Fibred Spaces

Chapter 1

Fibred Spaces 1.1

Fibred spaces

Let F be a smooth manifold on which a Lie group G smoothly and effectively acts. Then, every element g ∈ G defines a diffeomorphism Lg : F → F. Here and in what follows, the smoothness of manifolds and mappings means ” differentiable of class C ∞ ”. Definition 1.1.1 We call a local trivial differentiable fibred space a five-uplu (E, p, M, F, G), where M is a smooth manifold, p : E → M is a surjection and the following conditions are verified: (a) M can be covered by a family of open sets U, V, W, ... such that for every open U in this family, there exists a bijection ϕU : p−1 (U )→U ×F and p(ϕ−1 U (x, y)) = x, x ∈ U, y ∈ F. (b) If x ∈ U ∩ V and ϕU : p−1 (U ) → U × F, ϕV : p−1 (V ) → V × F, then ϕV,x ◦ ϕ−1 U,x : F → F is equal to Lg , g ∈ G. By ϕU,x it was denoted the restriction of the bijection ϕU to p−1 (x). We shall identify ϕV,x ◦ ϕ−1 U,x by the element g from G. (c) The application gU V : U ∩ V → G, given with gU V (x) = ϕV.x ◦ ϕ−1 U,x is smooth. Some remarks on this definition are imposed. The mapping ϕU,x : p−1 (x) → F is a bijection, p−1 (x) = Ex is called the fibre in the point x ∈ M of the considered fibred space. We shall choose on the set Ex the topology and the differentiable structure for which the application ϕU,x is a diffeomorphism. The uniqueness (up to the diffeomorphisms) of this structure is assured by the condition (b) from Definition 1.1.1. We denote by (Ui , ϕi )i∈I a smooth atlas of the manifold M having the property that the covering {Ui } is finer than the covering U, V, W, ... and let {Vj }j∈J be an atlas of the smooth manifold F . We consider then the open sets Ui × Vj , Ui ⊂ U and we call open in E the sets −1 W ij = ϕ−1 (U ). U (Ui × Vj ) ⊂ p

We obtain a topology on E for which the bijections ϕU , ϕV , ϕW , ... are homeomorphisms. We denote by n and m the dimensions of the manifolds M , respectively F 1

2

Chapter 1

and we define the application φij : W ij → Rn+m by Φij = (ϕi × ψj ) ◦ ϕij U , where ϕij is the restriction of the bijection ϕ to W . The ensemble {(W , Φ ) } U ij ij ij (i×j)∈I×J U constitutes a smooth atlas on E. We consider E endowed with the smooth structure defined by the preceding atlas. Then the applications ϕU , ϕV , ... are diffeomorphisms and p : E → M is a submersion. The mentioned smooth structure is unique and dim E = n + m. For the fibred space (E, p, M, F, G) there are given the designations: E – the total space, M – the base space, p – the projection of E on M , F – the type fibre, and the Lie group G – the structure group. The pairs (U, ϕU ) are called fibred charts, and the set of the fibred charts is called fibred atlas. The functions gU V : U ∩ V → G are called structure functions. They have the properties (1.2)

gU V (x)gV W (x) = gU W (x), x ∈ U ∩ V ∩ W,

(1.2)

gU U (x) = e (e is the neutral element from G), x ∈ M,

(1.3)

−1 gU V (x) = gV U (x), x ∈ U ∩ V,

(1.4)

−1 ϕ−1 U (x, y) = ϕV (x, gU V (x)y),

with the notation Lg (x) = gy, g ∈ G, y ∈ F. Taking ϕi (x) = (x1 , ...xn ), x ∈ M, ψj (y) = (y 1 , ..., y m ), y ∈ F, the coordinates of a point u ∈ W ij ⊂ E are (xk , y a ), k = 1, ..., n; a = 1, ..., m. Finally, we observe that a transformation of local charts on E is written in the form (1.5)

ij −1 −1 ◦ (ϕ−1 Φ0 ij ◦ Φij −1 = (ϕ0i × ψj0 ) ◦ Φij i × ψj ) V ◦ (ϕU )

or, taking into account the property (1.4), in the form (1.6)

−1 k −1 a k 0 (Φ0 ij ◦ Φij −1 )(xk , y a ) = (ϕ0i ◦ ϕ−1 i (x ), ψj (gV U (ϕi (x ))ψj (y ))).

