Form Preserving Diffeomorphisms on Open ... - Semantic Scholar

Form Preserving Diffeomorphisms on Open Manifolds J¨ urgen Eichhorn1 and Rudolf Schmid2,3 Annals of Global Analysis and Geometry, 14 (1996), 147-176

Contents 1 Introduction

1

2 Diffeomorphism groups on open manifolds

3

3 Form preserving diffeomorphisms Dωr

8

∞,r 4 The geometry of D∞,r and Dω,0 with applications

1

20

Introduction

It is well known that diffeomorphism groups play an important role in certain areas of mathematical physics, such as quantitative fluid dynamics and plasma physics, gauge theories and general Hamiltonian theory. Let (M n , g) be a compact oriented Riemannian manifold and µ the corresponding volume element. Denote by Dr = Dif f r (M ) the diffeomorphism group of M , completed in the r-Sobolev topology, and let Dµr ⊂ Dr be the subgroups of volume-preserving diffeomorphisms. Then Dµr ⊂ Dr is a closed submanifold of Dr and inherits the geometry from Dr . Geodesics on Dµr are roughly speaking the trajectories of Euler’s equations of incompressible fluids, [5]. If (M n , ω) is a symplectic manifold, one considers the subgroup Dωr of symplectic diffeomorphisms. This is again a closed submanifold of Dr . All constructions and known proofs about 1 Fachbereich Mathematik, University of Greifswald, Jahnstrasse 15a, 17487 Greifswald, Germany, e-mail: [email protected] 2 Department of Mathematics, Emory University, Atlanta, Georgia 30322, e-mail: [email protected] 3 Research supported by NSF grant # DMS-9303215 and Emory-Greifswald Exchange Program.

1

diffeomorphism groups essentially use the compactness of M n . For example, this concerns the definition and properties of the completed diffeomorphism group Dr , the Hodge-deRham isomorphism, the nonsingular pairing between H n (M ) and Hn (M ), the closedness of the image of a differential operator with injective symbol and many other facts. For open manifolds, which represent the majority of all manifolds, the techniques used in the compact case don’t work and almost nothing is known concerning diffeomorphism groups. In [7] we gave a canonical construction of Sobolev-manifolds of maps between open manifolds, in particular a construction of diffeomorphism groups. The basic new idea is to use the notion of bounded geometry. We assume that the Riemannian manifolds (M n , g) under consideration have bounded geometry of order k, that mean that M has positive injectivity radius and that the curvature tensor and all its covariant derivatives up to order k are uniformly bounded, i.e the following two conditions (I) and (Bk ) are satisfied: (I) : rinj (M ) = inf rinj (x) > 0 , x∈M

(Bk ) : |∇i R| ≤ Ci , 0 ≤ i ≤ k.

It is well known [11], that there are no topological obstructions against the existence of such metrics for all k, 0 ≤ k ≤ ∞. Our constructions contain the compact case as a special simple case and coincide for compact manifolds with earlier definitions of other authors. The main goal of this paper is to establish the diffeomorphism group approach to fluid dynamics and Hamiltonian mechanics for a very large class of open manifolds including all compact manifolds. The paper is organized as follows. In section 2 we recall some main facts about Sobolev spaces and recall the definition of our diffeomorphism group Dr (M n , g) with bounded geometry of order k, k ≥ r > n2 + 1. In section 3 we study the behavior of differential forms under the action of Dor (Dor the component of the indentity), which serves as preparation for our main Theorem 3.3. Theorem 3.3: Assume (M n , g) open satisfying the conditions (I) and (Bk ), k ≥ m ≥ r > n2 + 1 and the spectral condition inf σess (∆1 |(ker∆1 )⊥ ) > 0. Let ω be a C m -bounded closed q-form with inf x∈M |ω|2x > 0, and consider Dωr = {f ∈ r Dr |f ∗ ω = ω)}. Then the group Dω,o = Dor ∩ Dωr is a C k−r+1 submanifold of r Do . To apply a version of the ω-lemma and several continuity and density arguments, we need the density of Dr+s in Dr . But this is far from being evident, possibly it’s wrong in general. Dr consists of diffeomorphisms f with bounded differentials, |∇i df | ≤ Ci , 0 ≤ i ≤ r − 1, and completed with respect to our uniform structure, i.e. by completion with vector fields of Sobolev order r. It is easy to construct diffeomorphisms f with |∇i df | ≤ Ci , 0 ≤ i ≤ r − 1 but |∇r df | unbounded. Therefore we restrict ourselves to the completion of all diffeomorphisms f with |∇i df | ≤ Ci i = 0, 1, . . . and obtain D∞,r . Then D∞,r+s ⊂ D∞,r 2

(1)

is a dense and bounded inclusion. This is very important studying the geometry of D∞,r . For the components of the identity we even have the equalities Dor = r Do∞,r ; and Do∞,r = Dω,o . Hence Theorem 3.3 remains valid and we immediately get the following corollary: Corollary : Assume (M n , g) with (I) and (B∞ ) . Let ω be a C ∞ -bounded strongly nondegenerate q-form, q = n or q = 2, and assume the spectral con∞ r ∞ r dition above. Set Dω,o = lim←r Dω,o . Then {Dω,o , Dω,o | r > n2 + 1} is an ∞ ILH-Lie group and the Lie algebra of Dω,o consists of divergence free ( q = n), or locally Hamiltonian (q = 2) vector fields X with finite Sobolev norm |X|r for all r. The proof of Theorem 3.3 requires considerable analysis. We allow here volume forms ω = µ which are not coming from Riemannian metrics. Such forms reflect compressible fluids which we include in our considerations. The ∞,r final section is denoted to the geometry of D∞,r , Dω,o and to applications. For these diffeomorphism groups we calculate the Levi-Civita connection for the weak Riemannian metric as well as the curvature tensor and the sectional curvature. Our main application is the generalization of the famous existence and uniqueness theorem for Euler’s fluid equations of Arnold-Ebin-Marsden to arbitrary open manifolds satisfying (I) and (Bk ), k ≥ n2 + 1 and our spectral assumption. This is Corollary 4.6. Several forthcoming papers are devoted to the strong geometry, other diffeomorphism groups and to the Euler equations for compressible fluids.

2

Diffeomorphism groups on open manifolds

Let (M n , g) be an open complete Riemannian manifold. Consider the conditions (I) and (Bk ): (I)

(Bk )

rinj (M ) = inf rinj (x) > 0 x∈M

|∇i R| ≤ Ci , 0 ≤ i ≤ k,

where rinj denotes the injectivity radius, R = Rg the curvature tensor and | . | the pointwise norm. We call the manifold (M n , g) of bounded geometry up to order k if it satisfies (I) and (Bk ). For any open manifold, there are no obstructions against the existence of a metric g with bounded geometry of arbitrary high order (cf. Greene [11]). In the sequel we need some theorems concerning Sobolev spaces on open manifolds. Let (E, h, ∇h ) → M be a Riemannian vector bundle with metric connection ∇h . Then the Levi-Civita connection ∇g and ∇h define connections ∇ in all tensor bundles Trq ⊗E, in particular in Λq T ∗ M ⊗E, where Λq T ∗ M ⊂ T0q . We denote by Ωq (E) or Ω(Trq ⊗ E) ≡ Ω0 (Trq ⊗ E) the space of smooth q-forms or tensor fields with values in E, respectively. For the sake of brevity, we consider 3

only forms with values in E. The other case is quite similar. Let Ωqo (E) denote the subspace of forms with compact support. We define for p ∈ R, 1 ≤ p < ∞, and r a nonnegative integer Ωq,p r (E)

Z X r

q

= {φ ∈ Ω (E)| |φ|p,r :=

|∇

i

φ|px

! p1

d volx (g)

< ∞},

i=0

¯ q,p,r (E) = completion of Ωq,p Ω r (E) with respect to | . |p,r , q,p,r q ˆ Ω (E) = completion of Ωo (E) with respect to | . |p,r and Ωq,p,r (E) = {φ|φ measurable regular distributional form with |φ|p,r < ∞}. Furthermore, we define b,m

Ωq (E) = {φ| φ C m − form and

b,m

|φ| :=

m X

sup |∇i φ|x < ∞}

i=0 x∈M

and b,m

The space

◦

q Ω (E) = completion of Ωo (E) with respect to b,m

b,m

|.|.

Ωq (E) equals to the completion of b q m Ω (E)

with respect to

b,m

= {φ ∈ Ωq (E)| b,m |φ| < ∞}

| . |.

¯ q,p,r (E), Ωq,p,r (E), ˆ q,p,r (E), Ω Proposition 2.1 The spaces Ω q Ω (E) are all Banach spaces and there are inclusions

b,m ˆ q

b,m

ˆ q,p,r (E) ⊆ Ω ¯ q,p,r (E) ⊆ Ωq,p,r (E), Ω b,m ˆ q

⊂

Ω (E) 6=

b,m

Ωq (E).

ˆ q,2,r (E), Ω ¯ q,2,r (E), Ωq,2,r (E) are Hilbert spaces. 2 If p = 2 then Ω ˆ q,p,r (E), Ω ¯ q,p,r (E), Ωq,p,r (E) are different. In general the spaces Ω Proposition 2.2 If (M n , g) satisfies (I) and (Bk ), then ˆ q,p,r (E) = Ω ¯ q,p,r (E) = Ωq,p,r (E), 0 ≤ r ≤ k + 2. Ω We refer to [6] for the proof. 2

4

Ω (E),

Proposition 2.3 Assume (M n , g) open, complete, satisfying (I) and (B0 ). a) If r > np + m, then there are continuous embeddings ˆ q,p,r (E) ,→b,m Ω ˆ q (E), Ω ¯ q,p,r (E) ,→b,m Ωq (E). Ω b) If additionally, (M n , g) satisfies (Bk ), k ≥ r ≥ 1 and r− np > r0 ≥ then 0 0 Ωq,p,r (E) ,→ Ωq,p ,r (E)

n p0

, r > r0 ,

continuously. 2 Proposition 2.4 Assume (M n , g) satisfies (I) and (Bk ), and let (Ei , hi , ∇hi ) → M n be Riemannian vector bundles i = 1, 2, satisfying (Bk (Ei )) (i.e. |(∇h )j Rh | ≤ Cj , 0 ≤ j ≤ k), ri ≤ k, ri ≥ r¯, r1 − pn1 + r2 − pn2 ≥ r¯ − np . Then the pointwise tensor multiplication gives a continuous map Ω0,p1 ,r1 (E1 ) ⊗ Ω0,p2 ,r2 (E) → Ω0,p,¯r (E1 ⊗ E2 ). We refer to [8] for the proof. 2 0 Consider now (M n , g), (N n , h) open, complete satisfying (I) and (Bk ) and f ∈ C ∞ (M, N ). Then the differential df = f∗ = T f is a section of T ∗ M ⊗f ∗ T N . f ∗ T N is endowed with the induced connection f ∗ ∇h . The connections ∇g and f ∗ ∇h induce connections ∇ in all tensor bundles Tsq (M ) ⊗ f ∗ Tvu (N ). Therefore, ∇m df is well defined. Assume m ≤ k. We denote by C ∞,m (M, N ) the set of all f ∈ C ∞ (M, N ) satisfying bm

|df | :=

m−1 X

x∈M

sup |∇i df |x < ∞.

i=0

Let U be a uniformly locally finite cover of M by normal charts. Then ∂ α /∂xα f ν is well defined. A very simple condition for f to be in C ∞,m (M, N ) is given by 0

Proposition 2.5 Assume (M n , g), (N n , h) open, complete and satisfying (I) and (Bk ), f ∈ C ∞ (M, N ), 1 ≤ m ≤ k, all ∂ α /∂xα f ν uniformly bounded, |α| ≤ m. Then f ∈ C ∞,m (M, N ). We refer to [7] for the proof. 2 Let Y ∈ Ω(f ∗ T N ) ≡ C ∞ (f ∗ T N ). Then Yx can be written as (Yf (x) , x) and we define a map gY : M → N by gY (x) := (exp Y )(x) := exp Yx := expf (x) Yf (x) . Then the map gY defines an element of C ∞ (M, N ). More generally we have: 5

Proposition 2.6 Assume m ≤ k and rinj (N ), f ∈ C ∞,m (M, N ). Then,

b,m

|Y | =

Pm

i=0

supx∈M |∇i Y |x < δN
np + 1. Consider f ∈ C ∞,m (M, N ). According to proposition 2.3, for r > np + s Ωp,r (f ∗ T N ) ≡ Ω0,p,r (f ∗ T N ) ,→ b,s

b,s

Ω(f ∗ T N ),

|Y | ≤ D · |Y |p,r ,

(3)

1 p

R Pr

i p where |Y |p,r = ( Set for δ > 0, δ · D ≤ δN < i=0 |∇ Y | dvol) . rinj (N )/2 , 1 < p < ∞ Vδ = {(f, g) ∈ C ∞,m (M, N ) × C ∞,m (M, N )| there exists a Y ∈ Ωpr (f ∗ T N ) such that g = gY = exp Y and |Y |p,r < δ}.

