·ξ b(x, y, ξ)φ(y)f (x) dy dξ dx (2π)m Rm U U Z Z Z 1 = ei(x−y)·ξ b(x, y, ξ)f (x)φ(y) dx dξ dy (2π)m U Rm U for all φ, f ∈ Cc∞ (U). We see that B maps Cc∞ (U) into C ∞ (U) by the same argument as in Proposition 13. However now if φ ∈ Cc∞ (U) and supp φ = K then we know that πy−1 (K) ∩ suppx,y b = K 0 is a compact set and it is immediate from (4.14) that supp B(φ) ⊆ K 0 . Thus B : Cc∞ (U) −→ Cc∞ (U) and by the same argument B ∗ : Cc∞ (U) −→ Cc∞ (U).


101

By duality this means that B and B ∗ also induce maps C −∞ (U) −→ C −∞ (U). Let φ ∈ C ∞ (U), we know that Bφ is defined, but a priori this is a distribution. Let us check that Bφ is smooth, by checking that it is smooth on any compact set K ⊆ U. We know that πy−1 (K) ∩ suppx,y b = K 0 is also compact, so we can choose χ ∈ Cc∞ (U) such that χ(p) = 1 for all p ∈ K 0 . Then, on K, we have Bφ = Bχφ so Bφ is smooth in the interior of K. This shows that B : C ∞ (U m ) −→ C ∞ (U m ) and since B ∗ is of the same form, B ∗ : C ∞ (U) −→ C ∞ (U). Then by duality both B and B ∗ induce maps Cc−∞ (U) −→ Cc−∞ (U). Let us define aj (x, ξ) =

X i|α| (Dξα Dyα b(x, y, ξ)) x=y . α!

|α|=j

Note that aj ∈ S r−j (Rm ) so P it makes sense to define a(x, ξ) to be a symbol obtained by asymptotically summing aj . We will be done once we show that B − a(x, D) ∈ −∞ Ψ (U). Recall from Taylor’s theorem that, for each ` ∈ N, X 1 α ∂y b(x, x, ξ)(y − x)α + cα (x, y, ξ)(y − x)α α! |α|=` |α|≤`−1 Z 1 (1 − t)`−1 ∂yα b(x, (1 − t)x + ty, ξ) dt with cα (x, y, ξ) = X

b(x, y, ξ) =

0

and note the important fact that ∂yα b(x, y, ξ), and cα (x, y, ξ) all satisfy (4.12). Using (4.9) and integrating by parts we find that   Z Z X 1 1 1 α α ei(x−y)·ξ b(x, y, ξ) dξ = ei(x−y)·ξ  D ∂ b(x, x, ξ) dξ m m (2π) Rm (2π) Rm α! ξ y |α|≤`−1   Z X 1 1 α α i(x−y)·ξ  + e D ∂ cα (x, y, ξ) dξ (2π)m Rm α! ξ y |α|=`

=

`−1 X j=1

1 (2π)m

Z e Rm

i(x−y)·ξ

1 aj (x, ξ) dξ + (2π)m

Z

ei(x−y)·ξ R` (x, y, ξ) dξ

Rm

where R` (x, y, ξ) satisfies (4.12) with r replaced by r − `.


102

By definition of a(x, ξ), we know that there is an element Q` ∈ S r−` (U) such that a(x, D) =

`−1 X

aj (x, D) + Q` (x, D)

j=1

and hence we know that (4.16)

1 B − a(x, D) = K(x, y) = (2π)m

Z

ei(x−y)·ξ (R` (x, y, ξ) − Q` (x, ξ)) dξ

Rm

and R` (x, y, ξ) − Q` (x, ξ) satisfies (4.12) with r replaced by r − `. It is clear from the proof of Proposition 15 that the inequalities (4.13) for a given N, α, β hold for K(x, y) if ` is large enough. Since we are able to choose ` as large as we like, this means that the inequalities (4.13) hold for all N, α, β, but this means that B − a(x, D) ∈ Ψ−∞ (U) as required. This theorem lets us quickly deduce that we get the same class of pseudodifferential operators using amplitudes of symbols. And it lets us show that pseudodifferential operators with properly supported amplitudes are closed under adjoints and under composition. e r (U) coincide. Corollary 17. For all r ∈ R, the sets of distributions Ψr (U) and Ψ e r (U) have amplitude b(x, y, ξ). Choose χ ∈ C ∞ (U × U) such that Proof. Let K ∈ Ψ χ(x, y) = 0 if d(x, y) > 1 e r (U) with amplitude χ(x, y)b(x, y, ξ) and then we can write K as the sum of K1 ∈ Ψ e −∞ (U) with amplitude (1 − χ(x, y))b(x, y, ξ). The theorem guarantees that K2 ∈ Ψ e −∞ (U) = K1 ∈ Ψr (U) because it is properly supported, and we already know that Ψ Ψ−∞ (U). We will say that a pseudodifferential operator is properly supported if its Schwartz kernel is properly supported. It is easy to see that this occurs if and only if we can represent it using a properly supported amplitude. Corollary 18. If A ∈ Ψr (U) is properly supported then A∗ ∈ Ψr (U) is properly supported with symbol (4.17)

σtot (A∗ )(x, ξ) ∼

X i|α| Dα Dα σtot (A)(x, ξ). α! ξ x

|α|≥0

If σtot (A) is a classical symbol then so is σtot (A∗ ) and (4.18)

σr (A∗ ) = σr (A).


103

Proof. For any properly supported function χ ∈ C ∞ (U × U) identically equal to one in a neighborhood of the diagonal containing supp KA , the amplitude χ(x, y)σtot (A)(x, ξ) is a properly supported amplitude for A. It follows from the theorem that A∗ ∈ e r (U) = Ψr (U), with properly supported amplitude χ(y, x)σtot (A)(y, ξ), and symbol Ψ σtot (A∗ ) ∼

X i|α| (Dξα Dyα χ(y, x)σtot (A)(y, ξ)) x=y α!

|α|≥0

which, since χ(x, y) is equal to one in a neighborhood of the diagonal, implies (4.17). This formula in turn implies (4.18). Corollary 19. If A ∈ Ψr (U), B ∈ Ψs (U) are properly supported, the composition A ◦ B : Cc∞ (U) −→ Cc∞ (U) is a properly supported pseudodifferential operator A ◦ B ∈ Ψr+s (U) with symbol (4.19)

X i|α| Dξα σtot (A)(x, ξ)Dxα σtot (B)(x, ξ). σtot (A ◦ B) ∼ α! |α|≥0

If σtot (A) and σtot (B) are classical symbols then so is σtot (A ◦ B) and σr+s (A ◦ B) = σr (A)σs (B).

(4.20)

Proof. Let a(x, ξ) = σtot (A)(x, ξ) and b(x, ξ) = σtot (B)(x, ξ). We know that 1 B u(x) = (2π)m ∗

Z

eix·ξ b(x, ξ)b u(ξ) dξ

integrating against a test function φ ∈ Cc∞ (U) gives Z 1 e−ix·ξ φ(x)b(x, ξ)b u(ξ) dξ dx hBφ, ui = hφ, Bui = (2π)m and so

Z c Bφ(ξ) =

e−iy·ξ b(y, ξ)φ(y) dy.

This gives us a representation of B with an amplitude depending only on y. We can then write Z Z ix·ξ c (A ◦ B)(φ) = A(Bφ) = e a(x, ξ)Bφ(ξ) dξ = ei(x−y)·ξ a(x, ξ)b(y, ξ)φ(y) dy dξ.


104

e r+s (U) with amplitude a(x, ξ)b(y, ξ). Thus A ◦ B ∈ Ψ Next let us check that it is properly supported. If K ⊆ Rm is a compact set, then K 0 = πL (πR−1 (K) ∩ supp KB ) ⊆ Rm is compact and hence πR−1 (K) ∩ supp KA◦B = πR−1 (K 0 ) ∩ supp A is compact. Similarly, πL−1 (K) ∩ supp KA◦B = πL−1 (πR (πL−1 (K) ∩ supp KA )) ∩ supp B is compact, and so A ◦ B is properly supported. Thus, as before, we can multiply its amplitude by a cut-off function to make it a properly supported amplitude without affecting formula (4.15), which yields (4.19). If A and B are classical, then formula (4.19) shows that A ◦ B is classical, and that the principal symbols satisfy (4.20). To define pseudodifferential operators on manifolds, we need to know that an operator that looks like a pseudodifferential operator in one set of coordinates will look like a pseudodifferential operator in any coordinates. Theorem 20. Let U and W be open subsets of Rm and let F : U −→ W be a diffeomorphism. Let P ∈ Ψr (U) and let F∗ P : Cc∞ (W) −→ C ∞ (W) (F∗ P )(φ) = P (φ ◦ F ) ◦ F −1 then F∗ P ∈ Ψr (W). For each multi-index α there is a function φα (x, ξ) which is a polynomial in ξ of degree at most |α|/2, with φ0 = 1, such that the symbol of (F∗ P ) satisfies (4.21)

σtot (F∗ P )(F (x), ξ) ∼

X φα (x, ξ) (Dξα σtot (P ))(x, (DF )t (x)ξ). α!

|α|≥0

If σtot (P ) is classical, so is σtot (F∗ P ) and its principal symbol satisfies (4.22)

σr (F∗ P )(F (x), ξ) = σr (P )(x, DF t (x)ξ).

Notice that, as with a symbol of a differential operator, this shows that the principal symbol of a pseudodifferential operator on U is well-defined as a section of the cotangent bundle T ∗ U.


105

Proof. Let us denote F −1 by H. Note that Z 1 H(x) − H(y) = ∂t H(tx + (1 − t)y) dt = Φ(x, y) · (x − y) 0

where Φ is a smooth matrix-valued function. Since Φ(x, x) = limy→x Φ(x, y) = DH(x) and H is a diffeomorphism, the matrix Φ(x, y) is invertible in a neighborhood Z of the diagonal. Let χ ∈ Cc∞ (Z) be identically equal to one in a neighborhood of the diagonal. Let p = σtot (P ). We have (F∗ P )(φ)(x) = P (u ◦ F )(H(x)) Z Z 1 ei(H(x)−y)·ξ p(H(x), ξ)u(F (y)) dy dξ = (2π)m Rm U Z Z 1 = ei(H(x)−H(z))·ξ p(H(x), ξ)u(z) det DH(z) dz dξ (2π)m Rm W Z Z 1 t = ei(x−z)·Φ(x,z) ξ p(H(x), ξ)u(z) det DH(z) dz dξ. m (2π) Rm W Let us briefly denote the last integrand by γ(x, z, ξ), and decompose it as χγ + (1 − χ)γ. We know that (1−χ)γ is a pseudodifferential operator whose amplitude vanishes in a neighborhood of the diagonal, and hence is an element of Ψ−∞ (W). On the other hand, when integrating χγ we can make the change of variables η = Φ(x, z)t ξ and find Z Z det DH(z) 1 ei(x−z)·η χ(x, z)p(H(x), (Φ(x, z)t )−1 η)u(z) dz dη m (2π) Rm W det Φ(x, z) e r (W) with amplitude which is an element of Ψ pe(x, z, η) =

χ(x, z) p(H(x), (Φ(x, z)t )−1 η) det DF (H(z)) det Φ(x, z)

so we can apply (4.15) and establish (4.21), which in turn proves (4.22). Finally, let us consider pseudodifferential operators acting between sections of bundles. In the present context this is very simple since any vector bundle over Rm is trivial. Thus if E −→ Rm and F −→ Rm are vector bundles, a pseudodifferential operator of order r acting between sections of E and sections of F, A ∈ Ψr (Rm ; E, F ), is simply a (rank F ) × (rank E) matrix with entries in Ψr (Rm ). The extension of the discussion above of composition, adjoint, and mapping properties is straightforward.

106

4.5


Exercises

Exercise 1. Show that a distribution T on Rm supported at the origin is a differential operator applied to δ0 . Hint: First use continuity to find a number K such that hφ, T i = 0 for all φ that vanish to order K at the origin. Next, use this to view T as a linear functional on the finite dimensional vector space consisting of the coefficients of the Taylor expansion of order K of smooth functions at the origin. Exercise 2. Show that, if P is a differential operator and T is a distribution, supp(P (T )) ⊆ supp(T ) and sing supp(P (T )) ⊆ sing supp T. Exercise 3. Let f ∈ C ∞ (R) and let δ0 be the Dirac delta distribution at the origin. Express the distribution f ∂x3 δ0 as a C-linear combination of the distributions δ0 , ∂x δ0 , ∂x2 δ0 and ∂x3 δ0 . Exercise 4. a) Suppose that T ∈ C −∞ (R) satisfies ∂x T = 0 prove that T is equal to a constant C ∈ C. b) Let S ∈ C −∞ (R), show that there is a distribution R ∈ C −∞ (R) such that ∂x R = S. R Hint: Choose ψ ∈ Cc∞ (R) such that ψ dx = 1. Given φ ∈ Cc∞ (M ) show that R e For (a), show that φ dx ψ + ∂x φ. there exists φe ∈ Cc∞ (R) such that φ = R e Si defines φ dx hψ, T i, so C = hψ, T i. For (b), show that hφ, Ri = −hφ, hφ, T i = a distribution that satisfies ∂x R = S. Exercise 5. Suppose that T ∈ S 0 (Rm ) satisfies ∆T = 0 on Rm . Show that T is a polynomial. Hint: First show that F(T ) is a distribution supported at the origin. Exercise 6. Let U ⊆ Rm be an open set and K ∈ C −∞ (U × U). Recall that K defines an operator Cc∞ (U) −→ C −∞ (U). Suppose that K satisfies K : Cc∞ (U) −→ C ∞ (U) and K : Cc−∞ (U) −→ C −∞ (U) and that K is smooth except possibly on diagU , i.e., sing supp K ⊆ diagU . Show that sing supp Ku ⊆ sing supp u for all u ∈ Cc−∞ (U).