¯k , λk ) on G, such that gV U (ϕ−1 (xk )) ∈ U ¯k and we shall We consider a local chart (U i −1 k k s (x ))), s = 1, 2, ..., r = dim G. put λk (gV U (ϕi (x ))) = (AU V (ϕ−1 i Let ϕ¯ : G × F → F be the application which defines the action of G on F and ϕ −1 its local representation ϕ = ψj ◦ ϕ¯ ◦ (λ−1 k × ψj ). Then, the coordinate transformation (1.6) takes the form (1.7)

x ¯k = x ¯k (x1 , ..., xn ) a y¯ = ϕa (A1U V (xk ), ..., ArU V (xk ), y 1 , ..., y m ).

Example 1.1.1 The ensemble (M × F, pr1 , M, F, G) where pr1 is the projection on the first factor, is a fibred space, called product fibred space. Its fibred atlas is formed by only one chart (M, idM ×F ). Other examples, more significant, like the principal fibre bundles or the vector bundles, will be exposed in the next sections.

Fibred Spaces

1.2

3

Principal fibre bundles

In what follows we are dealing with an important class of fibred spaces. We mean those for which the type fibre F coincides with the structure group G, called principal fibre bundles . Definition 1.2.1 The fibred space (E, p, M, F, G) is called a principal fibre bundle if the type fibre F coincides with the structure group G and the action of G on itself is given by the left translation: Lg (a) = ga, g, a ∈ G. We shall denote a principal fibre bundle with (P, π, M, G). The Definition 1.2.1 is equivalent to Definition 1.2.2 We call principal fibre bundle a 4-uplu (P, π, M, G), where P and M are differentiable manifolds, π : P → M is a differentiable surjection, such that the properties a) and b) hold. (a) G is a group of transformations on the differentiable manifold P . That is, for every g ∈ G, the action Rg : P → P is Rg (u) = ug. The orbits of G are the local fibres and G acts on them simply transitive. (b) For every point x ∈ M there exists an open subset U in M and a differentiable mapping ϕU : π −1 (U ) → G such that ϕU (ug) = ϕU (u)g; the mapping ψU : π −1 (U ) → U × G, ψU (u) = (π(u), ϕU (u)) is a diffeomorphism. The structure functions are of the form gV U : U ∩ V → G, gV U (π(u)) = ϕV (u)(ϕU (u))−1 , u ∈ π −1 (U ∩ V ). Definition 1.2.3 Given two principal fibre bundles (P, π, M, G) and (P 0 , π 0 , M 0 , G0 ), we call morphism of these principal fibre bundles a pair of application (f, f 0 ) for which: (a) f : P → P 0 is a differentiable application. (b) f 0 : G → G0 is a Lie group morphism. (c) f (ug) = f (u)f 0 (g), ∀u ∈ P, g ∈ G. We associate to the morphism (f, f 0 ) the application f 00 : M → M 0 , f 00 (π(uG)) = π (f (u)G0 ), u ∈ P. It is easy to deduce that f 00 is differentiable. Two morphisms of principal fibre bundles are naturally compounded and it results a principal fibre bundles morphism. The pair (idP , idG ) : (P, π, M, G) → (P, π, M, G) is a morphism of principal fibre bundles. It results that the principal fibre bundles as objects and the principal fibre bundles morphisms provide a category denoted by Fibp . 0

Definition 1.2.4 The principal fibre bundles (P, π, M, G) and (P 0 , π 0 , M 0 , G0 ) are called isomorphic if there exists a morphism (f, f 0 ) of them which is isomorphism in the Fibp category. It is easy to deduce that the morphism (f, f 0 ) is an isomorphism in the Fibp category if and only if f is diffeomorphism and f 0 is isomorphism of Lie groups. In this case, the application f 00 is a diffeomorphism. An isomorphism (f, f 0 ) : (P, π, M, G) → (P, π, M, G) is called automorphism. Obviously, in this case, f 00 is the identity on M .