Theorem 2.1 V = {Vδ }0 np + 1. A choice of an orthonormal basis in each Tx M implies that |λ|min (df ), the minimum of the absolute value of the eigenvalues of the Jacobian of f , is well defined. Set Dp,r

= {f ∈ Ωp,r (M, M )|f is injective, subjective, preserves orientation and |λ|min (df ) > 0}. 6

Theorem 2.3 Dp,r is open in Ωp,r (M, M ); in particular, each component is a C k+1−r - Banach manifold , and for p = 2 it is a Hilbert manifold. 2 Theorem 2.4 Assume (M n , g), k, p, r as above. a) Assume f, g ∈ Dp,r , g ∈ comp(idM ) ⊂ Dp,r . Then g ◦ f ∈ Dp,r and g ◦ f ∈ comp(f ). b) Assume f ∈ comp(idM ) ⊂ Dp,r . Then f −1 ∈ comp(idM ) ⊂ Dp,r . c) comp(idM ) is a metrizable topological group. We refer to [7] for the proof. 2 We conclude this overview with the α - and ω - lemma. Theorem 2.5 (α-lemma) Assume r ≤ k, r > np + 1, f ∈ Dp,r . Then the right multiplication αf : Dop,r → Dp,r , αf (g) = g ◦ f , is of class C k+1−r . Theorem 2.6 (ω-lemma) Let k+1−(r+s) > s, f ∈ Dop,r+s ⊂ Dop,r , r > np +1. Then the left multiplication ωf : Dp,r → Dp,r , ωf (g) = f ◦ g is of class C s . The proofs are performed in [10]. 2 Remarks: 1. Dp,r ∩ C ∞,m (M, M ) is a group. This follows from the simple observation that f, g ∈ C ∞,m (M, M ) implies g ◦ f ∈ C ∞,m (M, M ). The latter is a consequence of |d(g ◦f )| ≤ |dg|·|df |, |∇d(g ◦f )| ≤ C ·(|∇dg|·|df |+|dg||∇df |) and so on. 2. Let f ∈ C ∞,r (M, M ) be a diffeomorphism such that the eigenvalues λ of the (diagonalized) Jacobian are bounded away from 0, i.e. satisfying x∈M

inf |λ|min (df ) > 0

(4) −1

is a diffeomorfor a uniformly locally finite cover by normal charts. Then f phism in C ∞,r (M, M ) and X ∈ Ωr (T M ) if and only if f ∗ X ∈ Ωr (f ∗ T M, f ∗ ∇). Moreover, Ωr (T M ) and Ωr (f ∗ T M, f ∗ ∇) are homeomorphic topological vector spaces. To prove this, we have to Rcompare ∇i X and (f ∗ ∇)i f ∗ X and start with i = 0. Then we must show M |Xf (x) |2f (x) dvolx (g) < ∞ if and R R only if M |X|2x dvolx (g) < ∞. For i = 1 we have to compare |∇f∗ e X|2f (x) R R dvolx (g) ≤ supx |df |x · |∇f∗ e/|f∗ e| X|2f(x) dvolx (g) with |∇X|2x dvolx (g). Similar for higher derivatives where b,r |df | and |∇i (parallel translation) | ≤ C for i ≤ k enter into the estimates. The assertion follow from the following Lemma 2.1 Let (M n , g) satisfy (B0 ) and f ∈ C ∞,r (M, M ) be a diffeomorphism satisfying (4) and h be a measurable function. Then Z Z |h(f (x))|2 dvolx (g) < ∞ ⇐⇒ |f (x)|2 dvolx (g) < ∞. (5) Moreover Z Z ν→∞ ν→∞ |hν (f (x))|2 dvolx (g) −→ 0 ⇐⇒ |hν (x)|2 dvolx (g) −→ 0 .

7

(6)

Proof: The if-directions of (5), (6) follow from 0 < C1 ≤ |dvolx (g)|x ≤ C2 and the existence of constants C, D > 0 such that diam(f (K)) ≤ C · diam(K) and D · diam(K) < diam(f (K)) for any compact set K ⊂ M . The other direction follows from the first by applying f −1 , (f −1 )∗ . 2 3. Proving Theorem 2.4, we used in [7] the equivalence of Ωr (T M ) and r Ω (f ∗ T M, f ∗ ∇) as Banach spaces which only holds for f ∈ comp(id). Actually, ν→∞ ν→∞ we only used |Xν |r −→ 0 if and only if |f ∗ Xν |f ∗ ∇,r −→ 0. This holds, as ∞ r we have seen for arbitrary f ∈ C (M, M ) ∩ D . Therefore Dr should be a topological group, not only Dor . 2

3

Form preserving diffeomorphisms Dωr

From now on we assume p = 2, (M n , g) with (I) and (Bk ), k ≥ m ≥ r > n2 + 1 and write Dr ≡ D2,r and Dor shall denote the component of the indentity, Dor = comp(id) ⊂ Dr . Let ω ∈b,m Ωq (M ) be a C m -bounded q-form on M and denote Dωr = {f ∈ Dr |f ∗ ω = ω}. Here we assume r ≤ m ≤ k. Then r Dω,o = Dωr ∩ Dor is a group. Assume additionaly that ω is closed. We want r to show that Dω,o is a good submanifold of Dor . The most important examples are ω = volume form µ or ω = symplectic form on a symplectic manifold. It is quite natural and helpful, in particular for integration theory of Hamiltonian systems, to assume on open symplectic manifolds that ω is adapted to a metric of bounded geometry by means of ω ∈b,m Ω2 (M ), 2 ≤ m ≤ k. This ensures the completeness of Hamiltonian vector fields, the transitivity of the flow and hence the existence of Liouville tori in the case of 12 dimM independent integrals in involution. Let I = [0, 1], it : M → I × M the embedding it (x) = (t, x) and furnish I × M with the product metric 1 00 g. Lemma 3.1 For every q ≥ 0 there exists a linear bounded mapping K :b,m Ωq (I × M ) →

b,m

Ωq (M )

such that dK + Kd = i∗1 − i∗0 . Proof: Since gI×M = 1 00 g, it : M → I × M is an isometric embedding and i∗t is bounded of any order. In particular, i∗t maps from b,m Ωq (X × M ) into b,m Ωq (M ). This follows easily applying the chain rule in ∇(i∗t ω), ∇2 (i∗t ω) ∂ and so on. Denote X0 = ∂t and for φ ∈b,m Ωq+1 (I × M ), φ0 (X1 ..., Xq ) := b,m φ(X0 , X1 , ..., Xq ). Then φ0 ∈ Ωq (X × M ) and b,m |φ0 | ≤ C ·b,m |φ|. We define Z 1 (Kφ)(X1 , . . . Xq ) := i∗t φ0 (X1 , . . . , Xq )dt 0

i.e. Z0 Kφ =

1

i∗t iX0 φ t. 8

R Exchanging integration 10 . . . dt and application of ∇, ∇2 , . . ., we see that K maps between b,m Ωq+1 (I × M ) and b,m Ωq (M ). The equation dK + dK = i∗1 − i∗0 is proved in [13], p. 135-136. 2 Lemma 3.2 Let f, g : M → N be C 1 -mappings and F : I × M → N a C 1 homotopy between f and g i.e. F (0, x) = f (x), F (1, x) = g(x). Let f ∗ , g ∗ : b,1 Ωq (N ) →b,1 Ωq (M ), F ∗ :b,1 Ωq (N ) →b,1 Ωq (I × M ) bounded and φ ∈b,1 Ωq (N ) closed. Then there holds (g ∗ − f ∗ )φ = dψ , for some ψ ∈b,1 Ωq−1 (M ). Proof: According to our assumption and Lemma 3.1, KF ∗ φ ∈b,1 Ωq−1 (M ) and (g ∗ − f ∗ )φ = ((F ◦ i1 )∗ − (F ◦ i0 )∗ )φ = (i∗1 F ∗ − i∗0 F ∗ )φ = (i1 − i0 )∗ F ∗ φ = (dK + Kd)F ∗ φ = d(KF ∗ φ). 2 Now we apply this to our situation M = N, (M n , g) with (I) and (Bk ), r ≤ r m ≤ k, r > n2 + 1 and ω ∈b,m Ωq (M ) closed, f ∈ Dω,0 = Dor ∩ Dωr . In 1 ∗ b,1 q b,1 q particular f ∈ C and f : Ω (M ) → Ω (M ) is bounded. The later two facts follow from the Sobolev embedding theorem. By construction of Dr and by the properties of comp(id) = Dor ⊂ Dr , any f ∈ Dor has a representation f = exp Xn ◦ . . . ◦ exp X1 , Xi+1 ∈ Ωr (exp Xi ◦ . . . ◦ exp X1 )∗ (T M, ∇g ) ∼ = Ωr (T M ) = Ω0,2,r (T M ). Start with the simplest case f = exp X, g = idM , or δ F (t, X) = expx tX, |X|r ≡ |X|2,r ≤ D , δ < rinj (M ), D from (2.2). Then b,1 |X| ≤ D · |X|r ≤ δ < rinj (M ); in particular |X|x , |∇X| < δ for all x ∈ M.