4.6. BIBLIOGRAPHY

107

Exercise 7. Let KA ∈ C −∞ (U × U) and suppose that the associated operator A : Cc∞ (U) −→ C −∞ (U) maps Cc∞ (U) −→ C ∞ (U). Show that: i)If KA ∈ Cc∞ (U × U), then A extends to a map C −∞ (U) −→ Cc∞ (U). ii)Conversely, if A : C −∞ (U) −→ Cc∞ (U) and A∗ : C −∞ (U) −→ Cc∞ (U), show that KA ∈ Cc∞ (U × U). Exercise 8. Give an example of a pseudodifferential operator A ∈ Ψ∗ (Rm ) (e.g., of order −∞) and a function φ ∈ Cc∞ (Rm ) such that supp A(φ) is not contained in supp φ. Exercise 9. For properly supported A ∈ Ψr (Rm ), B ∈ Ψs (Rm ) with classical symbols, show that the commutator [A, B] ∈ Ψr+s−1 (Rm ) and find its principal symbol. Exercise 10. Assume that A ∈ Ψr (Rm ) is properly supported with classical symbol. If σr (A) is real-valued (i.e., σr (A)(x, ξ) ∈ R for all x, ξ ∈ Rm ). Show that [A, A∗ ] = AA∗ − A∗ A ∈ Ψ2r−2 (Rm ). Exercise 11. a) Given a function a(x, ξ) ∈ C ∞ (Rm × Sm−1 ) uniformly bounded together with its derivatives, and a number r ∈ R show that there exists a pseudodifferential operator A ∈ Ψr (Rm ) with classical symbol such that σr (A)(x, ξ) |ξ|=1 = a(x, ξ). b) If A ∈ Ψr (Rm ) has classical symbol and σr (A) = 0 show that A ∈ Ψr−1 (Rm ).

4.6

Bibliography

Of the books we have been using so far, both Spin geometry by Lawson and Michelson and the second volume of Partial differential equations by Taylor, treat pseudodifferential operators. For distributions, one can look at Bony, Cours d’analyse, volume one of ¨ rmander, The analysis of linear partial differential operators, and Introduction Ho to the theory of distributions by Friedlander and Joshi. Richard Melrose has several excellent sets of lecture notes involving pseudodifferential operators at http://math.mit.edu/~rbm/Lecture_notes.html. Other good sources are the books Pseudodifferential operators by Michael Taylor and

108


Pseudodifferential operators and spectral theory by Mikhail Shubin, and the lecture notes Introduction to pseudo-differential operators by Mark Joshi (available online on the arXiv). For an in-depth treatment, one can look at the four volumes ¨ rmander. of The analysis of linear partial differential operators by Ho

Lecture 5 Pseudodifferential operators on manifolds 5.1

Symbol calculus and parametrices

Let M be an orientable compact Riemannian manifold, let E and F be complex vector bundles over M endowed with Hermitian metrics1 and let us denote by HOM(E, F ) −→ M × M the vector bundle whose fiber over a point (ζ, ζ 0 ) ∈ M × M is the vector space Hom(Eζ , Fζ 0 ). A distribution K ∈ C −∞ (M × M ; HOM(E, F )) is a pseudodifferential operator on M of order r, acting between sections of E and sections of F, K ∈ Ψr (M ; E, F ) if K is smooth away from the diagonal, i.e., sing supp K ⊆ diagM , and, for any coordinate chart φ : U −→ V of M and any smooth function χ ∈ Cc∞ (V), the distribution χ(x)K(x, y)χ(y) coincides (via φ) with a pseudodifferential operator on U, Kφ,χ ∈ Ψr (U; Rrank E , Rrank F ) 1

We assume orientability, compactness, and a choice of metrics for simplicity, but we could more invariantly work with operators on half-densities.

109

110

LECTURE 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

acting between the trivial bundles φ∗ E and φ∗ F and with classical symbol of order r. Although this definition requires us to check that Kφ,χ is a pseudodifferential operator for every coordinate chart, Theorem 20 shows that it is enough to check this for a single atlas of M. This same theorem guarantees that the principal symbols of the operators Kφ,χ together define σr (K) ∈ C ∞ (T ∗ M ; Hom(π ∗ E, π ∗ F )) with π : T ∗ M −→ M the canonical projection. The principal symbol is homogeneous away form the zero section of T ∗ M, so we do not lose any interesting information if we restrict it to the cosphere bundle S∗ M = {v ∈ T ∗ M : g(v, v) = 1}. From now on, we will think of the symbol as a section σr (K) ∈ C ∞ (S∗ M ; Hom(π ∗ E, π ∗ F )). Let us summarize some facts about pseudodifferential operators on manifolds that follow from our study of pseudodifferential operators on Rm . Theorem 21. Let M be an orientable compact Riemannian manifold and let E, F, and G be complex vector bundles over M endowed with Hermitian metrics. For each r ∈ R we have a class Ψr (M ; E, F ) of distributions satisfying 0

r < r0 =⇒ Ψr (M ; E, F ) ⊆ Ψr (M ; E, F ) with a principal symbol map participating in a short exact sequence (5.1) σ 0 −→ Ψr−1 (M ; E, F ) −→ Ψr (M ; E, F ) −−r→ C ∞ (S∗ M ; Hom(π ∗ E, π ∗ F )) −→ 0. Pseudodifferential operators are asymptotically complete: For any unbounded decreasing sequence of real numbers {rj } and Aj ∈ Ψrj (M ; E, F ) there is A ∈ Ψr1 (M ; E, F ) such that, for all N ∈ N, (5.2)

A−

N −1 X

Aj ∈ ΨrN (M ; E, F )

j=1

which we denote A∼

X

Aj .

5.1. SYMBOL CALCULUS AND PARAMETRICES

111

Pseudodifferential operators act on functions and on distributions A ∈ Ψr (M ; E, F ) =⇒ A : C ∞ (M ; E) −→ C ∞ (M ; F ) and A : C −∞ (M ; E) −→ C −∞ (M ; F ), they are closed under composition which is compatible with both the degree and the symbol A ∈ Ψr (M ; E, F ), B ∈ Ψs (M ; F, G) =⇒ A ◦ B ∈ Ψr+s (M ; E, G) and σr+s (A ◦ B) = σr (A) ◦ σs (B). The formal adjoint of a pseudodifferential operators is again a pseudodifferential operator of the same order and their principal symbols are mutually adjoint. Pseudodifferential operators contain differential operators Diff k (M ; E, F ) ⊆ Ψk (M ; E, F ) and the symbol of a differential operators coincides with its symbol as a pseudodifferential operator. Pseudodifferential operators also contain smoothing operators \ Ψr (M ; E, F ) ⇐⇒ A ∈ C ∞ (M × M ; HOM(E, F )) A ∈ Ψ−∞ (M ; E, F ) = r∈R

⇐⇒ A : C

−∞

(M ; E) −→ C ∞ (M ; F ) and A∗ : C −∞ (M ; F ) −→ C ∞ (M ; E).

Notice that, since M is compact, we do not need to distinguish between smooth functions and those with compact support (and similarly for distributions) and that all distributions are properly supported. We say that a pseudodifferential operator A ∈ Ψr (M ; E, F ) is elliptic if its principal symbol is invertible, i.e., σr (A) ∈ C ∞ (M ; Iso(π ∗ E, π ∗ F )). Thus an elliptic differential operator, such as a Laplace-type or Dirac-type operator, is also an elliptic pseudodifferential operator. Although invertibility of the symbol can not guarantee invertibility of the operator, e.g., the Laplacian on functions has all constant functions in its null space, it turns out that invertibility of the symbol is equivalent to invertibility of the operator up to smoothing operators. Theorem 22. A pseudodifferential operator A ∈ Ψr (M ; E, F ) is elliptic if and only if there exists a pseudodifferential operator (5.3) B ∈ Ψ−r (M ; F, E) s.t. AB − IdF ∈ Ψ−∞ (M ; F ),

BA − IdE ∈ Ψ−∞ (M ; E)

112


If B satisfies (5.3) we say that B is a parametrix for A. It is easy to see that A has a unique parametrix up to smoothing operators (see exercise 2). Proof. Since the symbol of A is invertible and the sequence (5.1) is exact, we know that there is an operator B1 ∈ Ψ−r (M ; F, E), with σ−r (B) = σr (A)−1 ∈ C ∞ (S∗ M ; Iso(π ∗ F, π ∗ E)). The composition A ◦ B1 is a pseudodifferential operator of order zero and σ0 (A ◦ B1 ) = σr (A) ◦ σ−r (B) = Idπ∗ E is the same symbol as the identity operator IdE ∈ Ψ0 (M ; E). Again appealing to the exactness of (5.1) we see that R1 = IdE −AB1 ∈ Ψ−1 (M ; E). Let us assume inductively that we have found B1 , . . . , Bk satisfying (5.4)

Bi ∈ Ψ−r−(i−1) (M ; F, E) and Rk = IdE −A(B1 + . . . + Bk ) ∈ Ψ−k (M ; E)

and try to find Bk+1 ∈ Ψ−r−k (M ; F, E) such that Rk+1 = IdE −A(B1 + . . . + Bk+1 ) ∈ Ψ−k−1 (M ; E). Note that we can write this last equation as Rk+1 = Rk − ABk+1 and since both Rk and ABk+1 are elements of Ψ−k (M ; E) exactness of (5.1) tells us that Rk+1 ∈ Ψ−k−1 (M ; E) ⇐⇒ σk (Rk − ABk+1 ) = 0. So it suffices to let Bk+1 be any element of Ψ−r−k (M ; F, E) with principal symbol σ−r−k (Bk+1 ) = σr (A)−1 ◦ σ−k (Rk ) which we know exists by another application of exactness of (5.1). Thus inductively we can solve (5.4) for all k ∈ N. Next use asymptotic completeness to find B ∈ Ψ−r (M ; F, E) such that X B∼ Bi

5.1. SYMBOL CALCULUS AND PARAMETRICES

113

and notice that, for any N ∈ N, IdE −AB = IdE −A(B1 + . . . + BN ) − A(B − B1 − . . . − BN ) ∈ Ψ−N (M ; E) hence IdE −AB ∈ Ψ−∞ (M ; E) as required. The same construction can be used to construct B 0 ∈ Ψ−r (M ; F, E) such that IdF −B 0 A ∈ Ψ−∞ (M ; F ) but then B − B 0 ∈ Ψ−∞ (M ; F, E) by exercise 2 and so we also have IdF −BA ∈ Ψ−∞ (M ; F ). An immediate corollary is elliptic regularity. Corollary 23. Let A ∈ Ψr (M ; E, F ) be an elliptic pseudodifferential operator. If u ∈ C −∞ (M ; E) then Au ∈ C ∞ (M ; F ) =⇒ u ∈ C ∞ (M ; E). More generally, sing supp Au = sing supp u for all u ∈ C −∞ (M ; E). Proof. It is true for any pseudodifferential operator A that sing supp Au ⊆ sing supp u, (see exercise 6 of the previous lecture) the surprising part is the opposite inclusion. Let B ∈ Ψ−r (M ; F, E) be a parametrix for A and R = IdE −BA ∈ Ψ−∞ (M ; E) then any u ∈ C −∞ (M ; E) satisfies u = BAu + Ru and since Ru ∈ C ∞ (M ; E) we have sing supp u = sing supp BAu ⊆ sing supp Au as required. In the next few sections we will continue to develop consequences of the existence of a parametrix.

114

5.2


Hodge decomposition

An elliptic operator in Ψ∗ (M ; E, F ) induces a decomposition of the sections of E into its null space, which is finite dimensional, and its orthogonal complement and of the sections of F into its image, which has finite codimension, and its orthogonal complement. This is a version of the Hodge decomposition for an arbitrary elliptic operator. We will establish this, starting with the case of smoothing perturbations of the identity, and then show that it implies the Hodge decomposition of differential forms. Theorem 24. Let M be a compact Riemannian manifold, and E a Hermitian vector bundle over M. Any R ∈ Ψ−∞ (M ; E) satisfies: i) The null space of IdE +R acting on distributions is a finite dimensional space of smooth sections of E, {u ∈ C −∞ (M ; E) : (IdE +R)u = 0} ⊆ C ∞ (M ; E) is finite dimensional. ii) We have a decomposition C −∞ (M, E) = Im(IdE +R) ⊕ ker(IdE +R∗ ) by which we mean that, whenever f ∈ C −∞ (M ; E), f ∈ Im(IdE +R) ⇐⇒ hf, uiE = 0 for all u ∈ ker(IdE +R∗ ). Proof. If u ∈ C −∞ (M ; E) and (IdE +R)u = 0 then u = −Ru ∈ C ∞ (M ; E). Since M is compact C ∞ (M ; E) ⊆ L2 (M ; E), so ker A ⊆ L2 (M ; E). In particular, if L ⊆ ker A is a bounded subset of L2 (M ; E) then L is also a bounded subset of C ∞ (M ; E) and so by the Arzela-Ascoli theorem every sequence has a convergent subsequence in C ∞ (M ; E) and hence in L2 (M ; E). This means that any orthonormal set in ker A must be finite, and so ker A is a finite dimensional space. It is easy to see that Im(IdE +R) = (ker(IdE +R∗ ))⊥ so let us show that Im(IdE +R) is closed. Assume that w ∈ C −∞ (M ; E) and there exists a sequence (un ) ⊆ C −∞ (M ; E) such that (IdE +R)un → w as distributions. Because smooth functions are dense in distributions (cf. Remark 6) we can assume without loss of generality that (un ) ⊆ C ∞ (M ; E) ∩ (ker(IdE +R))⊥ .