4

Chapter 1

Definition 1.2.5 A morphism (f, f 0 ) : (P, π, M, G) → (P 0 , π 0 , M 0 , G0 ) is called embedding if f is immersion and f 0 is monomorphism of Lie groups. Definition 1.2.6 We say that the principal fibre bundle (P, π, M, G) is a subbundle of the principal fibre bundle (P 0 , π 0 , M 0 , G0 ) if P ⊂ P 0 , G ⊂ G0 and (i, i0 ) with i : P → P 0 , i0 : G → G0 is a morphism of these principal fibre bundles. An example of subbundle of the principal fibre bundle (P, π, M, G) is (π −1 (U ), π|π−1 (U ) , U, G), where U is some open set in M . We shall denote it by (P, π, M, G)|U . Another example, which is basic for defining the notion of G–structure, will be presented in the following. Definition 1.2.7 A subbundle (P, π, M, G) of the principal fibre bundle (P 0 , π 0 , M 0 , G0 ) is called a reduction of the structure group G0 to its subgroup G if the smooth manifold M 0 is diffeomorphic with the manifoldM . Usually, one considers the reductions for which M 0 concides withM . In this case, we say that the principal fibre bundle (P, π, M, G) is a reduction to the group G0 . Definition 1.2.8 We say that two principal fibre bundles (P, π, M, G), (P 0 , π 0 , M, G0 ) are locally equivalent if for every point x ∈ M , there exists an open neighbourhood U of it such that the subbundles (P, π, M, G)|U and (P 0 , π 0 , M, G0 )|U are isomorphic. Example 1.2.1 The 4-uple (M × G, pr1 , M, G) is a principal fibre bundle, called trivial principal fibre bundle. Writting with all details the Definition 1.2.1 of a principal fibre bundle, one establi∼ shes that it is locally equivalent with a trivial principal fibre bundle, π −1 (U ) → U ×G. That is why one says that the principal fibre bundle is locally trivial. Then it is clear the importance of the preceding example. Before giving a second example of principal fibre bundle, we recall that a transformation of local coordinates on the total space P of the principal fibre bundle (P, π, M, G) is of the form  k ∂x ¯ k k 1 n = n, x ¯ =x ¯ (x , ..., x ), rank (2.1) ∂xi k 1 k r a a 1 y¯ = Φ (AU V (x ), ..., AU V (x ), y , ..., y m ), where Φa is the local representation of the left translation on G. Example 1.2.2 One of the most used example is that of the principal fibre bundle of the frames over a differentiable manifold. A system of n tangent vectors in a point x of a differentiable manifold M , linear independent, is called a frame in that point. The set L(M ) of the frames in all the points of the differentiable manifold M , can be naturally endowed with a differentiable structure. For that, let (Ui , ϕi ) be a local chart ¯i the set of the frames in the points of Ui . We consider the application on M and U ¯ ϕ¯i : Ui → IRn(n+1) defined by ϕ¯i (p, X p , ..., X p ) = (x1 , ..., xn , X n p , ..., X n p , ..., X 1 p , 1

n

1

1

n

..., X n p ), where (x1 , ..., xn ) = ϕi (p), p ∈ M and X 1 p , ..., X n p are the coordinates of n

a

a

the vector Xp in the local chart (Ui , ϕi ). If (Uj , ϕj ) is another local chart on M , with a

Fibred Spaces

5

0

0

the property Ui ∩ Uj 6= ∅ and ϕj (p) = (x1 , ..., xn ), then the change of the coordinates ϕj ◦ ϕ−1 is of the form i 0

0

0

xi = xi (x1 , ..., xn ), rank (∂xi /∂xj ) = n.