(7)

Theorem 3.1 Under these assumptions (exp X)∗ ω − ω ∈ dΩq−1,r−1 ,

(8)

where we write Ωq,r ≡ Ωq,2,r . Proof: According to Lemma 3.2, (exp X)∗ ω − ω = dK(expx tX)∗ ω. We have to show K(expx tX)∗ ω ∈ Ωq−1,r−1 . For i1 < . . . < iq−1 set I = (i1 , . . . , iq−1 ), eI = (ei1 , . . . , eiq−1 ), where e1 , . . . en is an orthonormal basis in Tx M . Then Z 1 [K(expx tX)∗ ω](eI ) = (i∗t [(expx tX)∗ ω]0 )(eI )dt = 0

=

Z

1

ω((expx tX)∗ 0

∂ , (expx tX)∗ (eI ))dt. ∂t 9

∂ At first we have to calculate (expx tX)∗ ∂t , and (expx tX)∗ ei . Consider ∂ ∂ d (expx tX)∗ ∂t . In [0, 1] × M we have ∂t |(t,x) = c(0) ˙ = dτ c(τ, x)|τ =0 , where ∂ d c(τ, x) = (t + τ, x). Then (expx tX)∗ ∂t = dτ (expx (t + τ )X)|τ =0 = Pt (tX) = t · Pt (X), where Pt (Y ) = parallel transport of Y along expx sX from x to expx tX. Consider now (expx tX)∗ Y, Y ∈ Tx M . Let c(τ ) be a curve representing Y, c(0) = x, c(0) ˙ = Y . Then c(t, τ ) = expc(τ ) tXc(τ ) is a 1-parameter family of geodesics with family parameter τ . This defines Jacobi fields Yτ (t) = ∂ ˙ ), ∂τ c(t, τ ) along each geodesic. Yτ (t) satisfies the initial conditions Yτ (0) = c(τ ∇ ∇ ∂ ∇ ∂ ∇ 0 Y τ (0) = ∂t Yτ (0) = ∂t ∂τ c(t, τ )|t=0 = ∂τ ∂t c(t, τ )|t=0 = ∂τ Xc(τ ) . In pard ticular, dτ expc(τ ) tXc(τ ) |τ =0 = Y0 (t). The condition (Bk ) implies (B0 ) and (B0 ) is equivalent to |κ| ≤ ∆, where κ is the sectional curvature. Then p p |Y0 (t) ≤ |Y0 (0)| cosh ∆ · |X| · t + |Y 0 0 (0)| · sinh ∆ · |X| · t p p = |Y | cosh ∆ · |X| · t + |∇Y X| sinh ∆ · |X| · t ≤ ≤ C1 (|Y | + |∇X| · |Y |) . (9)

For Y = e = unit vector, we obtain Y0 (t) ≤ C1 (1 + |∇X|). Using (7), (10), |tPt (X)| = t|X| and 0 ≤ t ≤ 1, we conclude X |K(exp tX)∗ ω|2 = |(K(exp tX)∗ ω)(eI )|2 ≤

(10)

(11)

I

≤

X

2(q−1)

|X|2 · |ω|2 · C1

(1 + δ)2(q−1) ≤ C · |X|2 .

I

|X|2 is integrable by assumption, hence |K(exp tX)∗ ω|2 is integrable too. Next, we have to study ∇(K exp X)∗ ω. In what follows, application of ∇ has to be understood in the distributional sense. Now X |∇(K(exp tX)∗ ω)2 | = |[∇ei (K(exp tX)∗ ω)](eI )|2 . (12) i,I

Therefore, we have to estimate [∇ei (K(exp tX)∗ ω)](eI ) = =

Z

1

[∇ei (ω(exp tX)∗ 0

=

Z

∂ , (exp tX)∗ · ](eI )dt = dt

1

[(∇ei ω)(tPt (X), (exp X)∗ (eI )) + ω(t∇ei Pt (X), (exp tX)∗ eI )+ 0

+ω(tPt (X), (∇ei (exp tX)∗ )(eI ))]dt. 10

(13)

Therefore |[∇ei (K(exp tX)∗ ω)](eI )| ≤ C · [|∇ω||X| · (1 + δ)q−1 + +|ω| · |∇Pt (X)|(1 + δ)q−1 + |ω||X||∇(exp tX)∗ |].

(14)

The first term of the right hand side of (14) is square integrable. Consider |∇e Pt (X)|. According to equation (4.8) of [7] |∇e Pt (X)| ≤ C1 [|∇X| + |X|2 (|∇ X| + 1) ≡ P1 (|X|, |∇X|). It is shown in [7], p. 289-290, that P1 (|X|, |∇X|) is square integrable. |∇e (exp tX)∗ | can be estimated by a polynomial P01 (|X|, |∇X|, |∇2 X|), where P01 has the following properties: 1. P01 is linear in |∇2 X|. 2. P01 is of second and lower order in |∇X|. 3. It contains no products of |∇X| with |∇2 X|. This has been proved in [7], p.267-268, and lemma 2.13 of [7]. Therefore, we must investigate whether products of the form |X| · |X|s , |X| · |X|s · |∇X|, |X| · |X|t · |∇X|2 , |X| · |X|v |∇2 X| are square integrable. This is clear since ((|X| · |X|s )2 ≤ |X|2 · δ 2 , (|X||X|s · |∇X|)2 ≤ |X|2 δ 2s (1 + δ)2 , (|X| · |X|t · |∇X|2 )2 ≤ |X|2 δ 2t (1 + δ)4 , (|X| · |X|v |∇2 X|)2 ≤ |∇2 X|2 · δ 2(v+1) . Hence, K(expx tX)∗ ω ∈ Ωq−1,1 . The next step would be to estimate X |∇2 K(exp tX)∗ ω|2 = |(∇2 (exp tX)∗ )(ei , ej )(eI )|2 . i,j,I

For an arbitrary tensor T we have (∇2 T )(X, Y ) ≡ ∇2Y X T = ∇Y ∇X T −∇∇Y X T . We generalize this to Lemma 3.3 Let T be a tensor. Then for u (∇u T )(X1 , . . . , Xu ) has a representation

≥

1, ∇uXu ...X1 T

≡

(∇u T )(X1 , . . . , Xu ) = ∇Xu · · · ∇X1 T +lower order iterated derivatives (15) including mixed derivatives of X1 , . . . , Xu−1 . 2 Now we prove by induction. Assume for i ≤ u − 1 ≤ r − 1 ∇i K(exp tX)∗ ω ∈ L2

(16)

(∇ek u · · · ∇ek 1 K(exp tX)∗ ω)(eI )

(17)

and consider

11

By the Leibniz rule, (3.11) splits into a sum of terms each of which can be estimated by C|∇j1 ω| |∇j2 Pt (X)| |∇j3 (exp tX)∗ | . . . |∇jq+1 (exp tX)∗ |,

(18)

where C = Cj1 ...jq+1 and j1 + . . . + jq+1 = u. Assume j1 = 0. The case j1 ≥ 1 easily reduces to our induction assumption. Now we have to consider |∇f Pt (X)| |∇j1 (exp tX)∗ | · · · |∇jq−1 (exp tX)∗ |,

(19)

j + j1 + · · · + jq−1 = u. Lemma 3.4 Assume µ ≤ m. Then |∇µ Pt (X)| ≤ Pµ , where Pµ = µ Pµ (|X|, |∇X|, . . . , |∇ X|) is a polynomial with the following properties: 1. Pµ is linear in |∇µ X|. 2. Pµ does not contain products |∇µ X| · |∇λ X|, λ ≥ 1. 3. The derivatives of order < µ can be arranged as X Ci0 i1 ···iµ−1 |X|i0 |∇X|i1 · · · |∇µ−1 X|iµ−1 (20) i1 +2i2 +···+(µ−1)iµ−1 ≤µ

4. Each term has at least |X| or same |∇j X| as factor. This is proposition 4.7 of [7]. 2 Lemma 3.5 Assume j ≤ m − 1. Then |∇j (exp tX)∗ | ≤ P0j (|X|, |∇X|, . . . , |∇j+1 X|), where P0j is a polynomial with the properties 1-3, replacing µ by j + 1. This is proposition 4.8 of [7]. 2 Now we can estimate the expressions (20). Assume j = 0 in (20). Then |Pt (X)| |∇j1 (exp tX)x | · · · |∇jq−1 (exp tX)∗ | ≤ ≤ C · |X| · P0j1 · · · P0jq−1 .

(21)

We have to show that this expression is square integrable. Clearly, (21) assumes q ≥ 2. The case q = 1 is particularly easy and will be discussed at the end. We assume q ≥ 2, r > n2 + 1 and start with the highest order derivatives |X| |∇j1 +1 X| · · · |∇jq−1 +1 X|.

(22)

X ∈ Ωr (T M ), hence ∇j X ∈ Ωr−j (T M ). We want to apply proposition 2.4 and have to establish n n n n r − + (r − (j + 1) − ) + · · · + (r − (jq−1 + 1) − ) ≥ r¯ − , r¯ ≥ 0. (23) 2 2 2 2 12

The lefthand side of (23) equals to n n ) − (j1 + · · · + jq−1 ) − (q − 1) = q(r − ( + 1)) − u + 1 ≥ (24) 2 2 n n n n q(r − ( + 1)) − (r − 1) + 1 ≥ (q − 1)(r − ( + 1)) + r − ( + 1) − r + 2 > 0 − 2 2 2 2 q(r −

i.e (22) is square integrable. The next step consists in replacing some of the factors of (22) by the monomials (21), i.e. we replace |∇j+1 X| by |X|i0 |∇X|i1 · · · |∇j X|ij , where j is one of the j1 , . . . , jq−1 . First we use |X|i0 ≤ δ i0 . The term |∇j+1 X| generates in (24) the term r − (j + 1) − n2 . This should be replaced by n n n ) + i2 (r − 2 − ) + · · · + ij (r − j − ) = 2 2 2 n n = (i1 + · · · + ij )(r − ) − (i1 + 2i2 + · · · + jij ) ≥ (r − ) − ¯j, 2 2 where i1 + · · · + ij ≥ 1 (if = 0 then we are immediately done) and ¯j = i1 + 2i2 + · · · + jij ≤ j + 1. Hence i1 (r − 1 −

i1 (r − 1 −

n n n n ) + · · · + ij (r − j − ) ≥ r − − (j + 1) , and ≥ − . 2 2 2 2

(25)

(24) and (25) once again guarantee that we obtain a square integrable expression. This procedure of replacing can be performed for any finite subset of {j1 , . . . , jq−1 } and we are done. Consider next the case j ≥ 1 in (20). Then we have to replace |X| by the polynomials Pj and j + j1 + · · · + jq−1 = u. (23) has to be replaced by n n n ) + (r − (j1 + 1) − ) + · · · + (r − (jq−1 + 1) − ) = (26) 2 2 2 n n n = q(r − ) − (j + j1 + · · · + jq−1 ) − (q − 1) = q(r − ( + 1) − u + 1 > 0 − . 2 2 2 Replacing the highest order derivatives by monomials does not affect the square integrability condition, as we have seen already. The P0j can have constant terms, but property 4 of Pj ensures in any case square integrability. Consider the case q = 1. Then in (19) only |∇u Pt (X)| ≤ Pu occurs. In [7], p. 289-290, we proved that Pµ is square integrable. Finally, we have to investigate the lower order derivatives of (15). Those terms without mixed derivatives of the vectors are already settled by induction. We assume the e1 , . . . , en being radially parallel translated and obtain |∇ei ej |x = 0. For higher order derivatives, this is wrong. Writing ei |exp sej = Ps,j (ei |x ), we see immediately that covariant derivatives of the parallel transaction P along expx (sej ) appear. These can be estimated by constants since we assumed (Bk ). Hence, we obtain expressions of the kind (19) where some of the factors are constants and j + j1 + · · · + jq−1 < u. These products (r − j −

13

are already done by assumption (or by the module structure proposition 2.4.). The induction is finished. Since in P0j the terms |∇j+1 X| appear we obtain, assuming X ∈ Ωr (T M ), (exp X)∗ ω − ω ∈ dΩq−1,r−1 . 2 Theorem 3.2 Assume (M n , g) with (I) and (Bk ), k ≥ m ≥ r > ω ∈b,m Ωq be closed and f ∈ Dor (M n , g). Then

n 2

+ 1. Let

f ∗ ω − ω ∈ dΩq−1,r−1 . Proof: From construction of Dr , f ∈ Dor implies f = exp Xu ◦ · · · ◦ exp X1 , Xi+1 ∈ Ωr ((exp Xi ◦ · · · ◦ exp X1 )∗ T M ) ∼ = Ωr (T M ), |Xi |r ≤ δ/D, r < rinj (M ). At first, it is clear that (exp Xi ◦ · · · ◦ exp X1 )∗ ω − ω ∈ db,1 Ωq−1 .