5.2. HODGE DECOMPOSITION

115

Let us first consider the case where (un ) is a bounded subset of L2 (M ; E). Here, as above, we know that (Run ) has a convergent subsequence, but then un = (IdE +R)un − Run must have a convergent subsequence, say un → u, and by continuity w = (IdE +R)u as required. On the other hand, if un is not bounded in L2 , then we have a subsequence with kun kL2 → ∞. Restricting to this subsequence and letting vn =

un kun kL2

we now have a sequence that is bounded in L2 (M ; E) and moreover (IdE +R)vn → 0. By the previous argument, vn must have a convergent subsequence, say vn → v and (IdE +R)v = 0. Since vn ∈ (ker(IdE +R))⊥ this requires that v = 0, but then this contradicts the fact that kvkL2 = lim kvn kL2 = 1. n→∞

Corollary 25. Let M be a compact Riemannian manifold, E and F Hermitian vector bundles over M. For any r ∈ R and any elliptic operator of order r, A ∈ Ψr (M ; E, F ), we have: i) The null space of A is finite dimensional. ii) The range of A is the orthogonal complement of the null space of A∗ , i.e., if f ∈ C ∞ (M ; F ) then f ∈ Im(A) ⇐⇒ hf, uiF = 0 for all u ∈ ker(A∗ ) An operator that satisfies parts (i) and (ii) of the corollary is called Fredholm. Proof. Let B ∈ Ψ−r (M ; F, E) be a parametrix for A and R = BA − IdE ∈ Ψ−∞ (M ; E),

S = AB − IdF ∈ Ψ−∞ (M ; F ).

The null space of A as an operator on C −∞ (M ; E) is contained in the null space of Id +R, hence by Theorem 24 is finite dimensional and made up of smooth functions. It is easy to see that Im A ⊆ (ker A∗ )⊥ , so assume that f ∈ (ker A∗ )⊥ and let us show that f ∈ Im(A). Using the theorem, we can write (5.5)

(ker A∗ )⊥ = (ker A∗ )⊥ ∩ ker(IdF +S ∗ ) ⊕ (ker A∗ )⊥ ∩ (ker(IdF +S ∗ ))⊥

so we only need to show that these two summands are in Im(A). For the first summand, notice that ker(IdF +S ∗ ) = ker(B ∗ A∗ ), and we have a short exact sequence (5.6)

A∗

0 −→ ker A∗ −→ ker B ∗ A∗ −−−→ Im(A∗ ) ∩ ker B ∗ −→ 0.

116


Since all of these spaces are finite dimensional, we can identify (ker A∗ )⊥ ∩ ker(IdF +S ∗ ) = Im A Im(A∗ )∩ker B ∗ ⊆ Im A. For the second summand in (5.5), IdF +S ∗ = B ∗ A∗ implies (ker(IdF +S ∗ ))⊥ ⊆ (ker A∗ )⊥ , so we can it as (ker A∗ )⊥ ∩ (ker(IdF +S ∗ ))⊥ = (ker(IdF +S ∗ ))⊥ = Im(IdF +S) = Im AB ⊆ Im A.

We can interpret this corollary, the fact that an elliptic operator A ∈ Ψr (M ; E, F ) is Fredholm as an operator A : C ∞ (M ; E) −→ C ∞ (M ; F ) or A : C −∞ (M ; E) −→ C −∞ (M ; F ), as telling us about solutions to the equation Au = f. Namely, we can solve this equation precisely when f ∈ C −∞ (M ; F ) satisfies the (finitely many) conditions hf, u1 i = 0, hf, u2 i = 0, . . . , hf, uk i = 0 where {u1 , . . . , uk } is a basis of ker A∗ and, in that case, there is a finite dimensional space of solutions, u0 + ker A ⊆ C −∞ (M ; E) with u0 a particular solution. By elliptic regularity these solutions are smooth if and only if f is smooth. We also know that the ‘cokernel’ of A as an operator on smooth sections coker(A) = C ∞ (M ; F )/A(C ∞ (M ; E)) or as an operator on distributional sections coker(A) = C −∞ (M ; F )/A(C −∞ (M ; E)) is naturally isomorphic to ker A∗ ⊆ C ∞ (M ; F ). The index of A as a Fredholm operator ind(A) = dim ker A − dim coker A = dim ker A − dim ker A∗ is finite and is the same when A acts on functions and when A acts on distributions. Another immediate consequence of this corollary is the Hodge decomposition.

5.2. HODGE DECOMPOSITION

117

Corollary 26. Let M be a compact Riemannian manifold, for any k ∈ N we have decompositions C ∞ (M ; Λk T ∗ M ) = d C ∞ (M ; Λk−1 T ∗ M ) ⊕ δ C ∞ (M ; Λk+1 T ∗ M ) ⊕ Hk (M ) C −∞ (M ; Λk T ∗ M ) = d C −∞ (M ; Λk−1 T ∗ M ) ⊕ δ C −∞ (M ; Λk+1 T ∗ M ) ⊕ Hk (M ). where the latter is a topological direct sum and the former an orthogonal direct sum. It follows that the de Rham cohomology groups coincide with the Hodge cohomology groups, and this remains true if the de Rham cohomology is computed distributionally. Proof. Applying Corollary 25 to the Hodge Laplacian we know that C ∞ (M ; Λk T ∗ M ) = Im(∆k ) ⊕ Hk (M ) ⊆ d C ∞ (M ; Λk−1 T ∗ M ) + δ C ∞ (M ; Λk+1 T ∗ M ) ⊕ Hk (M ). However it is easy to see that d C ∞ (M ; Λk−1 T ∗ M ) ⊥ δ C ∞ (M ; Λk+1 T ∗ M ) since hdω, δηiΛk T ∗ M = hd2 ω, ηiΛ∗ T ∗ M = 0, which establishes the decomposition for smooth forms. The decompositionfor smooth forms showsthat any distribution that vanishes on d C ∞ (M ; Λk−1 T ∗ M ) , δ C ∞ (M ; Λk+1 T ∗ M ) , and Hk (M ) must be identically zero. Since this is true for distributions in d C −∞ (M ; Λk−1 T ∗ M ) ∩ δ C −∞ (M ; Λk+1 T ∗ M ) , we also have the decomposition for distributional forms. The decomposition clearly shows that ker d : Ωk (M ) −→ Ωk+1 (M ) ∼ k = H (M ) Im (d : Ωk−1 (M ) −→ Ωk (M )) and similarly for distributions. The Hodge decomposition also holds on the Dolbeault complex (3.8), and more generally for an elliptic complex defined as follows: Let E1 , . . . , E` be Hermitian vector bundles over a compact Riemannian manifold M and suppose that we have differential operators Di ∈ Diff 1 (M ; Ei , Ei+1 ), D

D

D`−1

0 −→ C ∞ (M ; E1 ) −−−1→ C ∞ (M ; E2 ) −−−2→ . . . −−−−→ C ∞ (M ; E` ) −→ 0

118


such that Di ◦ Di+1 = 0. This complex is an elliptic complex if, for all ξ ∈ S∗ M, the sequence of leading symbols σ1 (D1 )(ξ)

σ1 (D`−1 )(ξ)

σ1 (D2 )(ξ)

0 −→ π ∗ E1 −−−−−−−→ π ∗ E2 −−−−−−−→ . . . −−−−−−−−→ π ∗ E` −→ 0 is exact. In this setting the same proof as above shows that we have a natural isomorphism of finite dimensional vector spaces ker Di ∼ ∗ ∗ + Di+1 Di ), = ker(Di Di−1 Im Di−1 i.e., the cohomology spaces of the complex are finite dimensional and coincide with the Hodge cohomology spaces.

5.3

The index of an elliptic operator

An elliptic operator acting on a closed manifold has a distinguished parametrix. Given A ∈ Ψr (M ; E, F ) elliptic, let us write C ∞ (M ; E) = ker A ⊕ Im(A∗ ),

C ∞ (M ; F ) = ker A∗ ⊕ Im(A)

and define G : C ∞ (M ; F ) −→ C ∞ (M ; E) by ( Gs = 0 if s ∈ ker A∗ Gs = u if s ∈ Im(A) and u is the unique element of Im(A∗ ) s.t. Au = s The operator G is known as the (L2 ) generalized inverse of A. It satisfies GA = IdE −Pker A ,

AG = IdF −Pker A∗

where Pker A and Pker A∗ are the orthogonal projections onto ker A and ker A∗ , respectively. Notice that these projections are smoothing operators, since we can write X KPker A (x, y) = φj (x)φ∗j (y), j (5.7) with {φj } a basis of ker A and {φ∗j } the dual basis and similarly for Pker A∗ . It turns out that G is itself a pseudodifferential operator, and we can express the index of A using the trace of a smoothing operator Tr : Ψ−∞ (M ; E) −→ C Z Tr(P ) = trE KP diag dvolg M

M

5.3. THE INDEX OF AN ELLIPTIC OPERATOR

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Theorem 27. Let A ∈ Ψr (M ; E, F ) be an elliptic operator and G : C ∞ (M ; F ) −→ C ∞ (M ; E) its generalized inverse, then G ∈ Ψ−r (M ; F, E) and ind(A) = Tr(Pker A ) − Tr(Pker A∗ ) =: Tr([A, G]). Proof. Let B ∈ Ψ−r (M ; F, E) be a parametrix for A and R = BA − IdE ∈ Ψ−∞ (M ; E),

S = AB − IdF ∈ Ψ−∞ (M ; F ).

Making use of associativity of multiplication, we have BAG = G + RG = B − BPker A∗ , GAB = G + GS = B − Pker A hence G = B − BPker A∗ − RG and G = B − Pker A B − GS, and substituting the latter into the former, G = B − BPker A∗ − RB + RPker A B + RGS. This final term satisfies RGS : C −∞ (M ; F ) −→ C ∞ (M ; E),

S ∗ G∗ R∗ : C −∞ (M ; E) −→ C ∞ (M : F )

which shows that RGS ∈ Ψ−∞ (M ; F, E), and so G − B ∈ Ψ−∞ (M ; F, E) and G ∈ Ψ−r (M ; F, E). Now note that, from (5.7), Tr(Pker A ) =

X kφj k2L2 (M ;E) = dim ker A

and similarly for Pker A∗ so the formula ind(A) = Tr(Pker A ) − Tr(Pker A∗ ) is immediate. Let us establish a couple of properties of the trace of smoothing operators that will be useful for understanding the index. Proposition 28. Let M be a compact Riemannian manifold and E, F Hermitian vector bundles. i) The operators of finite rank {K ∈ Ψ−∞ (M ; E, F ) : dim Im(K) < ∞} are dense in Ψ−∞ (M ; E, F ). ii) If P ∈ Ψ−∞ (M ; F, E) and A ∈ Ψr (M ; E, F ) then Tr(AP ) = Tr(P A).

120


Proof. i) Given P ∈ Ψ−∞ (M ; E, F ) so that KP ∈ C ∞ (M × M ; HOM(E, F )) let us show that we can approximate KP in the C ∞ -topology by finite rank operators. Using partitions of unity we can assume that KP is supported in an open set of M × M over which E and F are trivialized, so KP is just a smooth matrix valued function on an open set of Rm × Rm . Identifying this open set with an open set in Tm × Tm with Tm the m-dimensional torus, we can expand each of the entries in a Fourier series in each variable. The truncated series then are finite rank operators that converge uniformly with all of their derivatives to KP , and restricting their support we get finite rank operators on M that approximate P. ii) By continuity we can assume that P has finite rank and by linearity that P has rank one, say P φ = v(φ)s, for some v ∈ C ∞ (M ; F ∗ ), s ∈ C ∞ (M ; E). Then we have AP (φ) = v(φ)As, P A(φ) = v(Aφ)s Z so Tr(AP ) = v(As) dvolg = Tr(P A).