(2.2)

Here we state that the accent on the indices modifies also the indexed coordinates. This convention will be used almost 2in all follow. ¯i ) ⊂ IRn+n are open sets in IRn+n2 . We consider Let us observe that ϕ¯i (U on L(M ) the topology for which all the applications ϕ¯i are homeomorphisms. The [ ¯ ¯ pairs (Ui , ϕ¯i )i∈I provide a differentiable atlas on L(M ), since Ui = L(M ) and the applications ϕ¯j ◦ ϕ¯−1 are locally expressed in the form i 0

i0

i0

1

n

x = x (x , ..., x ), rank (2.3)

∂xi ∂xj

! = n,

0

0

Xi = a

∂xi i X , (a = 1, ..., n). ∂xi a

Hence ϕ¯j ◦ ϕ¯−1 are differentiable. Then, ϕ¯i are diffeomorphisms and the differeni tiable structure thus determined is unique. We endow L(M ) with this differentiable structure. We obtain the principal fibre bundle of frames taking L(M ) as total space, M as base space and π as the map which associates to every frame its origin. The structure group is the general linear group, GL(n, IR). The fibred chart associated to the local!chart (Ui , ϕi ) is (π −1 (Ui ), Φi ) where Φi : π −1 (Ui )→Ui ×GL(n, IR), !   0 ∂ ∂xi −1 i i i i Φi X = (p, X ), p∈Ui . We establish that (Φj ◦Φi )(p, X ) = p, X ∂xi p ∂xi a a a a ! 0 ∂xi so the structure functions g ij : Ui ∩ Uj → G are g ij (p) = . Hence, ∂xi ϕ(p)

(L(M ), π, M, GL(m, IR)) is a principal fibre bundle, briefly denoted by L(M ). We observe that the transformation of coordinates (2.3) is a particular case of (2.1). 2 Indeed (X i )∈IRn is an invertible matrix so (X i ) ∈ GL(n, IR) and the left translation a

a

Lg on GL(n, IR) is given by the left multiplication of g with the matrix g ij . We now use L(M ) for introducing Definition 1.2.9 We call G–structure on M every subbundle (P, π, M, G) of the principal fibre bundle L(M ). In other words, a reduction of the structure group GL(n, IR) of the principal fibre bundle L(M ) to the subgroup G ⊂ GL(n, IR) is a G–structure. Almost without exception, the remarkable geometric structures may be thought of as G–structures. The Riemannian structures are O(n)–structures, O(n) being the orthogonal group. The conformal structures are CO(n)–structures, CO(n) being the conformal group. Similarly, the simplectic structures are Sp(n)–structures, Sp(n) being the simplectic group. The almost product, almost complex, f –tensorial structures are G–structures. Next, we shall expound the standard construction by which to a principal fibre bundle (P, π, M, G) and to a manifold F on which the Lie group G acts differentiably and effectively to the left, it is associating a smooth fibred space.

6

Chapter 1

Let L : G × F → F, L(g, f ) = Lg (f ) = gf be the action of G on F , where the equalities means notations. We consider the product manifold P × F and define the action to the right of G on this manifold by (P × F ) × G → P × F , (u, f )g = (ug, g −1 f ). The quotient set with respect to the equivalence relation determined by this action, E = (P × F )/G is endowed with the quotient topology. The elements of this set are equivalence classes of (u, f ) ∈ P × F, denoted by [u, f ]. We have [u, f ] = {(ug, g −1 f ), g ∈ G}. The application pE : E → M , pE ([u, f ]) = π(u) is well defined, continuous and onto, because of π(ug) = π(u), ∀g ∈ G. Let (U, ψU ) be a bundle chart of (P, π, M, G). We set ψU (u) = (π(u), ϕU (u)) for every u ∈ p−1 (U ). Then the action of G on P × F is expressed by (u, f ) → (ug, g −1 f ) = (π(u), ϕU (u)g, g −1 f ). Define a mapping p−1 E (U ) → U × F, [u, f ] → (π(u), ϕU (u)f ). This mapping does not depend on the choice of the representatives of the class [u, f ]. Moreover, it is bijective. We can transport the differentiable structure of U ×F on p−1 E (U ) and then the mapping defined above is a diffeomorphism. The local differentiable structures obtained in this way determine a unique manifold structure on E. Indeed, if (U, ψU ) and (V, ψV ) are two bundle charts of (P, π, M, G) for every x ∈ U ∩ V and f ∈ F , we have: →