(27)

since F (t, x) = expx tXi ◦ · · · ◦ expx tX1 is a broken C 1 -bounded homotopy and Lemma 3.2 is applicable, 1 ≤ i ≤ u. We have shown (exp X)∗ ω − ω ∈ dΩq−1,r−1 .

(28)

The next step would be to show ( assuming (28)) that (exp X2 ◦ exp X1 )∗ ω − ω ∈ dΩq−1,r−1 .

(29)

We have (exp X2 ◦ exp X1 )∗ ω − ω = (exp X2 ◦ exp X1 )∗ ω − (exp X1 )∗ ω+ +(exp X1 )∗ ω − ω = (exp X1 )∗ ((exp X2 )∗ ω − ω) + (exp X1 )∗ ω − ω. According to (28), (exp X2 )∗ ω − ω, (exp X1 )∗ ω − ω ∈ dΩq−1,r−1 . Therefore, we would be done if we could show that φ ∈ dΩq−1,r−1 and exp X ∈ Dor imply (exp X)∗ φ ∈ dΩq−1,r−1 . Write φ = dψ, ψ ∈ Ωq−1,r ⊂ Ωq−1,r−1 . Then (exp)∗ dψ = d(exp X)∗ ψ. Hence we are done if we can show that ψ ∈ Ωq−1,r−1 , X ∈ Ωr (T M ), |X|r ≤ δ/D, δ < rinj (M ), imply (exp X)∗ ψ ∈ Ωq−1,r−1 . We start proving (exp X)∗ ψ ∈ L2 , i. e. (exp X)∗ ψ ∈ Ωq−1,0 . X |(exp X)∗ ψ|2x = |ψ(exp X)∗ eI |2x . I

14

(30)

As we have already seen, each term of the right hand side can be estimated by |ψ|2x (1 + δ)2(q−1) which is integrable since by assumption |ψ|2x is integrable. Next, we consider ∇((exp X)∗ ψ). X |∇(exp X)∗ ψ|2 = |(∇ei (ψ(exp X)∗ (eI )|2 . i,I

But by the chain and Leibniz rule, |(exp X)x | ≤ C · (1 + δ) and |∇ej (exp X)∗ | ≤ P01 |(∇e (ψ(exp X)∗ ))(eI )| can be estimated by C1 [|∇ψ|(1 + δ)q−1 + |ψ|(1 + δ)q−2 P01 ]. The first term is square integrable since |∇ψ|x is. The constant term and |X|, |∇X| cause no trouble since |X| ≤ δ, |∇X| ≤ δ and |ψ|x is square integrable. As we have already seen, the general case of ∇j ((exp X)∗ ψ) reduces to |∇j0 ψ| |∇j1 (exp X)x | · · · |∇jq−1 (exp X)x |, (31) j0 + j1 + · · · + jq−1 = j. But for q ≥ 1 and the highest order derivatives (r − 1 − j0 −

n n n ) + (r − (j1 + 1) − ) + · · · + (r − (jq−1 + 1) − ) = 2 2 2

n ) − (j0 + j1 + · · · + jq−1 ) − 1 − (q − 1) 2 n n = q(r − ( + 1)) − j ≥ q(r − ( + 1)) − (r − 1) 2 2 n n = (q − 1)(r − ( + 1)) + (r − ( + 1)) − (r − 1) 2 2 n n n = (q − 1)(r − ( + 1)) − ≥ 0 − . 2 2 2 = q(r −

(32)

Hence, the expression (31) (which contains no constant term) is square integrable. If we replace the highest order derivatives of the P0ji in (31) by lower ones we obtain once again square integrable expressions as we have already seen. ¿From our proof, it is clear that ψ ∈ Ωq−1,r−1 and f = exp Xu ◦ · · · ◦ exp X1 ∈ Dor imply f ∗ φ ∈ Ωq−1,r−1 . (33) Assume now (exp Xu ◦ · · · ◦ exp X1 )∗ ω − ω ∈ dΩq−1,r−1 . Then = + = +

(exp Xu+1 ◦ exp Xu ◦ · · · ◦ exp X1 )∗ ω − ω = (exp Xu+1 ◦ · · · ◦ exp X1 )∗ ω − (exp Xu ◦ · · · ◦ exp X1 )∗ ω + (exp Xu · · · exp X1 )∗ ω − ω = (exp Xu ◦ · · · ◦ exp X1 )∗ ((exp Xu+1 )∗ ω − ω) + (exp Xu ◦ · · · ◦ exp X1 )∗ ω − ω. 15

(34)

Then both terms of the right hand side of (34) are in dΩq−1,r−1 and we are done. 2. Corollary 3.1 Let f ∈ Dr ∩C ∞,r (M, M ) and f 0 ∈ comp(f ) , ω ∈b,m Ωq closed, k ≥ m ≥ r > n2 + 1. Then (f 0 )∗ ω − f ∗ ω ∈ dΩq−1,r−1 (f ∗ ∇) .

(35)

Proof: f 0 ∈ comp(f ) has a representation f 0 = exp Xu · · · exp X1 ◦ f,

(36)

Xi ∈ Ωr (T M ), |Xi |r ≤ δ/D < rinj (M ), where the Xi have to be understood as sections of (exp Xi−1 · · · exp X1 ◦ f )∗ T M . Now (exp Xu · · · exp X1 ◦ f )∗ ω − f ∗ ω = = f ∗ [(exp Xu · · · exp X1 )∗ ω − ω]. (exp Xu · · · exp X1 )∗ ω − ω ∈ dΩq−1,r−1 and f ∗ maps dΩq−1,r−1 homeomorphically to dΩq−1,r−1 (f ∗ ∇). 2 Now, we are ready to prepare our main theorem. For this, we must define nondegeneracy of forms on open manifolds. We say ω ∈b,0 Ωq is nondegenerate if for every φ ∈b,0 Ωq−1 there exists a uniquely determined C 0 -vector field X such that iX ω ≡ ω(X, . ) = φ( . ).

(37)

ω is said to be strongly nondegenerate if in addition inf |ω|2x > 0 .

x∈M

(38)

There is only a small choice. (37) means that X establishes for any x ∈ M an isomorphism between Tx M and Λq−1 Tx∗ M . Therefore φ must by a q-form with n components, i.e. an (n−1) or 1-form, q = n or q = 2. Hence, ω must be a volume form µ or symplectic form ω satisfying (38) if we additionally claim the closedness. The generalization in comparison with Ebin-Marsden, [5] consists in allowing arbitrary volume forms, not only such defined by a Riemannian metric. The restriction in comparison with Ebin-Marsden [5] consists in condition (38). Lemma 3.6 Assume k ≥ m ≥ r > n2 + 1 and µ ∈b,m Ωn a volume form with inf x∈M |µ|2x > 0. Then Φ : X → iX µ defines an isomorphism between Ωr−1 (T M ) and Ωq−1,r−1 . Proof: Let x ∈ M , define Xx = ξ 1 e1 + · · · + ξ n en by the equation µ(X, ei1 , . . . , ein ) = ξ 1 µ(e1 , ei1 , . . . , ein ) + · · · + ξ n µ(en , ei1 , · · · , ein−1 ) = 16

= φ(ei1 , · · · , ein−1 ), 1 ≤ i1 < · · · < in−1 ≤ n.

(39)

Choosing the arguments (e2 , . . . , en ), (e1 , e3 , . . . , en ), . . . , (e1 , . . . , en−1 ), we see that (39) is an n × n system with determinant   |µ| 0 . . . 0  0 −|µ| 0 · · · 0   6= 0 . det   0 . |µ| · · · 0  0 . . . . ±|µ| Therefore, X is uniquely defined, continuous and bounded. We have to show X ∈ Ωr−1 (T M ) if and only if φ = iX µ ∈ Ωq−1,r−1 . Assume φ ∈ Ωq−1,r−1 . Fix X x ∈ M and set e1 = |X| , complete this to an orthonormal basis e1 , . . . en and translate e2 , . . . , en radially. Then |φ|2x = |µ(X, e2 , . . . , en )|2x = |X|2x |µ(e1 , . . . , en )|2x , i.e. |X|2x =

|φ|2x |φ|2x ≤ . |µ|2x inf x∈M |µ|2x

(40)

|φ|2x is by assumption integrable, hence |X|2x is integrable too. Consider now ∇φ = ei ⊗ ∇ei φ. X |∇φ|2 = |(∇ei φ)(ei1 . . . , ein−1 )|2 . (41) i,i1 n2 + 1 and ω ∈b,m Ω2 a symplectic form with inf x∈M |ω|2x > 0. Then X → 7 iX ω establishes an isomorphism between Ωr−1 (T M ) and Ω1,r−1 . Let ∆q be the Laplace operator acting an q-forms, σe (∆q ) its essential spectrum and σe (∆q |(ker∆q )⊥ ) the essential spectrum of ∆q restricted to the orthogonal complement of its kernel, and inf σe (∆q |(ker∆q )⊥ ) its g.l.b. Now we can state our first main theorem. Theorem 3.3 Assume (M n , g) with (I) and (Bk ), k ≥ m ≥ r > n2 + 1 and ω ∈b,m Ωq , q = n or q = 2, a closed strongly nondegenerate form with r inf σe (∆1 |(ker∆1 )⊥ ) > 0. Then the group Dω,o = Dor ∩ Dωr is a C k−r+1 submanr ifold of Do . Proof: Consider the map ψ : Dor → [ω]r−2 := [ω + dΩq−1,r−1 ], ψ(f ) := f ∗ w. According to our spectral assumption we have dΩq−1,r−1 = dΩq−1,r−1 , where we have taken the closure in Ωq,r−2 , q = 2 or n. Hence, the affine space [ω]r−2 is a smooth Hilbert-manifold. We conclude from Theorem 3.2 that ψ is well defined. Dor has differentiability class C k−r+1 . A straightforward calculation shows that ψ has differentiability class C k−r+1 . The map ψ∗,id can easily be calculated (exp tX)∗ ω − ω = LX ω = diX ω + iX dω = diX ω , t→0 t