Now we can derive some of the most important properties of the index of an elliptic operator. Theorem 29. Let M be a compact Riemannian manifold, E, F, and G Hermitian vector bundles over M, and let A ∈ Ψr (M ; E, F ) and B ∈ Ψs (M ; F, G) be elliptic. i) B ◦ A ∈ Ψr+s (M ; E, G) is elliptic and ind(B ◦ A) = ind(B) + ind(A). ii) If Q ∈ Ψ−r (M ; F, E) is any parametrix of A then ind(A) = Tr(IdE −QA) − Tr(IdF −AQ) =: Tr([A, Q]). iii) If P ∈ Ψ−∞ (M ; E, F ) then ind(A) = ind(A + P ). iv) The index is a smooth homotopy invariant of elliptic operators. v) The index of A only depends on σr (A). Proof. i) Recall that we have a short exact sequence (5.6) A∗

0 −→ ker A∗ −→ ker B ∗ A∗ −−−→ Im(A∗ ) ∩ ker B ∗ −→ 0

5.3. THE INDEX OF AN ELLIPTIC OPERATOR

121

as these are all finite dimensional, we can conclude that dim ker(A∗ B ∗ ) = dim ker(B ∗ ) + dim((ker A∗ ) ∩ (Im B ∗ )). By the same reasoning dim ker(BA) = dim ker(A) + dim((ker B) ∩ (Im A)) so that ind(BA) is dim ker A + dim((ker B) ∩ (Im A)) − dim ker(B ∗ ) − dim((ker A∗ ) ∩ (Im B ∗ )) = dim ker A + dim((ker B) ∩ (ker A∗ )⊥ ) − dim ker(B ∗ ) − dim((ker A∗ ) ∩ (ker B)⊥ ) = dim ker A + dim(ker B) − dim ker(B ∗ ) − dim(ker A∗ ) = ind(A) + ind(B) as required. ii) From Theorem 27, we know that if G is the generalized inverse of A then Tr([A, G]) = ind(A) and from the proof of Theorem 27 we know that G − Q ∈ Ψ−∞ (M ; F, E). It follows that Tr([A, Q]) − Tr([A, G]) = Tr(IdE −QA) − Tr(IdF −AQ) − Tr(IdE −GA) + Tr(IdF −AG) = Tr((G − Q)A) − Tr(A(G − Q)) which vanishes by Proposition 28. iii) follows immediately from (ii) since any parametrix of A is a parametrix of A + P. iv) If t 7→ At ∈ Ψr (M ; E, F ) is a smooth family of elliptic pseudodifferential operators then we can construct a smooth family t 7→ Qt ∈ Ψ−r (M ; F, E) of parametrices and hence ind(At ) = Tr([At , Qt ]) is a smooth function of t. Since this is valued in the integers, it must be constant and independent of t. v) It suffices to note that if two operators A and A0 have the same elliptic symbol then tA + (1 − t)A0 is a smooth family of elliptic operators connecting A and A0 , so by (iv) they have the same index.

122

5.4


Pseudodifferential operators acting on L2

So far, we have been studying pseudodifferential operators as maps C ∞ (M ; E) −→ C ∞ (M ; F ) or C −∞ (M ; E) −→ C −∞ (M ; F ), now we want to look at more refined mapping properties by looking at distributions with some intermediate regularity, e.g., C ∞ (M ; E) ⊆ L2 (M ; E) ⊆ C −∞ (M ; E). Let us recall some definitions for linear operators acting between Hilbert spaces2 , H1 and H2 . It is convenient, particularly when differential operators are involved, to allow for linear operators to be defined on a subspace of H1 , P : D(P ) ⊆ H1 −→ H2 . We think of a linear operator between Hilbert spaces as the pair (P, D(P )) and only omit D(P ) when it is clear from context. We will always assume that D(P ) is dense in H1 . Thus for instance any pseudodifferential operator A ∈ Ψr (M ; E, F ) defines a linear operator between L2 spaces with domain C ∞ (M ; E), A : C ∞ (M ; E) ⊆ L2 (M ; E) −→ L2 (M ; F ). Of course, with this domain A actually maps into C ∞ (M ; F ), so we will be interested in extending A to a larger domain. An extension of a linear operator (P, D(P )) is an operator (Pe, D(Pe)), Pe : D(Pe) ⊆ H1 −→ H2 such that D(P ) ⊆ D(Pe) and Peu = P u for all u ∈ D(P ). An operator (P, D(P )) is bounded if it sends bounded sets to bounded sets. In particular, there is a constant C such that kP ukH2 ≤ CkukH1 for all u ∈ D(P ) and the smallest constant is called the operator norm of P and denoted kP kH1 −→H2 . 2

For simplicity, we will focus on Hilbert spaces based on L2 (M ; E), but pseudodifferential operators also have good mapping properties on Banach spaces such as Lp (M ; E), p 6= 2.

5.4. PSEUDODIFFERENTIAL OPERATORS ACTING ON L2

123

Recall that an operator (P, D(P )) is continuous at a point if and only if it is continuous at every point in D(P ) if and only if it is bounded. A bounded operator has a unique extension to an operator defined on all of H1 , moreover this extension is bounded with the same operator norm. If an operator is bounded, we will assume that its domain is H1 and generally omit it from the notation. The set of bounded operators between H1 and H2 is denoted B(H1 , H2 ). A compact operator is an operator that sends bounded sets to pre-compact sets, and is in particular necessarily bounded. We denote the space of compact operators between H1 and H2 by K(H1 , H2 ). An example of a smoothing operator is a smoothing operator. Indeed, for smoothing operators between sections of E and sections of F, the image of a bounded subset of L2 (M ; E) is bounded in C ∞ (M ; F ) and hence by Arzela-Ascoli compact in L2 (M ; F ). Lemma 30. Let M be a Riemannian manifold and E, F Hermitian vector bundles. If R ∈ Ψ−∞ (M ; E, F ), then (R, C ∞ (M ; E)) is a compact operator between L2 (M ; E) −→ L2 (M ; F ). Its unique extension to L2 (M ; E) is the restriction of R as an operator on C −∞ (M ; E). Since the identity map is trivially bounded, we can use this lemma to see that any operator of the form IdE +R with R ∈ Ψ−∞ (M ; E) defines a bounded operator on L2 (M ; E). There is a clever way of using this to show that any pseudodifferential operator of order zero defines a bounded operator on L2 . Proposition 31. If r ≤ 0 and P ∈ Ψr (M ; E, F ) then (P, C ∞ (M ; E)) defines a bounded linear operator between L2 (M, E) and L2 (M, F ). Proof. We follow Hörmander and reduce to the smoothing case as follows. Assume we can find Q ∈ Ψ0 (M, E), C > 0, and R ∈ Ψ−∞ (M, E) such that P ∗ P + Q∗ Q = C Id +R. Then we have kP uk2L2 (M,F ) = hP u, P uiF = hP ∗ P u, uiE ≤ hP ∗ P u, ui+hQ∗ Qu, ui = Chu, ui+hRu, ui ≤ (C+kRkL2 (M ;E)→L2 (M ;E) )kuk2L2 (M ;E) , so the proposition is proved once we construct Q, C, and R.

124


We proceed iteratively. Let σ0 (P ) denote the symbol of degree zero of P, e.g., σ0 (P ) = 0 if r < 0. Choose C > 0 large enough so that C − σ0 (P )∗ σ0 (P ) is, at each point of S ∗ M , a positive element of hom(E), then choose Q00 so that its symbol is a positive square root of C − σ0∗ (P )σ0 (P ), and let Q0 = 21 (Q00 + (Q00 )∗ ). Note that Q0 is a formally self-adjoint elliptic operator with the same symbol as Q00 . Suppose we have found formally self-adjoint operators Q0 , . . . , QN satisfying Qi ∈ Ψ−i (M ; E) and !2 N X P ∗P + Qi = C Id −RN , RN ∈ Ψ−1−N (M ; E). i=0

Then, if Q0N +1 ∈ Ψ−N −1 (M, E), we have P ∗P +

N X

!2 Qi + Q0N +1

− C Id

i=0

= −RN +

Q0N +1

N X i=0

! Qi

+

N X

! Qi

Q0N +1 ,

i=0

so in order for this to be in Ψ−2−N (M ; E), Q0N +1 must satisfy σ(Q0N +1 Q0 + Q0 Q0N +1 ) = σ(RN ). We can always solve this equation because σ(Q0 ) is self-adjoint and positive as shown in the following lemma. Lemma 32. Let A be a positive self-adjoint k × k matrix, and let ΦA (B) = AB + BA, then ΦA is a bijection from the space of self-adjoint matrices to itself. Proof. We can find a matrix S such that S ∗ = S −1 and S ∗ AS = D is diagonal. Define Ψ(B) = S ∗ BS and note that Ψ−1 ΦA Ψ = ΦΨ(A) so we may replace A with D = diag(λ1 , . . . , λk ). Let Est denote the matrix with entries ( 1 if s = i, t = j (Est )ij = 0 otherwise and note that ΦD (Est + Ets ) = (λs + λt )(Est + Ets ). Hence the matrices {Est + Ets } with s ≤ t form an eigenbasis of ΦD on the space of self-adjoint k × k matrices. Since, by assumption, each λs > 0 it follows that ΦD is bijective.

5.4. PSEUDODIFFERENTIAL OPERATORS ACTING ON L2

125

So using the lemma we can find σ(Q0N +1 ) among self-adjoint matrices and then define QN +1 = 21 (Q0N +1 + (Q0N +1 )∗ ), finishing the iterative step. Finally, we define Q by asymptotically summing the Qi and the result follows. Corollary 33. Let A ∈ Ψr (M ; E, F ), r ≤ 0, and let kσ0 (A)k∞ = sup |σ0 (A)(ζ, ξ)|, S∗M

then for any ε > 0 there is a non-negative self-adjoint operator Rε ∈ Ψ−∞ (M, E) such that, for any u ∈ L2 (M, E) (5.8)

kAuk2L2 (M,F ) ≤ (kσ0 (A)k2∞ + ε)kuk2L2 (M,E) + hRε u, uiE .

In particular, if r < 0 then A ∈ Ψr (M ; E, F ) defines a compact operator (A, C ∞ (M ; E)) between L2 (M ; E) and L2 (M ; F ). Proof. The existence of Rε satisfying the inequality (5.8) follows directly from the proof of the theorem. If kσ0 (A)k∞ = 0, we show that A is a compact operator by proving that the image of a bounded sequence is Cauchy. Let (uj ) be a bounded sequence in L2 (M ; E), say kuj k ≤ C for all j ∈ N. For each n ∈ N we know that (R 1 uj ) has a n

convergent subsequence in L2 (M ; E) and so by a standard diagonalization argument we can find a subsequence (u0j ) of (uj ) such that (R 1 u0j ) converges for all n. n

(Au0j )

It follows that is a Cauchy sequence in L2 (M ; F ). Indeed, given η > 0 choose N ∈ N such that j, k > N implies kRη (u0j − u0k )kL2 (M ;E) ≤ η. It follows that for such j, k kA(u0j −u0k )k2L2 (M ;F ) ≤ ηku0j −u0k kL2 (M ;E) +hRη (u0j −u0k ), (u0j −u0k )iL2 (M ;E) ≤ η(2C)+ηC. Hence (Au0j ) is Cauchy and A is compact. The graph of a linear operator (P, D(P )) is the subspace of H1 × H2 given by {(u, P u) ∈ H1 × H2 : u ∈ D(P )}. We say that a linear operator (P, D(P )) is closed if and only if its graph is a closed subspace of H1 × H2 . Any bounded operator has a closed graph, and the closed graph theorem says that an closed operator (P, D(P )) with D(P ) = H1 is bounded.

126


Let us point out that a closed operator need not have closed domain or closed range, however the null space of a closed operator does form a closed set. It is very convenient to deal with closed operators, especially when studying the spectrum of an operator or dealing with adjoints. For instance, the spectrum of a linear operator that is not closed consists of the entire complex plane! If A ∈ Ψr (M ; E, F ) then (A, C ∞ (M ; E)) has at least two closed extensions with domains: the maximal domain comes from restricting the action of A as a continuous operator on distributions, Dmax (A) = {u ∈ L2 (M ; E) : Au ∈ L2 (M ; F )}, the minimal domain comes from extending the action of A as a continuous operator on smooth functions, Dmin (A) = {u ∈ L2 (M ; E) : there exists (un ) ⊆ C ∞ (M ; E) s.t. lim un = u and (Aun ) converges}. In the latter case, we define Au = lim Aun and this is independent of the choice of sequence un as long as lim un = u and Aun converges. Since the graph of (A, Dmin (A)) is the closure of the graph of (A, C ∞ (M ; E)), it is clearly the minimal closed extension of A and (A, Dmax (A)) is also clearly the maximal closed extension of A. There are operators for which Dmin (A) 6= Dmax (A) (see exercise 8). However these must have positive order, since we have shown that operators of order zero are bounded (which implies that Dmin (A) = Dmax (A) = L2 (M ; E)). Next we show that elliptic operators also have a unique closed extension. Proposition 34. Let M be a compact Riemannian manifold and E, F Hermitian vector bundles over M, and r > 0. If A ∈ Ψr (M ; E, F ) is an elliptic operator then Dmin (A) = Dmax (A).

(5.9)

In fact the following stronger statement is true: For any Q ∈ Ψ−r (M ; F, E), (5.10)

Im(Q : L2 (M ; F ) −→ L2 (M ; E)) ⊆ Dmin (A).

Proof. Let us start by proving (5.10). Let u ∈ L2 (M ; F ) and v = Qu, we need to show that v ∈ Dmin (A). Choose (un ) ⊆ C ∞ (M ; F ), a sequence converging to u, and note that, since Q and AQ are bounded operators on L2 (M ; F ), we have vn = Qun → v, This implies v ∈ Dmin (A) as required.

Avn = AQun → Av.

5.5. SOBOLEV SPACES

127

Next, to prove (5.9), let B ∈ Ψ−r (M ; F, E) be a parametrix for A and w ∈ Dmax (A). Then w = BAw + Rw, R ∈ Ψ−∞ (M ; E), shows that w is a sum of an section in the image of B and a smooth section, and by (5.10) both of these are in Dmin (A).