−1 (x, f ) ∈ U ∩ V × F → [ψU (x, e), f ] ∈ E → −1 (ψU (x, e))f = (x, gV U (x)f ) ∈ U ∩ V × F,

−1 (π(ψU (x, e), ϕV

−1 where gV U (x) = (ψV ◦ ψU )(x, e). So, the changes of local charts on E are differentiable. We have now all the elements which permit us to affirm that (E, pE , M, F, G) is a differentiable fibred space. The structure functions of this fibred space coincide with those of (P, π, M, G). The following two theorems (for their proofs see [77]) underline the important role of the structure functions have.

Theorem 1.2.1 Let M be a manifold, {Ui } an open covering of it and G a Lie group. Assume there are given the differentiable functions g ji : Ui ∩ Uj → G, which verify the equalities (2.4)

gkj (x)g ji (x) = gki (x), x ∈ Ui ∩ Uj ∩ Uk .

In these conditions, there exists a principal fibred bundle, so (g ji ) are its structure functions associated to the covering {Ui }. Theorem 1.2.2 Let (P 0 , π 0 , M, G0 ) be a principal fibred bundle and G a Lie subgroup of G0 . The necessary and sufficient condition that there exists a reduction (P, π, M, G) of the structure group G0 to G is that there exists a system of structure functions (g 0 ji ) of (P 0 , π 0 , M, G0 ) subordinated to the open covering {Ui }, with values in G.

1.3

Vector bundles

An important class of fibred spaces is that of the vector bundles. Their simple local structure leads to many interesting results. Definition 1.3.1 We call vector bundle a differentiable fibred space (E, p, M, F, G) where F is a vectorial or linear space, and G is a group of automorphisms of F .

Fibred Spaces

7

If F is an infinite dimensional real or complex Banach space, we obtain the notion of real or complex Banach vector bundle. Next, we shall consider only vector bundles with as type fibre a real linear space of finite dimension m. In this case, we can take F = IRm and G = GL(m, IR) and denote the vector bundle of (E, p, M, IRm , GL(m, IR)) by (E, p, M ) or by ξ, η, ... The structure of a vector bundle is given by a fibred atlas {(Ui , ϕi , IRm )}i∈I , called vectorial atlas, where: 1. {Ui }i∈I is an open covering of the manifold M ; 2. The applications ϕi : p−1 (Ui ) → Ui × IRm are bijections and pϕ−1 i (x, f ) = x, for every x ∈ M, f ∈ F ; 3. For every (i, j) ∈ I × I so that Ui ∩ Uj 6= ∅ there exists a differentiable aplication −1 g ij : Ui ∩ Uj → GL(m, IR) so that ϕ−1 i,x = ϕj,x ◦ g ij (x), for every x ∈ Ui ∩ Uj , −1 where ϕi,x : F → p−1 (x) is the restriction of the application ϕ−1 to {x} × F. i The structure of vector bundle ξ = (E, p.M ) is a class of equivalent vectorial atlases. Two vectorial atlases being equivalent if their union is also a vectorial atlas. The structure of differentiable manifold of E is induced by the differentiable structure of M and by the structure of vector bundle (E, p, M ). That is, if A = {(Ui , ψi )i∈I } is an atlas on M and {(Ui , ϕi , F )}i∈I is a vectorial atlas (such a choice of a vectorial atlas is always possible), then {(p−1 (Ui ), hi }i∈I , where hi : p−1 (Ui ) → IRn × IRm , hi (u) = (ψi (p(u)), ϕi,p(u) (u)), is a differentiable atlas on the total space E. Let (Uj , ϕj , F ) be a vectorial chart and let Uj be the domain of a local chart (Uj , ψj ) on M such that Ui ∩ Uj 6= ∅. Let us calculate hj ◦ h−1 i . We have −1 −1 k k a k (hj ◦ h−1 i )(x , y ) = ((ψj ◦ ψi )(x ), g ji (ψi (x ))y). 0