ψ∗,id (X) = lim

18

X ∈ Tid Dor = Ωr (T M ). We conclude from Lemma 3.6, 3.7 that X 7→ iX ω maps into Ωq−1,r and conclude once again from our spectral assumption the closedness of dΩq−1,r in Ωq,r−2 , i.e. the map ψ∗,id : Tid Dor → dΩq−1,r−1 = Tω [ω]r−2 is surjective. A simple shifting argument using Tf Dor = Ωr (f ∗ T M, f ∗ ∇) and the fact that Ωr (T M ) is mapped homeomorphically to Ωr (f ∗ T M, f ∗ ∇) by means of f and ψ∗,f (X) = f ∗ (LX◦f −1 ω) shows that ψ∗ is surjective everywhere, i. e. r ψ is a submersion. Hence Dω,0 = ψ −1 (0) is a C k−r+1 -submanifold of D0r . 2 ∞,m We defined for C T (M, N ) an uniform structure U p,r . Consider ∞,m now C ∞,∞ (M, N ) = C (M, N ). Then we have an inclusion i : m C ∞,∞ (M, N ) → C ∞,m (M, N ) and i × i : C ∞,∞ (M, N ) × C ∞,∞ (M, N ) → C ∞,m (M, N ) × C ∞,m (M, N ) hence a well defined uniform structure U ∞,p,r = (i × i)−1 U p,r (cf. [17] p. 108-109). After completion we obtain once again the manifolds of mappings Ω∞,p,r (M, N ), where f ∈ Ω∞,p,r (M, N ) if and only if for every ε > 0 there exists an f˜ ∈ C ∞,∞ (M, N ) and an Y ∈ Ωp,r (f˜∗ T ∗ N ) such that f = exp Y and |Y |p,r ≤ ε. Moreover, each connected component of Ω∞,p,r (M, N ) is a Banach manifold and Tf Ω∞,p,r (M, N ) = Ωp,r (f ∗ T N ). As above we set D∞,p,r (M ) = {f ∈ Ω∞,p,r (M, M )|f is injective, subjective,

(44)

preserves orientation and |λ|min (df ) > 0}. We assume p = 2 and write Ω∞,r (M, N ) ≡ Ω∞,2,r (M, N ) and D∞,r (M ) ≡ D∞,2,r (M ). The only difference between our former construction is the fact that the spaces Ω∞,r and D∞,r are based on maps which are bounded up to arbitrary high order. For compact manifolds we have C ∞ (M, N ) = C ∞,r (M, N ) = C ∞,∞ (M, N ) , Ω∞,r (M, N ) = Ωr (M, N ) and D∞,r (M ) = Dr (M ) for all r. For open manifolds we have strong inclusions C ∞,∞ ⊂ C ∞,r and D∞,∞ ⊂ Dr . It is very easy to construct a diffeomorphism f ∈ C ∞,1 (R, R) such that f 6∈ C ∞,2 (R, R). This supports the conjecture that the inclusion Dr+s ,→ Dr , s ≥ 1 is not dense. It is very probable that the density fails as a careful analysis of our example shows. We settle this question in a forthcoming paper. Here we restrict ourselves to D∞,r . This space has the advantage that D∞,r+s is densely and continuously embedded into D∞,r . This follows easily from the corresponding properties for Sobolev spaces. Quite analogously we define the space Dω∞,r . The components of the identity have special nice properties: Lemma 3.8 Assume the condition for defining Dr and Dωr . Then ∞,r r Do∞,r = Dor , Dω,o = Dω,o .

Proof: We prove the first assertion. Let f ∈ Dor . Given any δ < rinj /D, there exists vector fields X1 , . . . , Xm ∈ Ωr (T M ), |Xµ |r < δ, µ = 1, . . . , m, f = exp Xm ◦ · · · ◦ exp X1 , b,1 |X| ≤ D|X|r . We are done if we can show that for X ∈ Ωr (T M ), |X|r < δ and given ε > 0 there exists a diffeomorphism fX ∈ ∗ C ∞,∞ and Y ∈ Ωr (fX T M ) = Ωr (T M ) with |Y |r < ε such that exp X = 19

exp Y ≡ expfX Y ◦ fX . But this is very easy. For ε1 arbitrary small, there exists a smooth vector field Y1 ∈ Co∞ (T M ) with compact support such that |X − Y1 | < ε1 . Choosing ε1 sufficiently small, there exists a unique vector field Y ∈ Ωr ((exp Y1 )∗ T M ) such that exp Y ≡ expexp Y1 Y ◦ exp Y1 = exp X and |Y |r ≤ Qr (ε1 ), where Qr is a poynomial with constant term. This follows from the closing geodesic triangle argument of [7]. Hence, for ε1 sufficiently small we have |Y |r < ε. We set fX = exp Y1 . For f = exp Xm ◦ · · · ◦ exp X1 we apply the techniques of the proof for Dor being a group of [7] and obtain for any given small ε > 0 a representation f = expf˜ Y ◦ f˜ with f˜ ∈ C ∞,∞ , Y ∈ Ωr (f˜∗ T M ), |Y |r < ε and f˜ is built up from the fXµ ∈ C ∞,∞ . r r The same arguments hold for Dω,o by choosing the vector fields in Tid Dω,o . 2 Remarks: 1. A detailed proof of the lemma 3.8 would occupy dozens of pages but the arguments needed are all in [7]. 2. The essential reason for the special r good property of Dor and Dω,o is that id ∈ C ∞,∞ (M, M ). For some diffeomorphisms in other components this is in general wrong. r Therefore Theorem 3.3 remains valid if we replace Dor by Do∞,r and Dω,o by ∞,r Dω,o , but the density properties imply: Corollary 3.2 Assume (M n , g) with (I), (B∞ ) and ω ∈b,m Ωq for all m, closed and strongly nondegenerate, q = n or q = 2 with inf σe (∆1 |(ker∆1 )⊥ ) > 0 . Let ←r

∞ r ∞ r Dω,o =lim Dω,o . Then {Dω,0 , Dω,0 | r > n2 + 1} is an ILH-Lie - group in the ∞ sense of [16] and the Lie algebra of Dω,0 consists of divergence free (q = n) or locally Hamiltonian (q = 2) vector fields X, respectively, with |X|r < ∞ for all r. 2

4

∞,r The geometry of D∞,r and Dω,0 with applications

Most of the interesting diffeomorphism groups are endowed with a natural Riemannian metric which is usually a weak one. For further applications, it is useful and important to know the corresponding Riemannian geometry, i.e. the curvature and the geodesics. In this section, we study the general diffeomorphism groups D∞,r ⊂ Dr and the subgroups of form preserving diffeomorphisms Dω∞,r ⊂ Dωr . Later on, we restrict ourselves to the component of the indentity. There are several papers where this geometry already has been studied. We mention [5], [4], [14], [18], [19] and [20]. But in all of these papers, only the case of compact manifolds M n has been studied. Moreover, serious analytical problems arising in this investigation mostly have been suppressed. Completions have not been considered. In the noncompact case, one has to solve these difficulties. Without that, everything becomes wrong. If we start with Dr , we would have to consider Dr+s with k + 1 − (r + s) > s, s ≥ 2, r > n2 + 1 in order 20

to apply some version of the ω-lemma. Finally, one obtains curvature formulas for tangent vectors tangent to Dr at f ∈ Dr+s . Then one would like to extend these formulas to all f ∈ Dr . But this is impossible since Dr+s ⊂ Dr is probably not dense. Therefore, one has, at least at the final stage, to restrict everything to D∞,r+s , D∞,r . Then we have D∞,r+s ⊂ Dr densely. Moreover, in this case {D∞,∞ , D∞,r , r > n2 + 1} is an ILH-group if k = ∞. In our considerations, we consider a finite tower of this ILH-group. As a matter of fact, the final formulas e.g. for sectional curvature coincide with those of the compact case, as it should be. We present here a rapid, very short presentation. All details would exceed the framework of such a paper. Calculations which are parallel to the compact case are completely suppressed. Many of them are contained in [4] and [20]. We start by defining the Levi-Civita connection for Dr . Later on, we restrict everything to D∞,r . Let M n be a manifold. Then a connection is given by a field of horizontal subspaces of T T M or a covariant derivative ∇ or a connector map K : T T M → T M , respectively. They are related by ∇X Y = K(Y∗ (X)) and K = projection onto the vertical subspaces along the horizontal ones. Locally, K can be expressed explicitly by the Christoffel symbols Γkij (cf. 2.4 of [12]). A connector K can be independently characterized by the following properties. K is a map T 2 M → T M which satisfies the following conditions. For all X ∈ T M, K : TX (T M ) → Tπ(X) M is linear.

(45)

For all x ∈ M, K : T (Tx M ) → Tx M is linear.

(46)

For all X ∈ T M, HX := ker K|TX T M is horizontal.

(47)

Assume now (M n , g) open with (I) and (Bk ), k > n2 + 2. Let ∇ denote the Levi-Civita connection with associated connector map K. The tangent manifold T M can be endowed with a canonical metric, the so called Sasaki metric given by: gT M (X, Y ) = gM (π∗ X, π∗ Y ) + gM (KX, KY ), X, Y ∈ TZ T M. Lemma 4.1 The Sasaki metric gT M satisfies (I) and (Bk−1 ). Proof . (Bk−1 ) follows immediately from [1], p.130, equation (2). We proved in [9] (I) for principal fiber bundles P (M, G) with respect to the Kaluza-Klein metric gω (X, Y ) = gM (π∗ X, π∗ Y ) + hω(X), ω(Y )ig , gM with (I) and (Bk ), ω with (Bk ). A similar proof can also be performed in our case here. 2 Corollary 4.1 For k − 2 ≥ r >

n 2

Ωr (M, T M )

+1 and

21

Ωr (T M, T 2 M ),

Ω∞,r (M, T M )

and

Ω∞,r (M, T 2 M )

are well defined. 2 We add two simple remarks concerning the geometry of T M with respect to the Sasaki metric. The horizontal lifts of geodesics of M coincide with the horizontal geodesics of T M . Vertical geodesics in Tx M are straight lines in the euclidean vector space (Tx M, gM,x ). For the sequel, we recall a fundamental lemma from [7]. δ Let f ∈ C ∞,r (M, N ) , g1 = gY1 = exp Y1 , Y1 ∈ Ωr (f ∗ T N ) , |Y1 |r < D , n r ∗ δ < rinj (N )/4 , g2 = exp Y2 , Y2 ∈ Ω (g1 T N ) , |Y2 |r < δ/D , k ≥ r > 2 + 1, where D comes from (2.2). Since Ωr (f ∗ T N ), Ωr (g1∗ T N ) and b,1 Ω(f ∗ T N ), b,1 Ω(g1∗ T N ) are equivalent Banach spaces (cf. p. 284 and 292 of [7],) there exists such a common D. Then there exists a unique Z ∈ Ωr (f ∗ T N ) such that exp Z = exp Y2 . Lemma 4.2 There exists a polynomial Qr without constant term such that |Z|r ≤ Qr (|∇i Y1 |L2 , |∇j Y2 |L2 )

(48)