5.5

Sobolev spaces

The second part of Proposition 34 strongly suggests that the L2 -domain of an elliptic pseudodifferential operator of order r is independent of the actual operator involved. This is true, and the common domain is known as the Sobolev space of order r. We define these spaces, for any s ∈ R, by H s (M ; E) = {u ∈ C −∞ (M ; E) : Au ∈ L2 (M ; E) for all A ∈ Ψs (M ; E)}. We can think of H s as the sections ‘whose first s weak derivatives are in L2 ’, and if s ∈ N this is an accurate description. Notice that s > 0 =⇒ H s (M ; E) ⊆ L2 (M ; E) ⊆ H −s (M ; E). It will be useful to have at our disposal a family of operators (5.11)

R 3 s 7→ Λs ∈ Ψs (M ; E), σs (Λs ) = idE ∈ C ∞ (S∗ M ; hom(π ∗ E))

that are elliptic, (formally) self-adjoint, strictly positive, and moreover satisfy ∞ Λ−s = Λ−1 s on C (M ; E).

These are easy to construct. First define Λ0 = IdE . Next for s > 0, start with any operator A ∈ Ψs/2 (M ; E) with σs/2 (A) = idE and then define Λs = A∗ A + IdE . Finally, for s < 0, take Λ−s to be the generalized inverse from section 5.3 and note that since Λs is (formally) self-adjoint and injective, Λ−s = Λ−1 s . (When there is E more than one bundle involved, we will denote Λs by Λs .) Using these operators, let us define an inner product on H s (M ; E) by (5.12)

/C

H s (M ; E) × H s (M ; E) (u, v)

/ hu, viH s (M ;E)

= hΛs u, Λs viL2 (M ;E)

128


and the corresponding norm kuk2H s (M ;E) = hu, uiH s (M ;E) . It is easy to see that this inner product gives H s (M ; E) the structure of a Hilbert space. Theorem 35. Let M be a compact Riemannian manifold and E, F Hermitian vector bundles over M. A pseudodifferential operator A ∈ Ψr (M ; E, F ) defines bounded operators A : H s (M ; E) −→ H s−r (M ; F ) for any s ∈ R. For a distribution u ∈ C −∞ (M ; E), the following are equivalent: i) Au ∈ L2 (M ; E) for all A ∈ Ψr (M ; E), i.e., u ∈ H r (M ; E) ii) There exists P ∈ Ψr (M ; E, F ) elliptic such that P u ∈ L2 (M ; F ). iii) u is in the closure of C ∞ (M ; E) with respect to the norm k·kH r (M ;E) . Moreover, every elliptic operator P ∈ Ψr (M ; E, F ) defines a norm on H s (M ; E), kukP,s = kP ukH s−r (M ;F ) + kPker P ukH s (M ;E) which is equivalent to the norm k·kH s (M ;E) . In particular, the Hilbert space structure on H s (M ; E) is independent of the choice of family Λs as another choice of family yields equivalent norms. Proof. Showing that the map A : C ∞ (M ; E) ⊆ H s (M ; E) −→ H s−r (M ; F ) is bounded is equivalent to showing that the map ∞ 2 2 ΛFs−r ◦ A ◦ ΛE −s : C (M ; E) ⊆ L (M ; E) −→ L (M ; F )

is bounded. But this is an operator of order zero, and so is bounded by Proposition 31. It follows that, for any A ∈ Ψr (M ; E, F ) and any s > 0 there is a constant C > 0 such that kAukH s−r (M ;F ) ≤ CkukH s (M ;E) . If A = P is elliptic, and G is the generalized inverse of P, then since it has order −r we have kGvkH s−r (M ;E) ≤ kvkH r (M ;F ) . In particular, we have kP ukH s−r (M ;F ) ≥ CkGP ukH s (M ;E) = ku − Pker P ukH s (M ;E)

5.5. SOBOLEV SPACES

129

hence kukH s (M ;E) ≤ C(kP ukH s−r (M ;F ) + kPker P ukH s−r (M ;E) ), which proves that kukP,s is equivalent to kukH s (M ;E) , since ker P is a finite dimensional space and all norms on finite dimensional spaces are equivalent. This also establishes the equivalence of (i) and (ii). For any s ∈ R, choose s > r and note that H s (M ; E) is the maximal domain of Λs−r : C ∞ (M ; E) ⊆ H r (M ; E) −→ H r (M ; F ) the same argument as in Proposition 34 shows that the minimal domain equals the maximal domain and this shows that C ∞ (M ; E) is dense in H s (M ; E). In this way we can refine our understanding of regularity of sections from C ∞ (M ; E) ⊆ L2 (M ; E) ⊆ C −∞ (M ; E) to, for any s > 0, C ∞ (M ; E) ⊆ H s (M ; E) ⊆ L2 (M ; E) ⊆ H −s (M ; E) ⊆ C −∞ (M ; E). In fact, the natural L2 -pairing which exhibits C −∞ (M ; E) as the dual of C ∞ (M ; E) and L2 (M ; E) as its own dual puts H s (M ; E) in duality with H −s (M ; E). Formally, for every s ∈ R there is a non-degenerate pairing /C

H s (M ; E) × H −s (M ; E) (u, v) /

hu, viH s ×H −s = hΛ−s u, Λs viL2 (M ;E)

and so we can identify (H s (M ; E))∗ = H −s (M ; E). (Notice that this is different from the pairing (5.12) with respect to which H s is its own dual.) Also notice that there are natural inclusions s < t =⇒ H t (M : E) ⊆ H s (M ; E) and indeed we can isometrically embed H t (M ; E) into H s (M ; E) by the operator Λs−t . Notice that since these are pseudodifferential operators of negative order, this inclusion is compact! Put another way, if we have a sequence of distributions in H s (M ; E) that is bounded in H t (M ; E), then it has a convergent sequence in H s (M ; E). This is an L2 -version of the usual Arzela-Ascoli theorem on a compact set K: a sequence of functions in C 0 (K) that is bounded in C 1 (K) has a convergent sequence in C 0 (K). It is easy to see that for any ` ∈ N0 C ` (M ; E) ⊆ H s (M ; E) whenever s > `

130


interestingly there is a sort of converse embedding, known as the Sobolev embedding theorem, (see exercises 9 and 10) H s (M ; E) ⊆ C ` (M ; E) if ` ∈ N0 ,

` > s − 12 dim M.

This lets us see that C ∞ (M ; E) =

\

H s (M ; E), and C −∞ (M ; E) =

s∈R

[

H s (M ; E)

s∈R

so that the scale of Sobolev spaces {H s (M ; E)}s∈R is a complete refinement of the inclusion of smooth sections into distributional sections. With the machinery we have built so far, it is a simple matter to refine Corollary 25 to Sobolev spaces. Theorem 36. Let M be a compact Riemannian manifold, E and F Hermitian vector bundles over M. Let P ∈ Ψr (M ; E, F ) be an elliptic operator of order r ∈ R. We have a refined elliptic regularity statement: u ∈ C −∞ (M ; E), P u ∈ H t (M ; F ) =⇒ u ∈ H t+r (M ; E) for any t ∈ R. Also, for any s ∈ R, P defines a Fredholm operator either as a closed unbounded operator P : H s+r (M ; E) ⊆ H s (M ; E) −→ H s (M ; F ) or as a bounded operator P : H s+r (M ; E) −→ H s (M ; F ) and its index is independent of s. Explicitly this means that: i) The null spaces of P and P ∗ in either case are finite dimensional (in fact these spaces are themselves independent of s.) ii) We have an orthogonal decomposition H s (M ; F ) = P (H s+r (M ; E)) ⊕ ker P ∗ . A consequence of this theorem is a refined version of the Hodge decomposition. Namely, for any s ∈ R, H s (M ; Λk T ∗ M ) = d H s+1 (M ; Λk−1 T ∗ M ) ⊕ δ H s+1 (M ; Λk+1 T ∗ M ) ⊕ Hk (M ).

5.6. EXERCISES

5.6

131

Exercises

Exercise 1. Make sure you understand how to prove all parts of Theorem 21. Exercise 2. Let A ∈ Ψr (M ; E, F ) be an elliptic pseudodifferential operator. Show that if BL , BR ∈ Ψ−r (M ; F, E) satisfy BL A − IdE ∈ Ψ−∞ (M ; E) and ABR − IdF ∈ Ψ−∞ (M ; F ) then BL − BR ∈ Ψ−∞ (M ; F, E). Exercise 3. Show that any linear operator A : Rm −→ Rn is Fredholm and compute its index. Exercise 4. (Schur’s Lemma) Suppose (X, µ) is a measure space and K is a measurable complexvalued function on X × X satisfying Z Z |K(x, y)| dµ(x) ≤ C1 , |K(x, y)| dµ(y) ≤ C2 , X

X

for all y and for all x respectively. Show that the prescription Z Ku(x) = K(x, y)u(y) dµ(y) X

defines a bounded operator on Lp (X, µ) for each p ∈ (1, ∞) with operator norm 1/p

1/q

kKkLp →Lp ≤ C1 C2 , where q satisfies

1 p

+

1 q

= 1.

Hint:It suffices to bound |hKu, vi| for u and v with kukLp ≤ 1, kvkLq ≤ 1. Start with the expression Z Z |hKu, vi| = K(x, y)u(y)v(x) dµ(y)dµ(x) X

X

and apply the inequality ab ≤

ap p

+

bq q

with a = tu and b = t−1 v where t is a positive real number to be determined. Minimize over t to get the desired bound. Exercise 5. Show that the linear operator T : C ∞ ([0, 1]) ⊆ L2 ([0, 1]) −→ L2 ([0, 1]) given by T u = ∂x u is not bounded and not closed.

132


Exercise 6. Let H1 = L2 (R), H2 = R and consider the operator Z u(x) dx Au = R

from H1 to H2 with domain D(A) = Cc∞ (R). Prove that the closure of the graph of A is not the graph of a linear operator. Hint: For each n ∈ N, let ( 1 if |x| ≤ n un (x) = n 0 if |x| > n Show that un → 0 in L2 (R) and Aun = 2 for all n. Why does this solve the exercise? Exercise 7. Let A be a linear operator A : D(A) ⊆ H −→ H. Given λ ∈ C, show that A − λ Id is closed if and only if A is closed. Exercise 8. Show that the operator A = ∂x sin(x)∂x on the circle is symmetric, and that Dmin (A) 6= Dmax (A). Exercise 9. One can also define Sobolev spaces on Rm where there is a natural choice for the operators Λs in (5.11), namely Λs f (x) = F −1 (1 + |ξ|2 )s/2 F(f ) and then as before H s (Rm ) is the closure of Cc∞ (Rm ) with respect to the norm kf ks = kΛs f kL2 (Rm ) . Let s ∈ R and k ∈ N0 such that k < s − 12 m show that there is a constant C such that sup sup |Dα f | ≤ Ckf ks . |α|≤k ζ∈Rm

Hint: First note that kφkL∞ (Rm ) ≤ kF(φ)kL1 (Rm ) for any φ ∈ S(Rm ). Next note that kF(Dα f )kL1 (Rm ) = kξ α F(f )kL1 (Rm ) ≤ k(1 + |ξ|2 )|α|/2 F(f )kL1 (Rm ) Finally, write (1 + |ξ|2 )k/2 = (1 + |ξ|2 )s/2 (1 + |ξ|2 )(k−s)/2 and use Cauchy-Schwarz. Exercise 10. Prove the Sobolev embedding theorem for integer degree Sobolev spaces: If s ∈ N0 there is a continuous embedding of H s (M ; E) into C ` (M ; E) for any ` ∈ N0 satisfying ` < s − 21 dim M.

5.6. EXERCISES

133

Hint: Let f ∈ H s (M ; E), use the fact that multiplication by a smooth function is a pseudodifferential operator of order zero to reduce to the case where f is supported in a coordinate chart φ : U ⊆ Rm −→ M that trivializes E. Since s is a natural number the Sobolev norm on H s (M ; E) is equivalent to a norm k·kD with D a differential operator, use this to show that f ◦ φ ∈ H s (U; Crank E ). Use the previous exercise to conclude that f has ` continuous derivatives. One can use a technique called ‘interpolation’ to deduce the general Sobolev embedding theorem from this integer degree version. Exercise 11. (The adjoint of a linear operator) Let H1 and H2 be C-Hilbert spaces and A : D(A) ⊆ H1 −→ H2 a linear operator. The (Hilbert space) adjoint of A, is a linear operator from H2 to H1 defined on D(A)∗ = {v ∈ H2 : ∃w ∈ H1 s.t. hAu, viH2 = hu, wiH1 for all u ∈ D(A)} by the prescription hAu, vi = hu, A∗ vi for all u ∈ D(A), v ∈ (D(A))∗ . (Note that we always assume our linear operators have dense domains, so A∗ v is determined by this relation.) If H1 = H2 , a linear operator (A, D(A)) is self-adjoint if D(A)∗ = D(A) and hAu, viH1 = hu, AviH1 for all u, v ∈ D(A). i) Show that the graph of (A∗ , D(A)∗ ) is the orthogonal complement in H2 × H1 of the set {(−Au, u) ∈ H2 × H1 : u ∈ D(A)} so in particular (A∗ , D(A)∗ ) is always a closed operator (even if A is not a closed operator). ii) If (A, D(A)) is a closed operator, show that (D(A))∗ is a dense subspace of H2 (e.g., by showing that z ⊥ D(A)∗ =⇒ z = 0) so we can define the adjoint of (A∗ , D(A)∗ ). Prove that ((A∗ )∗ , (D(A)∗ )∗ ) = (A, D(A)). ii) Suppose H1 = L2 (M ; E), H2 = L2 (M ; F ) where M is a Riemannian manifold and E, F are Hermitian vector bundles, and A ∈ Ψr (M ; E, F ) for some r ∈ R. Let A† denote the formal adjoint of A, show that the Hilbert space adjoint of the linear operator A : C ∞ (M ; E) ⊆ L2 (M ; E) −→ L2 (M ; F ) is the linear operator A† : Dmax (A† ) ⊆ L2 (M ; F ) −→ L2 (M ; E).

134


Hint: Recall the definition of the action of A† on distributions. iii) Let H1 = H2 = L2 (M ; E). Show that if A ∈ Ψr (M ; E) is elliptic and formally self-adjoint, then the unique closed extension of A as an operator on L2 (M ; E) is self-adjoint. Exercise 12. Let H1 and H2 be C-Hilbert spaces and A : D(A) ⊆ H1 −→ H2 a linear operator. Recall that A is Fredholm if: i) ker A is a finite dimensional subspace of H1 ii) ker A∗ is a finite dimensional subspace of H2 iii) H2 has an orthogonal decomposition H2 = Im(A) ⊕ ker A∗ (i.e., Im(A) is closed). Prove that an operator is Fredholm if and only if it has an inverse modulo compact operators. That is, there exists B : D(B) ⊆ H2 −→ H1 such that AB − IdH2 and BA − IdH1 are compact operators. Hint: One can follow the arguments above very closely, replacing smoothing operators by compact operators.

5.7

Bibliography

Same as Lecture 4.

Lecture 6 Spectrum of the Laplacian 6.1

Spectrum and resolvent

Let H be a complex Hilbert space and A : D(A) ⊆ H −→ H a linear operator. We divide the complex numbers into the resolvent set of A ρ(A) = {λ ∈ C : A − λ IdH has a bounded inverse} and its complement, the spectrum of A, Spec(A) = C \ ρ(A). The resolvent of A is the family of bounded linear operators /

ρ(A) λ

/ R(λ)

B(H)

= (A − λ IdH )−1

This is, in a natural sense, a holomorphic family of bounded operators1 . Notice that the graph of the resolvent satisfies {(u, R(λ)u) : u ∈ H} = {((A − λ IdH )v, v) : v ∈ H} and hence λ ∈ ρ(A) implies that the operator A − λ IdH is closed, and so by exercise 7 of the previous lecture, that A is closed. In other words if A is not closed then Spec(A) = C. 1

See for instance, Rudin Functional Analysis, for the meaning of a holomorphic vector-valued function.

135

136

LECTURE 6. SPECTRUM OF THE LAPLACIAN

Since a complex number λ is in the spectrum of A if A − λ IdH fails to have a bounded inverse, there is a natural division of Spec(A) by taking into account why there is no bounded inverse: the point spectrum or eigenvalues of A consists of Specpt (A) = {λ ∈ C : A − λ IdH is not injective }, the continuous spectrum of A consists of Specc (A) = {λ ∈ C : A−λ IdH is injective and has dense range, but is not surjective}, everything else is called the residual spectrum, Specres (A) = {λ ∈ C : A − λ IdH is injective but its range is not dense}. If λ ∈ Specpt (A) then ker(A − λ IdH ) is known as the λ-eigenspace and will be denoted Eλ (A), its elements are called eigenvectors. Recall that the adjoint of a bounded operator A ∈ B(H) is the unique operator A ∈ B(H) satisfying ∗

hAu, viH = hu, A∗ viH for all u, v ∈ H. Just as on finite dimensional spaces, for any operator eigenvectors of different eigenvalues are always linearly independent and if the operator is self-adjoint then they are also orthogonal (and the eigenvalues are real numbers). Let us recall some other elementary functional analytic facts. Proposition 37. Let H be a complex Hilbert space and T ∈ B(H). If T = T ∗ then kT kH→H = sup |hT u, uiH | = kuk=1

max |λ|. λ∈Spec(T )

Proof. Let us abbreviate |T | = supkuk=1 |hT u, uiH |. Cauchy-Schwarz shows that |T | ≤ kT kH→H . To prove the opposite inequality, let u, v ∈ H have length one and assume that hT u, viH ∈ R (otherwise we just multiply v by a unit complex number to make this real). We have 4hT u, viH = hT (u + v), (u + v)iH − hT (u − v), (u − v)iH ≤ |T |(ku + vk2H + ku − vk2H ) = 2|T |(kuk2H + kvk2H ) = 4|T | and hence kT kH→H = sup kT ukH = sup kukH =1

sup hT u, viH ≤ |T |.

kukH =1 kvkH =1

6.1. SPECTRUM AND RESOLVENT

137

Next, replacing T by −T if necessary, we can assume that supkukH =1 hT u, uiH > 0. Choose a sequence of unit vectors (un ) such that limhT un , un i = |T | and note that k(T − |T | Id)un k2H = kT un k2H + |T |2 − 2|T |hT un , un i ≤ 2|T |2 − 2|T |hT un , un i → 0 implies that (T − |T | Id) is not invertible, so |T | ∈ Spec(T ). Finally, we need to show that if λ > kT kH→H then λ ∈ ρ(T ), i.e., that (T − λ IdH ) is invertible. It suffices to note that A = λ1 T has operator norm less than one and hence the series X (−1)k Ak k≥0

converges in B(H) to an inverse of IdH −A. With those preliminaries recalled, we can now establish the spectral theorem for compact operators. Theorem 38. Let H be a complex Hilbert space and K ∈ K(H) a compact operator. The spectrum of K satisfies Spec(K) ⊆ Specpt (K) ∪ {0} with equality if H is infinite dimensional, in which case zero may be an eigenvalue or an element of the continuous spectrum. Each non-zero eigenspace of K is finite dimensional and the point spectrum is either finite or consists of a sequence converging to 0 ∈ C. If K is also self-adjoint there is a complete orthonormal eigenbasis of H and we can expand K in the convergent series X K= λPλ λ∈Spec(K)

where Pλ is the orthogonal projection onto Eλ (K). Proof. First notice that if a compact operator on a space V is invertible, then the unit ball of V is compact and hence V is finite dimensional. Thus if H is infinite dimensional, then zero must be in the spectrum of K. Let λ = 6 0 then K − λ IdH = −λ(IdH − λ1 K) and we know that IdH − λ1 K is a Fredholm operator of index zero (since we can connect it to IdH along Fredholm

138


operators). Thus its null space is finite dimensional and if it is injective it is surjective so by the closed graph theorem has a bounded inverse. This shows that Spec(K)\{0} consists of eigenvalues with finite dimensional eigenspaces. Notice that, for any ε > 0 the operator K restricted to    span  

[ λ∈Spec(K) |λ|>ε

 Eλ (K) 

is both compact and invertible, hence this space must be finite dimensional. Thus if the spectrum is not finite, the eigenvalues form a sequence converging to zero. Finally, if K is self-adjoint, let H0 be the orthogonal complement of   [ span  Eλ (K) . λ∈Specpt (K)

Notice that K preserves H0 , since if v ∈ Eλ (K) and w ∈ H0 we have hv, KwiH = hKv, wiH = λhv, wi = 0. Therefore K H0 is a compact self-adjoint operator with Spec(K H0 ) = {0} and so must be the zero operator. But this means that H0 is both in the null space of K and orthogonal to the null space of K, so H0 = {0}. Similarly, notice that X K− λPλ λ∈Spec(K) |λ|>ε

is a self-adjoint operator with spectrum contained in [−ε, ε] ⊆ R. It follows that its operator norm is at most ε which establishes the convergence of the series to K. This immediately yields the spectral theorem for elliptic operators. Theorem 39. Let M be a closed manifold, E a Hermitian vector bundle over M, and P ∈ Ψr (M ; E) an elliptic operator of positive order r ∈ R. If ρ(P ) 6= ∅ then Spec(P ) consists of a sequence of eigenvalues diverging to infinity, with each eigenspace a finite dimensional space of smooth sections of E. If P is self-adjoint then ρ(P ) 6= ∅, the eigenvalues are real, the eigenspaces are orthogonal, and there is an orthonormal eigenbasis of L2 (M ; E).

6.2. THE HEAT KERNEL

139

Proof. If ζ ∈ ρ(P ) then (P − ζ IdE )−1 ∈ Ψ−r (M ; E) is a compact operator on L2 (M ; E) and we can apply the spectral theorem for compact operators. So it suffices to note that P v = λv ⇐⇒ (P − ζ IdE )−1 v = (λ − ζ)−1 v. If P is self-adjoint then Spec(P ) ⊆ R, so ρ(P ) 6= ∅. This is the case in particular for the Laplacian of a Riemannian metric. A classical problem is to understand how much of the geometry can be recovered from the spectrum of its Laplacian. This was popularized by an article of Mark Kac entitled ‘Can you hear the shape of a drum?’.

6.2

The heat kernel

The fundamental solution of the heat equation ( (∂t + ∆)u = 0 u t=0 = u0 is a surprisingly useful tool in geometric analysis. In particular, it will help us to show that the spectrum of the Laplacian contains lots of geometric information. We will discuss an approach to the heat equation known as the Hadamard parametrix construction, following Richard Melrose. For simplicity we will work with the scalar Laplacian, but the construction extends to any Laplace-type operator with only notational changes. On any compact Riemannian manifold, the heat operator e−t∆ : C ∞ (M ) −→ C ∞ (R+ × M ) is the unique linear map satisfying ( (∂t + ∆)e−t∆ u0 = 0 e−t∆ u0 t=0 = u0 Its integral kernel H(t, x, y) is known as the heat kernel and it is a priori a distribution on R+ × M × M, but we will soon show that it is a smooth function for t > 0 and understand its regularity as t → 0. Once we know H(t, x, y) then we can

140


solve the heat equation, even with an inhomogeneous term, since ( (∂t + ∆)v = f (t, x) =⇒ v t=0 = v0 (x) Z Z tZ H(t − s, x, y)f (s, y) dvolg (y)ds. v(t, x) = H(t, x, y)v0 (y) dvolg (y) + 0

M

M

It is straightforward to write e−t∆ as an operator on L2 (M ) by using the spectral data of ∆. Indeed, if {φj } is an orthonormal basis of L2 (M ), with ∆φj = λj φj , then we have X H(t, x, y) = e−tλj φj (x)φj (y) for all t > 0. j

However it is hard to extract geometric information from this presentation. On Euclidean space, the heat kernel is well-known (see exercise 7) 1 |x − y|2 HRm (t, x, y) = exp − . (4πt)m/2 4t Notice that HRm is smooth in t, x, y as long as t > 0. As t → 0, HRm vanishes exponentially fast away from diagRm = {x = y} and blows-up as we approach {0} × diagRm . Notice that tm/2 HRm is still singular as we approach this submanifold since the argument of the exponential as we approach a point (0, x, x) depends not just on x, but also on the way we approach. For instance, for any z ∈ Rm , the limit of tm/2 H(t, x, y) as t → 0 along the curve (t, x + t1/2 z, x) is |z| . exp − 4 To understand the singularity at {0} × diagRm we need to take into account the different ways of approaching this submanifold. We do this by introducing ‘parabolic’ polar coordinates (parabolic because the t-direction should scale differently from the x − y direction). Define SP = {(θ, ω) ∈ R+ × Rm : θ + |ω|2 = 1} and notice that any point (t, x, y) ∈ R+ × Rm × Rm can be written p t = r2 θ, x = x, y = x − rω, with r = t + |x − y|2 .


141

Indeed, (r, θ, ω, x) are the parabolic polar coordinates for R+ × Rm × Rm . In these coordinates, we can write tm/2 HRm as |ω|2 exp − 4θ and, since θ and ω never vanish simultaneously, this is now a smooth function! We say that these coordinates have resolved the singularity of the heat kernel. Taking a closer look at the parabolic polar coordinates, notice that these are coordinates only when r 6= 0. Each point in the diagonal of Rm at t = 0 corresponds to r = 0 and to a whole SP . A better way of thinking about this is that we have constructed a new manifold with corners (the heat space of Rm ) HRm = R+ × SP × Rm with a natural map HRm

β

(r, θ, ω, x)

/ /

R+ × Rm × Rm (r2 θ, x, y − rω)

which restricts to a diffeomorphism HRm \ {r = 0} −→ (R+ × Rm × Rm ) \ ({0} × diagM ). We refer to HRm (together with the map β) as the parabolic blow-up of R+ ×Rm ×Rm along {0} × diagM . Now let (M, g) be a Riemannian manifold. As a first approximation to the heat kernel of ∆g consider d(x, y)2 1 exp − . G(t, x, y) = (4πt)m/2 4t Analogously to the discussion above, this is a smooth function on R+ ×M 2 except at {0} × diagM , and we will resolve this singularity by means of a ‘parabolic blow-up’. What should replace the diagonal at time zero? Consider the space R+ × T M with its natural map down to M, π : R+ × T M −→ T M −→ M. At each point p ∈ M define SP (p) = {(θ, ω) ∈ R+ × Tp M : t + |ω|2g = 1} and then SP (M ) =

[ p∈M

SP (p).