0 ≤ i, j ≤ r. This is equation (5.13) of[7]. 2 Corollary 4.2 Assume |Y1 |r , |Y2 |r < 1. Then there exists a constant C1 > 0 such that |Z|r ≤ C1 (|Y1 |r + |Y2 |r ).2 (49) Remark: One has to overcome a small difficulty in defining Ωr (g ∗ T N ) since g is only C 1 but not smooth. This causes no serious problems. We discuss it in a greater context on differential operators with Sobolev coefficients elsewhere. 2 Proposition 4.1 Assume k − 1 ≥ r > n2 + 1. Then there exists a C k−1−r+1 = C k−r embedding φ : T Ωr (M, M ) → Ωr (M, T M ). (50) Proof. Let Xf ∈ Tf Ωr (M, M ) = Ωr (f ∗ T M ). Xf can be written as Xf (x) = (Xf (x) , x) and we define φ(Xf )(x) := Xf (x) . We have to show the following. ˜ ∈ C ∞,r (M, T M ) and a Z ∈ Ωr ((X) ˜ ∗T 2M ) Given any ε > 0, there exists an X with |Z|r < ε such that φ(Xf ) = exp Z. (51) Given ε > 0, there exists f˜ ∈ C ∞,r (M, M ) and Y ∈ Ωr (f ∗ M ) such that f = exp Y and |Y |r < ε. Consider the geodesic segment from (exp Y )(x) to f˜(x) in M which is given by exp t(−P Y ) where P Y is the parallel translation of Y from 22

f˜(x) to (exp Y )(x). Let P1−t Xf be the parallel translation of Xf (x) from f (x) to f˜(x) along this geodesic segment, 0 ≤ t ≤ 1, x ∈ M, P1 Xf = Xf . Then P0 Xf is a vector field along f˜. Denote by P˜t (P0 Xf ) the parallel translation of P0 Xf from f˜(x) to f (x) = (exp Y )(x). P˜t is inverse to P1−t and {Pt (P0 Xf )}0≤t≤1 is a geodesic in (T M, gT M ) covering (exp tY )(x), 0 ≤ t ≤ 1. Its initial point is P0 Xf and its initial vector Y2 is π −1 (Y )∩ {horizontal subspace of P0 Xf }. Hence, we obtain a vector field Y2 along the map M → T M, x 7→ (P0 Xf )(x), π(P0 Xf ) = f˜. The vector field P0 Xf along f˜ can be approximated arbitrary good in the | . |r g norm by a smooth bounded one Pg 0 Xf , |P0 Xf − P0 Xf |r < ε. Choosing ε small enough, there exists a unique vector field Y1 ⊂ T T M along Pg 0 Xf such that exp Y1 = P0 Xf , i. e. we have Pg 0 Xf

exp Y1

→

P0 Xf ,

P0 X f

exp Y2

→

Xf .

It follows from the (highly nontrivial) estimates performed in [7] that |Y1 |r < C1 · ε,

|Y2 |r < C2 · ε.

Choosing ε small enough, there exists a unique vector field Z along Pg 0 Xf such that exp Z = Xf . (52) Letting ε → 0 implies |Z|r → 0, according to (49) above. (52) is the desired representation for Xf . Each component of T Ωr (M, M ) is a C k−1−r+1 = C k−r Hilbert manifold. The proof of the C k−r property of φ follows from considering charts. It is rather technical, but quite canonical. 2 Iterating the procedure, we obtain Corollary 4.3 Assume k − 2 ≥ r > C k−1−r -embedding

n 2

+ 1. Then there exists a canonical

ψ : T 2 Ωr (M, M ) → Ωr (M, T 2 M ).2

(53)

Remark: φ is not surjective. Let (M n , g) = (Rn , gRn ), T Rn = Rn × Rn and e be the section x ∈ Rn 7→ (x, (1, 0, . . . , 0)) ∈ Tx Rn . Then e ∈ C ∞,r (Rn , T Rn ), for arbitrary r, in particular e ∈ Ωr (Rn , T Rn ). If e ∈ im φ then e should be a Sobolev vector field covering idRn . This is impossible since e is not even square integrable. Proposition 4.2 T Dr is C k−r -diffeomorphic to {Xf ∈ im φ ⊂ Ωr (M, T M )|Xf covers f ∈ Dr (M )}.

(54)

T 2 Dr is C k−r−1 -diffeomorphic to {V ∈ im ψ ⊂ Ωr (M, T 2 M )| V covers Xf f rom (54)}. 23

(55)

Proof. The assertion follows immediately from the openness of Dr ⊂ Ωr (M, M ), T Dr ⊂ T Ωr (M, M ), T 2 Dr ⊂ T 2 Ωr (M, M ). 2 Remark: All assertions above remain true if we replace Ωr (M, N ) by Ω∞,r (M, N ), and Dr (M ) by D∞,r (M ). 2 In the sequel, we identify T Dr or T 2 Dr with their corresponding images. Now we would like to define for V ∈ T 2 Dr ⊂ Ωr (M, T 2 M ) ¯ ) := K ◦ V K(V

(56)

¯ by and a covariant derivative ∇ ¯ X¯ Y¯ := K( ¯ Y¯x (X)). ¯ ∇

(57)

¯ is defined by left multiplicaBut, there arise several serious difficulties. K ¯ we should have an tion by K. To obtain a certain differentiability class of K, ω-lemma. In the compact case, all considered manifolds are Hilbert manifolds of class C ∞ , an α - and ω - lemma are very easily established. Ωr (M, M ) is of class C k−r+1 , Ωr (M, T M ) of class C k−1−r+1 = C k−r , and Ωr (M, T 2 M ) of class C k−r−1 . The same holds for Ω∞,r (M, M ), Ω∞,r (M, T M ), and Ω∞,r (M, T 2 M ), assuming k − 2 ≥ r > n2 + 1. Therefore, considering the C s -property of ωK , ωK (V ) = K ◦ V , this does not make sense if k − r − 1 < s. It does make sense for k − 1 − (r + s) > s ≤ 1. ¯ ) := K ◦ Proposition 4.3 Assume k−1−(r+s) > s ≥ 1, r > n2 +1. Then K(V V = ωK (V ) is a C s -map ωK : T 2 Dr+s → T Dr or ωK : T 2 D∞,r+s → T D∞,r , respectively. Proof. The proof consists of two steps. First, one shows that K ◦ V is an element of T Dr and secondly that the map V 7→ K ◦ V = ωK (V ) is of class C s . For the first step we must show that there exists an Xg ∈ T Dr such that (K ◦ V )(x) = Xg(x) . There are several possibilities to do that. Now V ∈ T 2 Dr+s covers an Yg ∈ T Dr+s , g ∈ Dr+s , i. e. V (x) ∈ TYg(x) T M . Since k − 1 − (r + s) > s ≥ 1, k − 1 > r + 2s, and K has bounded derivatives at least up to order r + s we have KV ∈ Ωr+s (g ∗ T M ). Now approximate g by g˜ ∈ C ∞,r+s (M, M ) ∩ Dr+s (M ), translate KV parallel to g˜, thus obtaining P KV , approximate finally P KV by a smooth Pg KV ∈ Ωr+s (˜ g ∗ T M ). The usual ∗ r+s g 2 triangle argument produces a Z ∈ Ω (P KV T M ) such that exp Z = KV , and Xg = exp Z, i.e. KV is well defined in our category. The C s property of ωK can be proved quite analogously as in [10], theorem 3.4.b. The properties ¯ follow immediately from the corresponding properties of K. 2 (45) - (47) for K Corollary 4.4 If k = ∞ then Dr , T Dr , and T 2 Dr are smooth manifolds and ¯ is an everywhere defined smooth connection map. K This follows from proposition (4.3) and the fact that the Christoffel symbols and hence K are bounded of arbitrarily high order. 2 24

¯ ∈ C 2 (T Dr+s ) be a C 2 -vector field. We say X ¯ is right invariant if Now let X ¯ ) = X(id) ¯ X(f ◦f ¯ ¯ ) = X ◦ f. for all f ∈ Dr+s . Defining X := X(id), we can write X(f Lemma 4.3 Let Xf ∈ Tf Dr+s = Ωr+s (f ∗ M ). Then Xf ◦ f −1 is a vector field along idM and Xf ◦ f −1 ∈ Ωr+s (T M ). In particular, Xf ◦ f −1 is a C 2 -vector field if r + s > n2 + 2. Proof. The proof is quite analogous to that of lemma (2.1). There we assumed f to be smooth and bounded. In our case, f is not smooth, but according to the definition of Dr+s , f = exp Y , with Y ∈ Ωr+s (f˜∗ T M ), and f˜ ∈ C ∞,r+s (M, M )∩ Dr+s , such that |Y |r+s < ε, and the existence of the constants C, D remains valid. 2 ¯ Y¯ right invariant C 2 -vector fields on Dr+s . Then Lemma 4.4 Let X, ¯ Y¯ ](f ) = [X, Y ] ◦ f, [X,

(58)

¯ X¯ Y¯ )(f ) = ∇X Y ◦ f, (∇

(59)

¯ X¯ Y¯ − ∇ ¯ Y¯ X ¯ = [X, ¯ Y¯ ]. ∇

(60)

These are the lemmas 11.8 - 11.10 of [4] and their proof is quite formal. 2 ¯ ) = 0 follows (∇ ¯ A¯ B)(f ¯ )=0 We remark that for A¯ ∈ C 2 (T Dr+s ) with A(f ¯ B¯ A(f ¯ ) = [B, ¯ A](f ¯ ). ¯ ∈ C 2 (T Dr+s ). In particular, ∇ for all B ¯ Y¯ ∈ C 2 (T Dr+s ), r + s > Lemma 4.5 Let X,

n 2

+ 2. Then

¯ X¯ Y¯ − ∇ ¯ Y¯ X)(f ¯ ) = [X, ¯ Y¯ ](f ). (∇

(61)

¯ r , Y¯ r be the right invariant vector fields with Proof. Let f ∈ Dr+s and X r r ¯ ¯ ¯ ¯ ¯ Y¯ ] = [X ¯− X (f ) = X(f ), Y (f ) = Y (f ). Then, using the remark above, [X, r ¯ r ¯ r r ¯r r r ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X , Y ] + [X , Y − Y ] + [X , Y ] , but (X − X )(f ) = 0 = (Y − Y )(f ) and ¯ −X ¯ r , Y¯ ](f ) = −∇ ¯ Y¯ (X ¯ −X ¯ r ))(f ) , [X ¯ r , Y¯ − Y¯ r ](f ) = ∇ ¯ X¯ r (Y¯ − Y¯ r ))(f ) , and [X r r ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Y¯ r X ¯ r )(f ) [X, Y ](f ) = (∇X¯ r (Y − Y ))(f ) −∇Y¯ (X − X ))(f ) +(∇X¯ r Y¯ r )(f ) −(∇ . Moreover, according to the R-bilinearity of a connection ¯ Y¯ ](f ) = ∇ ¯ Y¯ X(f ¯ )+∇ ¯ X¯ r Y¯ (f ) + ∇ ¯ Y¯ X ¯ r (f ) − ∇ ¯ X¯ r Y¯ r (f ). [X,

(62)

¯ r − X)(f ¯ ) = 0 and (Y¯ r − Y¯ )(f ) = 0 imply (X ¯ X¯ r Y¯ (f ) = ∇ ¯ X¯ Y¯ (f ) ∇

(63)

¯ Y¯ r X ¯ r (f ) = ∇ ¯ Y¯ X ¯ r (f ), ∇

(64)

and

25

respectively. We infer from (62) - (64) ¯ Y¯ ](f ) = (∇X¯ Y¯ − ∇Y¯ X)(f ¯ ). 2 [X, Now, we define a weak Riemannian metric on Dr+s , D∞,r+s and establish ¯ is the corresponding Levi-Civita connection. Set for V, W ∈ Tf Dr+s = that ∇ r+s ∗ Ω (f T M ) Z hV, W i = g0 (V, W ) = (Vf (x) , Wf (x) )f (x) dvolx (g) = M

=

Z

(V, W ) ◦ f (x) dvolx (g) .