142


The map π restricts to a map π : SP (M ) −→ M which is a locally trivial fibration with fibers SP . Let us define the heat space of M as the disjoint union G HM = (R+ × M 2 \ {0} × diagM ) SP M, endowed with a smooth structure as a manifold with corners2 where we attach SP M to R+ ×M 2 \{0}×diagM by the exponential map of g. There is a natural ‘blow-down map’ β : HM −→ R+ × M 2 which is the identity on R+ × M 2 \ {0} × diagM and is equal to π on SP M. This means that given a normal coordinate system for R+ × M 2 at (0, p, p), U ⊆ R+ × Tp M × Tp M −→ R+ × M 2 we get a coordinate system around β −1 (0, p, p) by introducing parabolic polar coordinates as before, but now on the fibers of Tp M. Thus, starting with a coordinate system induced by expp : Tp M −→ M, say ζ = (ζ1 , ζ2 , . . . , ζm ), we get a coordinate system on HM, r, (θ, ω), ζ ∈ R+ × SP × Rm with the identification of r = 0 with SP (expp (ζ)) and the identification of points with r 6= 0 with (r2 θ, expp (rω + ζ), expp (ζ)) ∈ (R+ × M 2 \ {0} × diagM ). Recall that the distance between expp (rω + ζ) and expp (ζ) is given by |rω|2g(p) + O(|rω|4g(p) ) so that our approximation to the heat kernel in these coordinates is ! |rω|2g(p) + O(|rω|4g(p) ) 1 exp − (4πr2 θ)m/2 4r2 θ !! |ω|2g(p) r2 |ω|4g(p) 1 = exp − +O (4πr2 θ)m/2 4θ θ and this is r−m times a smooth function, so we have again resolved the singularity. This computation shows that G approximates the Euclidean heat kernel to leading order as t → 0. That is enough to see that, for any u0 ∈ C ∞ (M ), Z G(t, x, y)χ(dg (x, y))u0 (y) dvolg (y) → u0 (x) M 2

For a careful proof, see Proposition 7.7 in Richard B. Melrose, The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics, 4. A K Peters, Ltd., Wellesley, MA, 1993.


143

where χ(dg (x, y)) is a cut-off function near the diagonal . In fact, we will proceed much as in the construction of an elliptic parametrix and find a sequence of functions Aj such that the operator e (θ, ω), x) ∼ β ∗ (χ(dg (x, y))G(t, x, y)) H(r,

X

tj/2 Aj (r, (θ, ω), x),

j≥0

e (θ, ω), x) = O(r∞ ) as r → 0. Here we are using the fact that, satisfies β ∗ (∂t +∆)H(r, since β is the identity away from the blown-up face, it makes sense to pull-back the differential operator ∂t + ∆; in a moment, we will compute this in local coordinates. e (θ, ω), x). We will assume inLet us denote β ∗ (χ(dg (x, y))G(t, x, y)) by G(r, ductively that we have found A0 , A1 , . . . , AN such that ! N X e (θ, ω), x) β ∗ (∂t + ∆) G(r, rj Aj (r, (θ, ω), x) = r−n+N +1 EN (r, (θ, ω), x) j≥0

with EN a smooth function on the blown-up space, vanishing to infinite order at t = 0 away from the diagonal, and then we need to find AN +1 (r, (θ, ω), x) such that e N +1 r−n+N +1 EN (r, (θ, ω), x) + β ∗ (∂t + ∆) r(N +1) GA vanishes to order −n + N + 2 at the blown-up boundary face {r = 0}, i.e., h i e N +1 rn−N −1 β ∗ (∂t + ∆) r(N +1) GA = −EN (0, (θ, ω), x). r=0

To carry out this construction, it is convenient to use a different set of coordinates, ‘projective coordinates’ τ=

√

t,

ζ Y =√ , t

x

in which we can compute ∗

β ((t∂t )f (t, x, y)) =

1 τ ∂τ 2

−

1 2

X

Yj ∂Yj β ∗ (f (t, x, y))

β ∗ (∂zj f (t, x, y)) = τ1 (∂Yj + τ Vj )β ∗ (f (t, x, y)) for some vector fields Vj on HM. In these coordinates, |Y | 1 ∗ 2 2 + O(τ |Y | ) β G(t, x, y) = exp − (4π)m/2 τ m 4

144


so altogether we have to solve N + 1 − m −|Y |2 /4 1 ∆Y − 2 Y · ∂Y + e AN +1 = −EN (0, Y, x). 2 and taking the Fourier transform in Y, this becomes ρ=|η|

(|ξ|2 + 12 ξ · ∂ξ + C)(f (ξ)) = h(ξ) −−−−→ (ρ∂ρ + 2ρ2 + C)(f (ξ)) = h(ξ) and one can check that this has a unique solution which decays rapidly as ρ → ∞ ξ . Taking the inverse Fourier transform we get the and depends smoothly on ξb = |ξ| required AN +1 . e (θ, ω), x) be a funcHaving inductively constructed all of the Aj , now let H(r, me tion on HM such that r H(r, (θ, ω), x) is smooth on HM, vanishes to infinite order at t = 0 away from the P diagonal, and at the blown-up boundary face has Taylor expansion G(r, (θ, ω), x) rj Aj . (The existence of such a function follows from the classical Borel’s theorem, much like asymptotic completeness of symbols above.) e is our parametrix for the heat kernel. It solves the heat This function H equation to infinite order as r → 0, in that \ \ e∈ tn C ∞ (R+ × M 2 ). rn C ∞ (HM ) = S = β ∗ (∂t + ∆)H n∈N

n∈N

Next we use it to find the actual solution to the heat equation. Notice that we can think of a function F ∈ C ∞ (R+ × M 2 ) as defining an operator /

F

C ∞ (M ) u

/

R M

C ∞ (R+ × M ) F (t, x, y)u(y) dy

but we can also think of it as defining an operator Op(F )

C ∞ (R+ × M ) u

/

RtR 0

M

/

C ∞ (R+ × M )

F (s, x, y)u(t − s, y) dyds

and the advantage is that then it makes sense to compose operators. The composition of the operators corresponding to the functions F1 and F2 is easily seen to be an operator of the same form, but now associated to the function Z tZ F1 ∗ F2 (t, x, y) = F1 (s, x, z)F2 (t − s, z, y) dzds. 0

M


145

With respect to this product the identity is associated to the distribution δ(t)δ(x−y) and so we can recognize the heat equation for the heat kernel ( (∂t + ∆)H(t, x, y) = 0 if t > 0 H(0, x, y) = δ(x − y) as the equation Op((∂t + ∆)H(t, x, y)) = Id . Our parametrix satisfies e = Id + Op(S) Op(β ∗ (∂t + ∆)H) so we need to invert Id + Op(S). This can be done explicitly by a series ! X (Id + Op(S))−1 = Op (−1)j S ∗ S ∗ . . . ∗ S . j≥0

One can check that this series converges to Id + Op(Q) for some \ Q∈ tn C ∞ (R+ × M 2 ), n∈N

and hence e + (H e ∗ β ∗ Q). β ∗H = H In particular we see that e∈ β ∗H − H

\

rn C ∞ (HM ).

n∈N

Theorem 40. Let (M, g) be a compact Riemannian manifold, HM its heat space with blow-down map β : HM −→ R+ × M 2 The pull-back of H(t, x, y), the heat kernel of ∆g , satisfies β ∗ H ∈ r−m C ∞ (HM ). The trace of the heat kernel has an asymptotic expansion as t → 0, Z X −t∆ Tr e = H(t, x, x) dvolg ∼ t−m/2 tk/2 ak M

k≥0

with a0 = (4π)−m/2 Vol(M ) and each of the ak are integrals over M of universal polynomials in the curvature of g and its covariant derivatives, Z ak = Uk (x) dvolg . M

146


The coefficients ak are known as the local heat invariants of (M, g). One can use a symmetry argument to show that all of the ak with k odd are identically zero, but the other ak contain important geometric information. Proof. The only thing left to prove is the statement about the coefficients ak . Its clear from the construction above that the coefficients Aj are built up in a universal fashion from the expansion of the symbol of ∆. But as this symbol is precisely the Riemannian metric, and we know that its Taylor expansion at a point in normal coordinates has coefficients given by the curvature and its covariant derivatives, the result follows with Uk (x) = Ak (0, (1, 0), x).

6.3

Weyl’s law

The following lemma is an example of a ‘Tauberian theorem’. Lemma 41 (Karamata). If µ is a positive measure on [0, ∞), α ∈ [0, ∞), then Z ∞ e−tλ dµ(λ) ∼ at−α as t → 0 0

implies Z 0

`

a dµ(λ) ∼ `α as ` → ∞. Γ(α + 1)

Proof. We start by pointing out that Z Z α −tλ t e dµ(λ) = e−λ dµt (λ) with µt (λ)(A) = tα µ(t−1 A), so we can rewrite the hypothesis as Z ∞ Z a −λ λα−1 e−λ dλ lim e dµt (λ) = a = t→0 Γ(α) 0 and the conclusion as Z lim t→0

a χ(λ) dµt (λ) = Γ(α)

Z

∞

λα−1 χ(λ) dλ

0

with χ(λ) equal to the indicator function for [0, `]. But this last equality is true for χ(λ) = esλ for any s ∈ R+ , hence by density for any χ ∈ C0∞ (R+ ) and hence for Borel measurable functions, in particular for the indicator function of [0, `].

6.3. WEYL’S LAW

147

If we apply this to the measure dµ(λ) =

P

δ(λ − λi ), we can conclude.

Theorem 42 (Weyl’s law). Let (M, g) be a compact Riemannian manifold and let N (Λ) be the ‘counting function of the spectrum’ of the Laplacian, X

N (Λ) =

dim Eλ (∆g ).

λ∈Spec(∆) λ 0, a bounded self-adjoint operator on L2 (M ). Hence its operator norm is equal to the modulus of its largest eigenvalue. In the same way ke−t∆ −

N X

e−tλj Pλj kL2 (M )→L2 (M ) = e−tλN +1 .

j=0

Let us denote e

−t∆N

−t∆

=e

−

N X

e−tλj Pλj

j=0

and point out that this is the heat operator of the Laplacian restricted to the orthogonal complement of span{φ0 , . . . , φN }. In particular, the semigroup property applied to its integral kernel HN yields Z −s∆N −t∆N −(s+t)∆N HN (s, x, z)HN (t, z, y) dz. ⇐⇒ HN (s + t, x, y) = e =e e M

Thus Z HN (t + 2, x, y) =

HN (1, x, z1 )HN (t, z1 , z2 )HN (1, z2 , y) dz1 dz2 , M2

and we have |Dxα Dyβ HN (t + 2, x, y)| ≤ kDxα HN (1, x, z1 )kL2 (M ) ke−t∆N kL2 →L2 kDyβ HN (1, z2 , y)kL2 (M ) ≤ Cα,β e−tλN +1 . 4

See Gregor Weingart, A characterization of the heat kernel coefficients. arXiv.org/abs/ math/0105144

6.5. DETERMINANT OF THE LAPLACIAN

149

Proposition 44. Let (M, g) be a compact Riemannian manifold and {φj } an orthonormal eigenbasis of the Laplacian of g such that λj ≤ λj+1 . The series N X

e−tλj φ(x)φ(y)

j=0

converges uniformly to the heat kernel H(t, x, y) in C ∞ (M ). In particular, for any t>0 X Tr(e−t∆ ) = e−tλj and for large t, | Tr(e−t∆ − Pker ∆ )| ≤ e−tλ1 . Remark 7. One can show much more generally that the trace we defined on smoothing operators coincides with the functional analytic trace given by summing the eigenvalues.

6.5

Determinant of the Laplacian

The asymptotic expansion of the trace of the heat kernel allows us to define the determinant of the Laplacian. This is an important invariant with applications in physics, topology, and various parts of geometric analysis. First we imitate the classical Riemann zeta function and define X ζ(s) = λ−s λ∈Spec(∆) λ6=0

where eigenvalues are repeated in the sum according to their multiplicity. Here s is a complex variable, and Weyl’s law shows that this sum converges absolutely (and hence defines a holomorphic function) if Re(s) > 21 dim M. Notice that on the circle the eigenfunctions are eiθn with eigenvalue n2 , so the non-zero eigenvalues are the squares of the integers, each with multiplicity two. It follows that the ζ function of the circle is X ζ(s) = 2 n−2s = 2ζRiemann (2s) n∈N

essentially equal to the Riemann zeta function! Returning to general manifolds, we can use the identity Z ∞ dt 1 −s µ = ts e−tµ Γ(s) 0 t

150


valid whenever Re µ > 0 to write 1 ζ(s) = Γ(s)

Z

∞

ts Tr(e−t∆ − Pker ∆ )

0

dt t

whenever Re(s) > 12 dim M. Indeed, Proposition 44 shows that the integral Z ∞ dt ts Tr(e−t∆ − Pker ∆ ) t 1 converges as t → ∞, for any s ∈ C, and Theorem 40 shows that the integral Z 1 dt ts Tr(e−t∆ − Pker ∆ ) t 0 converges as t → 0, if Re s > m/2. Moreover, it is straightforward to show that ζ(s) is a holomorphic function of s on the half-plane Re s > m/2 where the integral converges. Another use of Theorem 40 shows that the integral ! Z 1 N X dt ts Tr(e−t∆ ) − t−m/2 tk/2 ak t 0 k=0 −1 converges, and defines a holomorphic function, on the half plane Re s > m−N . To 2 incorporate the projection onto the null space, it suffices to replace ak with ( ak + dim ker ∆ if k = m e ak = ak otherwise

On the other hand, for Re s > m/2, we can explicitly integrate Z

1 s

−m/2

t t 0

N X k=0

N

k/2

t

dt X e ak e ak = t s + (k − m)/2 k=0

Thus, for any N ∈ N, we have a function Z ∞ 1 dt ζN (s) = Tr(e−t∆ − Pker ∆ ) Γ(s) t 1 ! ! Z 1 N N X X dt e a k tk/2e ak + ts Tr(e−t∆ − Pker ∆ ) − t−m/2 + t s + (k − m)/2 0 k=0 k=0 −1 which is meromorphic on Re s > m−N and coincides with ζ(s) for Re s > m/2. 2 Since N was arbitrary we can conclude:

6.6. THE ATIYAH-SINGER INDEX THEOREM

151

Theorem 45. Let (M, g) be a compact Riemannian manifold of dimension m. The ζ function admits a meromorphic continuation to C, with at worst simple poles located at (m − k)/2, k ∈ N and with residue e ak

Γ( m−k ) 2 where e ak is the coefficient of t(k−m)/2 in the short-time asymptotics of Tr(e−t∆ − Pker ∆ ). It follows easily that the meromorphic continuation of ζ(s) is always regular at s = 0. Notice that, if the Laplacian had only finitely many positive eigenvalues {µ1 , . . . , µ` } then ∂s

hX

µ−s j

i

=− s=0

X

log µj = − log

Y

µj “ = − log det ∆”.