(65)

M

The integral (65) converges according to the definition of Tf Dr+s . Using charts in Dr+s , it is easy to see that g0 is a weak C s metric. Remember k −(r +s)−1 > s. Remark: g0 only gives the L2 - topology of each tangent space. To obtain the actual, i. e. the Sobolev topology, we should work with the corresponding strong Riemannian metric. A forthcoming paper is devoted to this task. 2 ¯ Y¯ , Z¯ be right invariant C 1 vector fields on Dr+s . Then Lemma 4.6 Let X, ¯ Z¯ X, ¯ Y¯ ) + g0 (∇ ¯ Z¯ Y¯ , X) ¯ = Zg ¯ 0 (X, ¯ Y¯ ). g0 (∇

(66)

This is lemma 11.12 from [4]. 2 ¯ Y¯ , Z¯ ∈ C 2 (T Dr+s ), f ∈ Dr+s , r > Lemma 4.7 Let X,

n 2

+ 1, s ≥ 2. Then

¯ Z¯ X, ¯ Y¯ )(f ) + g0 (∇ ¯ Z¯ Y¯ , X)(f ¯ ) = Zg ¯ 0 (X, ¯ Y¯ )(f ). g0 (∇

(67)

The same holds for D∞,r+s , f ∈ D∞,r+s . ¯ )◦ Proof. In [4], equation (67) was proved under the assumption that X(f ¯ ) ◦ f −1 ∈ C ∞ (T M ). What was really needed there was f −1 , Y¯ (f ) ◦ f −1 , Z(f ¯ ) ◦ f −1 , Y¯ (f ) ◦ f −1 , Z(f ¯ ) ◦ f −1 ∈ C 1 (T M ). But, we conclude from lemma X(f −1 ¯ ¯ ¯ ) ◦ f −1 ∈ C 2 (T M ). Hence, the proof of 4.3 above X(f ) ◦ f , Y (f ) ◦ f −1 , Z(f [4] carries over to our case. It essentially uses lemma 4.6. 2 ¯ is the Levi-Civita connection for C s vector fields on Corollary 4.5 ∇ n ∞,r D , r > 2 + 1. ¯ is defined for C s vector fields on D∞,r+s , r > n + 1, s ≥ 2, k − (r + Proof. ∇ 2 s)−1 > s. Equation (67) shows that under these conditions it is the Levi- Civita connection for such vector fields and g0 restricted to D∞,r+s . But D∞,r+s is dense in D∞,r and we can extend (67) to D∞,r . 2 Remark: This is the step where we must go from Dr to D∞,r . Dr+s is very probably not dense in Dr and we cannot conclude as in corollary (4.5). 2 26

Proposition 4.4 Let V¯0 ∈ T D∞,r , γ¯V¯0 be the geodesic in D∞,r with initial condition V¯0 . Then γ¯V¯0 : I → D∞,r is given by the map t 7→ (x 7→ γV¯0 (x) (t)). where γV¯0 (x) is the geodesic in M with initial condition V¯0 (x) ∈ T M . ¯ γ¯˙ γ¯˙ V¯ )(t)(x) = K( ¯ γ¨¯ V¯ (t))(x) = K ◦ γ¨¯ V¯ (t)(x) = K(γ¨¯ V¯ (x) (t)) = 0, 2 Proof. (∇ ¯ 0 0 0 0 V 0 ∞,r Hence, a curve γ(t) in D is a geodesic in D∞,r if and only if the associated curve γ(t)(x) is a geodesic in M for all x ∈ M . Finally, we study the curvature of D∞,r . Let µ be the volume element of n (M , g). Define for any f ∈ D∞,r a function ρ : M → R+ by ((f −1 )∗ µ)(x) = ρ(x)µ(x). Then with µ = dvolx g Z g0 (Vf , Wf ) = (Vf (x) , Wf (x) )f (x) dvolx (g) = Z = (Vf ◦ f −1 , Wf ◦ f −1 )x ρ(x) dvolx (g). We obtain a very simple, suggestive formula for the sectional curvature. Let ¯ Y¯ be right invariant vector fields of class C 2 , and X = X(id), ¯ X, Y = Y¯ (id). Then, as we have already seen ¯ X¯ Y¯ )(f ) = (∇X Y ) ◦ f. (∇ ¯ This implies for the curvature tensor R ¯ X(f ¯ ), Y¯ (f ))Z(f ¯ ) = R(X, Y )Z ◦ f. R( ¯ ), Y¯ (f ) be linearly independent and Let X(f ¯ ) ∧ Y¯ (f )|2 = g0 (X(f ¯ ), X(f ¯ )) · g0 (Y¯ (f ), Y¯ (f )) − g0 (X(f ¯ ), Y¯ (f ))2 . |X(f ¯ ), Y¯ (f ), Xx = X(f ¯ ) ◦ f −1 |x , Yx = Y¯ (f ) ◦ f −1 |x Then with σ = linear hull of X(f ¯ K(σ)

¯ ) ∧ Y¯ (f )|−2 g0 (R(X(f ¯ ), Y¯ (f ))Y¯ (f ), X(f ¯ )) = = |X(f Z ¯ ) ∧ Y¯ (f )|−2 = |X(f K(Xx , Yx )ρ(x)dvolx (g),

(68)

where K(Xx , Yx ) = g(R(Xx , Yx )Yx , Xx ). As well known, the sectional curva¯ is a tensor, i.e. R ¯ depends only on X(f ¯ ), Y¯ (f ) ture depends only on σ and R and not on the whole vector field. Hence (68) also makes sense for arbitrary vectors Xf , Yf ∈ Tf D∞,r with Xx = Xf ◦ f −1 |x , Yx = Yf ◦ f −1 |x . We obtain immediately from (68). Proposition 4.5 If the sectional curvature of (M n , g) is always ≥ 0 or always ≤ 0, respectively, then the same holds for the sectional curvature of D∞,r . 2 27

Remarks: 1. One could define for Dr the sectional curvature by means of (68). But this would not have an immediate geometrical meaning. 2. The integral in (68) converges since g(R(Xx , Yx )Yx , Xx ) ≤b |R||X|2x · |Y |2x , with X, Y ∈ L2 and ρ is bounded. The latter property arises from f ∈ D∞,r . 2 n Example. Let H−1 be the hyperbolic space of constant curvature −1 . The n the sectional curvature of D∞,r (H−1 ) is everywhere < 0 . ∞,r r Now, we want to study the geometry of Dω,0 ≡ Dω,0 and start with ω = µ n the volume form of (M , g). Then it follows immediately from the proof of theorem (3.3) that ∞,r Tf Dµ,0 = {X|X ◦ f −1 ∈ Ωr (T M ), div(X ◦ f −1 ) = 0}. ∂ For a vector Y denote by Y [ the corresponding 1-form, i.e. if Y = η i ∂u i then Y [ = ηi dui and (Y [ )] = Y . Divergence freeness of Y is equivalent to ∞,r r δY [ = 0. Under our assumption on σe , Dµ,0 = Dµ,0 is a submanifold of ∞,r r D0 = D0 . Hence, it is endowed with an induced weak Riemannian metric. We want to describe explicitly the corresponding Levi-Civita connection. Consider the Laplace operator ∆ = δd : Ωr (M ) → Ωr−2 (M ). The spectral assumption ¯ im δ, ¯ and im ∆ ¯ are closed. Moreover, ker∆ ∩ L2 = {0} implies that im d, n according to vol(M , g) = ∞, and ∆ has a bounded inverse ∆−1 .

Proposition 4.6 Let for V ∈ Tid D0∞,r , Pid (V ) := V − (d∆−1 δ(V [ ))] . Then ∞,r Pid is the orthogonal projection Pid : Tid D0∞,r → Tid Dµ,0 associated to g0 . Proof. One has to show im Pid ⊂ Tid D∞,r ,

(69)

2 Pid = Pid ,

(70)

Pid (Tid D

∞,r

)=

Tid Dµ∞,r ,

W ∈ ker Pid , V ∈ im Pid implies g0 (V, W ) = 0.

(71) (72)

Proof. (69) - (72) are easy calculations which are performed in the proof of Proposition 11.26 of [4]. They hold under our assumptions also in the noncompact case. 2 Consider the strong metric gr0 in Tid (D∞,r ) = Ωr (T M ). gr0 (V, W )

=

Z X r

(∇i V, ∇i W )x dvolx (g).

i=0

For r ≤ k, r = 2s even, this is equivalent to Z X s

(((1 + ∆)i V [ )] , ((1 + ∆)i W [ )] )x dvolx (g).

i=0

28

This has been proved in [3], [6], but there is an essential mistake in the proof. Nevertheless, the result is true. It is equivalent to Garding’s inequality. One has to show that the constants in Garding’s inequality can be chosen uniformly since we have (uniformly) bounded geometry. A complete proof for the generalized Divac operator is contained in [2]. Another equivalent strong metric gr is given by the extension of gr (V, W ) = g0 (V, W ) + g0 ((∆r (V [ )] , W ) to Ωr (T M ). Here we use essentially the self-adjointness of the powers of ∆ and the completeness of (M n , g). Proposition 4.7 Pid is an orthogonal projection with respect to gr , in particular, it is continuous with respect to the Ωr -topology. Proof. Once again, we have to prove for V ∈ Tid Dµ∞,r , W ∈ ker Pid , gr (V, W ) = 0. This has been done in [4]. 2 Proposition 4.8 The metric g0 is right invariant on Dµ∞,r . Proof. We have already seen for Vf , Wf ∈ Tf D∞,r , Vid = Vf ◦ f −1 , Wid = Wf ◦ f −1 , Z g0 (Vf , Wf ) = (Vid,x Wid,x )x ρ(x) dvolx (g). But for f ∈ Dµ∞,r , ρ(x) ≡ 1. 2 ∞,r ∞,r Let now for f ∈ Dµ,0 , V ∈ Tf ∈ Dµ,0 Pf V := Pid (V ◦ f −1 ) ◦ f. We want to show that this extended P is a C 2 - morphism of fiber bundles ∞,r ∞,r → T D P : T D0∞,r |Dµ,0 µ,0 . For this, we have to assume k − r ≥ 2. The problem is that the map f 7→ f −1 is only continuous. Proposition 4.9 Assume k − r ≥ 2, r > of fiber bundles.

n 2

+ 1. Then P is a C 2 - morphism

∞,r Proof. As we have already seen, for k − r ≥ 2, r > n2 + 1, T D0∞,r and T Dµ,0 are manifolds of at least class C 2 . At first, it is clear that Pf is continuous with respect to the Ωr -topology since Pid has this property and this property is preserved by right multiplication with f −1 and after that with f . We ∞,r see immediately that Pf2 = Pf . Moreover, Pf is the projection onto Tf Dµ,0 ∞,r ⊥g0 ∞,r . with respect to g0 : Let W ∈ (Tf Dµ,0 ) and X ∈ T D0∞,r |T Dµ,0 Then

29

g0 (Pf X, W ) = g0 (Pid (X ◦ f −1 ) ◦ f, W ) = g0 (Pid (X ◦ f −1 ) ◦ f ◦ f −1 , W ◦ f −1 ) = g0 (Pid (X ◦ f −1 ), W ◦ f −1 ) = 0 , according to propositions (4.8), (4.6) and the ∞,r ⊥g0 fact that W ◦ f −1 ∈ (Tid Dµ,0 ) . The latter follows again from (52). Now, ∞,r ∞,r ∞,r T D0 |Dµ,0 /T Dµ,0 is of class k − r since each element of the quotient is of ∞,r class k − r and T Dµ,0 is a submanifold of T D0∞,r . On D0∞,r g0 is a weak metric ∞,r ⊥g0 ∞,r ∞,r of class k − r. Hence, the isomorphism (T Dµ,0 ) → T D0∞,r |Dµ,0 /T Dµ,0 is of class k − r and therefore also the projector P associated to the decomposition ∞,r T D∞,r |D∞,r ∼ ⊕ (T D∞,r )⊥g0 = TD 0

µ,0

µ,0

µ,0

is of class k − r. 2 ˜ := P ◦ ∇ ¯ is the Levi - Civita connection for g0 |D∞,r . Proposition 4.10 ∇ µ,0 ∞,r+s Proof. Let X, Y ∈ C 2 (T Dµ,0 ). Then

˜ XY − ∇ ˜ Y X = P (∇ ¯ XY − ∇ ¯ Y X) = P ([X, Y ]) ∇ ∞,r+s ∞,r+s ) since Dµ,0 is a submanifold of D0∞,r+s . Hence, and [X, Y ] ∈ C 1 (T Dµ,0 ˜ XY − ∇ ˜ Y X = [X, Y ], i.e. ∇ ˜ is torsion free. Let P ([X, Y ]) = [X, Y ], and ∇ ∞,r+s 2 X, Y, Z ∈ C (T Dµ,0 ). We consider them as vector fields on D0∞,r+s , defined ∞,r+s ¯ is the Levi - Civita connection . Then, since ∇ only on the submanifold Dµ,0 of g0 , ¯ Z X, Y ) + g0 (∇ ¯ Z Y, X). Zg0 (X, Y ) = g0 (∇ (73) ∞,r+s ⊥g0 ¯ ¯ The components in (T Dµ,0 ) of ∇Z X, ∇Z Y produce 0 in (73) since the second component of the scalar products on the right hand side of (73) belong ∞,r+s to T Dµ,0 . Hence,

¯ Z X), Y ) + g0 (P (∇ ¯ Z Y ), X). Zg0 (X, Y ) = g0 (P (∇ i. e. ˜ Z X, Y ) + g0 (∇ ˜ Z Y, X).2 Zg0 (X, Y ) = g0 (∇ ∞,r+s ∞,r Using density arguments, we extend all formulas from Dµ,0 to Dµ,0 . Now, we are able to calculate the curvature tensor and the sectional curva∞,r ture of Dµ,0 . We performed these calculations but independently Smolentzew [20] published the same formula, but there they are only established for M n compact and without considering completions. The result can be summarized in r Theorem 4.1 Let X, Y ∈ Tid Dµ,0 be orthonormal, r > n2 + 1. then Z  1 1 K(X, Y ) = − (X, [[X, Y ]], Y ])x − ([X, [X, Y ]], Y )x 2 2 M 3 − ([X, Y ], [X, Y ])x − (Pid (∇X X), Pid (∇Y Y ))x (74) 4 1 + (Pid (∇X Y + ∇Y X), Pid (∇X Y + ∇Y X))x ] dvolx (g) . 4

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Here (·, ·)x = gx ( , ) and ∇ is the Levi-Civita connection of (M n , g). A calculational proof is contained in [20], p. 170-171. 2 Now, let µ 6= dvolx (g) be a volume form in b,k Ωn , such that inf x∈M |µ|g,x > 0. Then all Sobolev spaces Ωi (f ∗ T M, f ∗ ∇g , dvolx (g)) and Ωi (f ∗ T M, f ∗ ∇g , µ) are equivalent, i ≤ r, and the metrics g0 and g0,µ Z g0,µ (X, Y ) = (X, Y )g,f (x) µ, M

are quasi isometric. If we write µ = ρ · dvolx (g) and g˜ = ρ2/(2+n) then g0,µ (X, Y ) = g˜0 (X, Y ). i.e. g0,µ is the weak metric associated to the Riemannian metric g˜. Clearly, D0∞,r (˜ g ) = D0∞,r (g) as manifolds and we endow D0∞,r with the weak metric g˜0 . Now, all constructions above can be repeated and we obtain a formula as (74), ∞r . but g replaced by g˜. This should be the canonical metric and curvature of Dµ,0 ∞,r Next, we turn to Dω,0 , ω a strongly nondegenerate symplectic 2-form. Until now, ω is adapted to a metric g of bounded geometry only in the sense that the degree of boundedness and the strong nondegeneracy of ω is measured with respect to g. Given any such pair (ω, g), there are many other metrics g 0 such that ω has the same nondegeneracy and boundedness properties with respect to g 0 . Therefore, there is a considerable arbitrariness for g. We restrict g by assuming ω to be the K¨ahler form of an almost K¨ ahler structure (g, J ) i.e. ω(X, Y ) = g(J X , Y) where g has the properties (I) and (Bk ), J is an almost complete structure and dω = 0. Then all constructions ∞,r ∞,r of above can be performed, Dω,0 is a submanifold of D0∞,r and Tid Dω,0 = ∞,r r {X ∈ Ω (T M )|LX ω = 0} . Furthermore Dω,0 is endowed with the right invariant weak Riemannian metric Z g0 (X, Y )id = (X, Y )x dvolx (g). ˜ is given by The corresponding Levi-Civita connection ∇ ˜ X Y )(id) = Pid (∇X Y ). (∇ The same calculation as for Theorem (4.1) leads to ∞,r Theorem 4.2 Let X, Y ∈ Tid Dω,0 , X ⊥ Y, with |X|L2 = |Y |L2 = 1. Then the sectional curvature K(X, Y ) is given by Z  1 1 K(X, Y ) = − (X, [[X, Y ], Y ])x − ([X, [X, Y ]], Y )x − 2 2 3 − ([X, Y ], [X, Y ])x − (Pid (∇X X), Pid (∇Y Y ))x + (75) 4 1 + (Pid (∇X Y + ∇Y X), Pid (∇X Y + ∇Y X))x ] dvolx (g).2 4

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Finally , we turn to the well known application in hydrodynamics. We recall the Euler equations for an incompressible, homogeneous fluid without viscosity  ∂u ∂t + ∇u(t) u(t) = grad p , (76) div u(t) = 0 , where u = u(x, t) is a time dependent C 1 vector field on (M n , g), ∇ = ∇g , div = divdvolx (g) . Additionally, we assume u(t) ∈ Ωr (T M ) for all t which means that the fluid moves very slowly at infinity , r > n2 + 1. According to Proposition 3.10 of [10], u(t) defines a 1-parameter family of diffeomorphisms ft defined by dfs |s=t = u(t) ◦ ft . ds The ft remain in the indentity component of Dµ∞,r , since f0 = id , div u = 0, and µ = dvolx (g). Theorem 4.3 Assume (M n , g) with (I) and (Bk ), inf σe (∆1 |(ker ∆1 )⊥ ) > 0, k − 2 ≥ r > n2 + 1. Then u(t) satisfies the Euler equations (76) iff {ft }t is a ∞,r geodesic in Dµ,0 . Proof. Under the above assumptions, the proof is the same as in the compact case. Assume (76) and apply Pid to it. This yields   ∂u + Pid (∇u u) = 0 Pid ∂t
∂u and, since div u = 0, Pid ∂u ∂t = ∂t , ∂u = −Pid (∇u u). ∂t

(77)

We differentiate (76) and the equation ft ◦ ft−1 = id, and obtain d2 f |s=t ds2

∂u(t) ◦ f (t) = ∂t = u∗ (t) ◦ u(t) ◦ ft − Pid (∇u(t) u(t)) ◦ ft ,

= u∗ (t) ◦ u(t) ◦ f (t) +

∞,r−1 where we indentify Pid (∇u(t) u(t))◦ft with its horizontal lift in T 2 Dµ,0 . Using

Pid (∇u(t) u(t)) ◦ ft = P (∇u(t) u(t) ◦ ft ) ˜ = P ◦ K, ¯ we obtain and applying K  2  ˜ d f |s=t ¯ ∗ (t) ◦ u(t) ◦ ft − Pid (∇u(t) u(t) ◦ ft ] = K = P K[u ds2 = P (∇u(t) u(t) ◦ ft ) − P (∇u(t) u(t) ◦ ft ) = 0 , ∞,r i.e. {ft } is a geodesic in Dµ,0 . We omit the converse direction and refer to [15], 187-188. 2

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Corollary 4.6 Assume (M n , g) with (I), (Bk ), inf σe (∆1 |(ker ∆1 )⊥ ) > 0. (These conditions are automatically satisfied if M n is compact.) Then for small t the Euler equations (76) have a unique solution u ∈ Ωr (T M ) if k ≥ r > n2 + 1. Proof. This follows from the local existence of geodesics in Hilbert manifolds. 2 In several forthcoming papers we will investigate other diffeomorphism subgroups, the strong geometry and the case of a compressible fluid.

References [1] Blair, D. E., Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Berlin, 1976. [2] Bunke, U., Dirac Operatoren auf offenen Mannigfaltigkeiten, PhD thesis, Greifswald University, 1991. [3] Dodzink, J., Sobolev spaces of differential forms and de Rham - Hodge isomorphism, J. Diff. Geom. 16 (1981), 63-73. [4] Ebin, D. G., Espace des metriques Riemanniennes et movement des fluids via les varietes d’applications, Lecture Notes, Paris, 1972. [5] D. G. Ebin and J.E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Mathematics 92, 1970, 102-163. [6] J. Eichhorn, Elliptic operators on noncompact manifolds, Teubner Texte zur Mathematik 106 (1988), 4-169. [7] J. Eichhorn, The manifold structure of maps between open manifolds, Annals of Global Analysis and Geometry 3, 1993, 253-300. [8] J. Eichhorn, A priori estimates and Sobolev spaces on open manifolds, Banach Center Publications 27, (1992), 141-146. [9] Eichhorn, J., Gauge theory on open manifolds of bounded geometry, Int. J. of Modern Physics 7 (1993), 3927-3977. [10] Eichhorn, J., Diffeomorphism groups on noncompact manifolds, Preprint, Greifswald, 1993, to appear. [11] R. Greene, Complete metrics of bounded curvature on noncompact manifolds, Arch. Math. 31, 1978, 89-95. [12] Gromoll, D., Klingenberg, W., Meyer, W., Riemannian Geometrie im Grossen, Lecture Notes in Mathematics #55, Berlin, 1968.

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[13] H. Holman and H. Rummler, Alternierende Differentialformen, BI Mannheim, 1981. [14] Lukatsky, A. M., On the curvature of the diffeomorphisms group, Annals of Global Analysis and Geometry (1993), 135-140. [15] Marsden, J., Ebin, D. E., Fischer, A. E., Diffeomorphism groups, hydrodynamics and relativity, Proc. 13th Biennial Seminar of Canadian Math. Congress J. R. Vanstone (ed.), 135-279, Montreal, 1972. [16] R. Schmid, Infinite Dimensional Hamiltonian Systems, Monographs and Textbooks in Physical Sciences, Biblionopolis, Napoli, 1987. [17] H. Schubert, Topologie, Stuttgart, 1966. [18] Smolentzew, N. K., The curvature of diffeomorphism groups and the space of volume elements, Sibir. Math. J. 33 No. 4, (1992), 135-140. [19] Smolentzew, N. K., The biinvariant metric on the group of symplectic ∂ diffeomorphisms and the equation ∂t ∆F = {∆, F, F }, Sibir. Math. J. 27, No. 1 (1986), 150-156. [20] Smolentzew, N. K., The curvature of classical diffeomorphism groups, Sibir. Math.J. 35, No. 1 (1994), 169-176.

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