Inspired by this computation, Ray and Singer defined −∂s ζ(s) s=0 det ∆ = e and, as mentioned above, this has found applications in topology, physics, and geometric analysis.

6.6

The Atiyah-Singer index theorem

In this section we will sketch how one can use the constructions of this lecture to prove the Atiyah-Singer index theorem. Recall that the objective is to find a topological formula for ind P where P ∈ Ψk (M ; E, F ) is an elliptic operator acting between sections of Hermitian vector bundles E and F. A deep result known as Bott periodicity reduces the general problem to finding a formula for the index of a generalized Dirac operator as treated in Lecture 3, in fact it would be enough to handle twisted signature operators.

6.6.1

Chern-Weil theory

The connection with topology is through the theory of characteristic classes, which takes a vector bundle with some structure (e.g., an orientation or a complex structure) and defines cohomology classes.

152


Given a rank n R-vector bundle E −→ M and a connection ∇E , recall (from section 2.5) that the curvature of ∇E is a two-form valued in End(E), RE ∈ Ω2 (M ; End(E)). In a local trivialization of E, say by a local frame {sj }, the connection is described by a matrix of one-forms ∇E sa = ωab sb , or in terms of the Christoffel symbols, ωab = Γbia dxi . In this frame we have (RE )ba = dωab − ωcb ∧ ωac , which we can express in matrix notation as RE = dω − ω ∧ ω. Taking the exterior derivative of both sides, we find the ‘Bianchi identity’ dRE = −dω ∧ ω + ω ∧ dω = −RE ∧ ω + ω ∧ RE = [ω, RE ] Now let P : Mn (R) −→ R be an invariant polynomial, i.e., a polynomial in the entries of the matrix such that P (A) = P (T −1 AT ) for all T ∈ GLn (R). Because of this invariance, it makes sense to evaluate P on sections of End(E) and in particular we get P (RE ) ∈ Ωeven (M ). Since differential forms of degree greater than dim M are automatically zero, it makes equally good sense to evaluate any invariant power series on RE . Lemma 46. If P is an invariant power series and RE is the curvature of the connection ∇E , then P (RE ) is a closed differential form and its cohomology class is independent of the connection. Proof. By linearity, it suffices to prove this for P homogeneous, say of order q. For any q matrices, A1 , . . . , Aq , define p(A1 , . . . , Aq ) to be the coefficient of t1 t2 · · · tq in P (t1 A1 + . . . + tq Aq ). Thus p(A1 , . . . , Aq ) is a symmetric multilinear function satisfying p(A, . . . , A) = q!P (A) and moreover the invariance of P implies that p(T −1 A1 T, . . . , T −1 Aq T ) = p(A1 , . . . , Aq ) for all T ∈ GLn (R). For any one-parameter family of matrices T (s) in GLn (R), with T (0) = Id, we have q X 0 = ∂s s=0 p(T −1 A1 T, . . . , T −1 Aq T ) = p(A1 , . . . , [Ai , T 0 (0)], . . . , Aq ) = 0 i=1

6.6. THE ATIYAH-SINGER INDEX THEOREM

153

and note that T 0 (0) is an arbitrary element of Mn (R). We can apply this identity to see that P (RE ) is closed, dP (RE ) = q!dp(RE , . . . , RE ) = q(q!)p(dRE , RE , . . . , RE ) = q(q!)p([ω, RE ], RE , . . . , RE ) = 0. e E is another connection on E. We know that their differNext assume that ∇ ence is an End(E) valued one-form, e E − ∇E = α ∈ Ω1 (M ; End(E)). ∇ E Define ∇E t = ∇ + tα, and note that this is a connection on E with curvature given locally by

RtE = d(ω + tα) − (ω + tα) ∧ (ω + tα) = RE + t(dα − [ω, α]) − 21 t2 [α, α]. A computation similar to that above yields ∂t P (RtE ) = q(q!)dp(α, RtE , . . . , RtE ) and so P (R1E )−P (R0E )

Z = q(q!)

1

dp(α, RtE , . . . , RtE )

Z

dt = d q(q!)

0

1

p(α, RtE , . . . , RtE )

dt .

0

(Another way of thinking about this is that the connections ∇E t can be amalgamated into a connection ∇ on the pull-back of E to [0, 1] × M, then the fact that P (∇) is closed implies that ∂t P (RtE ) is exact.) We will denote by [P (E)] the cohomology class of P (RE ). Recall that the symmetric functions on matrices of size n are functions of the elementary symmetric polynomials τj defined by det(t Id +A) = tn + τ1 (A)tn−1 + . . . + τn−1 (A)t + τn (A), so, e.g., τ1 (A) = Tr(A), and τn (A) = Det(A). Hence the cohomology class [P (E)] is a a polynomial in the cohomology classes [τi (E)] ∈ H 2i (M ). Moreover, notice that if we choose a metric on E and a compatible connection ∇E , then in a local orthonormal frame {sa } for E the matrix (RE )ba will be skewsymmetric and hence τi (RE ) = 0 if i is odd. The non-vanishing classes [τ2i (E)] ∈ H 4i (M )

154


are called the Pontryagin classes of E and they generate all of the characteristic classes of E. If we restrict the class of matrices we consider then there are sometimes more examples of invariant polynomials. For instance, if we assume that E is an oriented bundle then we can assume that RE is valued in skew-symmetric matrices and, if P is a polynomial on skew-symmetric matrices, we can define P (RE ) as long as P (T −1 AT ) = P (A) for all T ∈ SOn (R). This yields one new invariant, the Pfaffian characterized (up to sign) by the equation Pff(A)2 = Det(A). Given a skew-symmetric matrix A = (aij ), let X α= aij dxi ∧ dxj ∈ Ω2 (Rn ) i 0. (One can also prove this by showing that the derivative of the left hand side with respect to t vanishes, and then taking the limit of the left hand side as t → ∞.) They noted that this gives a very non-explicit formula for the index, Z + + − (6.1) ind(ðE ) = Um,E (x) − Um,E (x) dvolg , M

and conjectured that after a ‘fantastic cancellation’, the universal polynomial + − Um,E (x) − Um,E (x)

depends only on the curvatures and on no covariant derivatives. Note that the formula (6.1) already gives interesting information about the index of a Dirac-type operator. For instance, it vanishes in odd dimensions, it is additive for connected sums, and multiplicative for finite covers. The fantastic cancellation, now known as the local index theorem was subsequently shown in a couple of different ways. First Patodi was able to show that the cancellation takes place for a couple of specific operators by a direct computation. Then Gilkey gave a proof, simplified by Atiyah-Bott-Patodi, that first characterizes the form-valued invariants of a Riemannian structure and Hermitian bundle that are ‘natural’ in that they are invariant under pull-back, as precisely the elements of the ring generated by the Pontrjagin forms of T M and the Chern forms of E. The + − invariance properties of Um,E (x) − Um,E (x) are enough to show that it is natural, and then evaluation of some simple examples determines the polynomial explicitly. − + The other approach allows a direct computation Um,E (x) − Um,E (x) using the ‘supersymmetry’ of Clifford algebras. This was discovered by Getzler who showed that after taking the supersymmetry into account, one could use the classical formula of Mehler for the heat kernel of the harmonic oscillator. In one dimension this says that

P1 = −∂x2 + x2 =⇒ −tP1

e

1 1 2 2 exp − (x cosh 2t − 2xy + y cosh 2t) , (x, y) = √ 2 sinh 2t 2π sinh 2t

but this can be generalized to higher dimensional Laplacians acting on vector bundles.

6.7. EXAMPLES

157

Applied to the Dirac operator on a spin manifold, this yields R/4πi 1/2 + − Um,S/ (x) − Um,S/ (x) = Evm det sinh(R/4πi) and hence

Z

+

ind(ð ) =

b M ). A(T M

For a generalized Dirac operator associated to a Z2 -graded Dirac bundle over an even dimensional manifold, recall that we have e +R bE End(E) = Cl(T ∗ M ) ⊗ EndCl(T ∗ M ) (E), RE = 1 cl R 4

bE , the twisting curvature, a section of EndCl(T ∗ M ) (E). Similarly the grading with R γ of E decomposes into cl (Γ) ⊗ γ b with γ b ∈ EndCl(T ∗ M ) (E). The local index theorem in this case is the fact that ! ! e R/4πi + − bE /2π) tr γ b exp(iR . Um,E (x) − Um,E (x) = Evm det1/2 e sinh(R/4πi) E b The differential form tr γ b exp(iR /2π) is known as the relative Chern form of E, it is closed and its cohomology class is independent of the choice of Clifford connection on E and will be denoted Ch0 (E). The index theorem in this case is Z + b M ) Ch0 (E). A(T ind(ðE ) = M

6.7

Examples

For the Gauss-Bonnet operator, recall that we have even ðeven (M ) −→ Ωodd (M ) dR : Ω

and that the index of ðeven dR is equal to the Euler characteristic of M. The index theorem in this case is the Chern-Gauss-Bonnet theorem Z even χ(M ) = ind(ðdR ) = Pff(T M ). M

For the signature operator, recall that we use the Hodge star to split Λ∗C T ∗ M = Λ∗+ T ∗ M ⊕ Λ∗− T ∗ M

158


and then the restriction of the de Rham operator to Λ∗+ T ∗ M is the signature operator ∗ ∗ ∗ ∗ ð+ sign : Λ+ T M −→ Λ− T M.

The index of ð+ sign is the signature of the manifold M. The index theorem in this case is the Hirzebruch signature theorem Z + sign(M ) = ind(ðsign ) = L(M ). M

For the ∂ operator on a Kähler manifold, recall that we have a splitting M Λp,q M Λk T ∗ M ⊗ C = p+q=k

with respect to which d splits into d = ∂ + ∂ : Ωp,q M −→ Ωp+1,q M ⊕ Ωp,q+1 (M ) and then operator ð+ RR =

√

∗

2 (∂ + ∂ ) : Ω0,even M −→ Ω0,odd (M )

is a generalized Dirac operator whose index is the Euler characteristic of the Dolbeault complex. The index theorem in this case is the Grothendieck-Riemann-Roch theorem Z + ind(ðRR ) = Td(T M ⊗ C). M

For a general elliptic pseudodifferential operator P ∈ Ψ∗ (M ; E, F ), choose a connection ∇E with curvature −2πiωE and a connection ∇F with curvature −2πiωF . Define ω(t) = (1 − t)ωE + tσ(P )−1 ωF σ(P ) +

1 t(1 2πi

− t)(σ −1 (P )∇σ(P ))2

and then set 1 f Ch(σ(P )) = − 2πi

Z

1

Tr(σ −1 (P )∇σ(P )eω(t) ) dt ∈ Ωodd (S∗ M )

0

This is the ‘odd Chern form’ defined by the symbol. The Atiyah-Singer index theorem is the formula Z f ind(P ) = Ch(σ(P )) Td(T M ⊗ C). S∗ M

6.8. EXERCISES

6.8

159

Exercises

Exercise 1. Show that zero is an eigenvalue of infinite multiplicity for d : Ω∗ (M ) −→ Ω∗ (M ), that d has no other eigenvalue, and that there is an infinite dimensional space of forms that are not in this eigenspace. Exercise 2. Verify directly that the spectrum of the Laplacian on R is [0, ∞) and that there are no eigenvalues. Exercise 3. Compute the spectrum of the Laplacian on the flat torus R2 /Z2 . Compute the spectrum of the Laplacian on 1-forms on the flat torus. Exercise 4. Compute the spectrum of the sphere S2 with the round metric. Exercise 5. Show that AB and BA have the same spectrum, including multiplicity, except possibly for zero. Exercise 6. Show that Eλ (∆k ) form an acyclic complex. Exercise 7. Use the Fourier transform to show that the integral kernel of the heat operator on Rm is given by |ζ−ζ 0 |2 1 − 4t . e Ke−t∆ (t, ζ, ζ 0 ) = (4πt)m/2 Exercise 8. Use the heat kernel to give an alternate proof that a de Rham class contains a harmonic form.

6.9

Bibliography

A great place to read about spectral theory is the book Perturbation theory for linear operators by Kato, I also like Functional analysis by Rudin. Our construction of the heat kernel is due to Richard Melrose and can be found in Chapter 7 of The Atiyah-Patodi-Singer index theorem. One should also look at Chapter 8 to see how to carry out Getzler rescaling geometrically.

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