We shall denote by Maa (x) the matrix of the linear application (g ji ◦ ψi−1 )(x). Then, we can write k a k0 1 n a0 a (hj ◦ h−1 i )(x , y ) = (x (x , ..., x ), Ma (x)y ). 0

0

In other words, the transformations of coordinates (xk , y a ) → (xk , y a ) on the differentiable manifold E are of the form: 0

0

0

xk = xk (xk ), rank (∂xk /∂xk ) = n 0 0 0 y a = Maa (x)y a , kMaa (x)k ∈ GL(m, IR).

(3.2)

We observe that the transformation of coordinates (3.2) are particular cases of (1.7). The properties (1.1), (1.2) of the structure functions imply (3.3) (3.4)

00

0

00

Maa0 (x)Maa (x) = Maa (x), 0

0

Maa (x) = δaa ,

ψi−1 (x) = ψj−1 (x0 ) ∈ Ui ∩ Uj ∩ Uk ψi−1 (x) ∈ Ui = Uj .

Some properties of the differential structure of the base manifold M are propagated to the differential structure of the total space E. For example, if M is paracompact, so is E. The same thing holds for the existence of the partition of unity on M. To justify these affirmations, we shall suppose that the atlas on M , considered above, is maximal, that is A is the union of all the equivalent atlases which give the differential structure of M . Then, the atlas {p−1 (Ui ), hi }i∈I on the total space E is generally not maximal.

8

Chapter 1

Theorem 1.3.1 [77] Let (E, p, M ) be a vector bundle. If the base manifold M is paracompact, then the manifold E is paracompact, too. Proof. Let u1 , u2 ∈ E and x1 = p(u1 ), x1 = p(u2 ). Let be the 1 and D2 [open sets D[ such that x1 ∈ D1 , x2 ∈ D2 and D1 ∩ D2 = ∅. We set D1 = Uj , D 2 = Uj . It j∈I1

j∈I2

results there exist j1 ∈ I1 , j2 ∈ I2 such that x1 ∈ Uj1 , x2 ∈ Uj2 and Uj1 ∩ Uj2 = ∅. Then p−1 (Uj1 ) ∩ p−1 (Uj2 ) = ∅ and u1 ∈ p−1 (Uj1 ), u2 ∈ p−1 (Uj2 ). So, the manifold E is a Hausdorff space. [ p−1 (Ui ). Now, let {Dj }j∈I be an open covering of E and let us set Dj = i∈Ij

The open covering {Ui }i∈I admits an open refinement {Vk }k∈K . That is, for every k ∈ K, there exists i(k) ∈ I, such that Vk ⊂ Ui(k) . It results that p−1 (Vk ) ⊂ p−1 (Ui(k) ). Consequently, there exists an open set Dj(k) , such that p−1 (Ui(k) ) ⊂ Dj(k) . Hence {p−1 (Vk )}k∈K is an open refinement of the covering {Dj }. We shall show now that this covering is local finite. If u ∈ E and x = p(u), there exists an open neighbourhood U of x which intersects only a finite number of the open sets Vk . Let them be V1 , ..., Vn . It results, by contradiction, that p−1 (U ) intersects only the open sets p−1 (V1 ), ..., p−1 (Vn ). Hence, the manifold E is paracompact. Corollary 1.3.1 Let (E, p, M ) be a vector bundle with the paracompact base M. Then the total space E admits differentiable partitions of the unity. Proof. According to the Theorem 1.3.1, the manifold E is paracompact. By a general result, [40], E admits differentiable partitions of the unity. Proposition 1.3.1 Let {fj }j∈J be a differentiable partition of the unity on the base manifold M of a vector bundle (E, p, M ), subordinated to an open covering {Ui }i∈I . Then {fjv = fj ◦ p}j∈J is a differentiable partition of the unity on the total space E, subordinated to the open covering {p−1 (Ui )}i∈I. Proof. It is obvious that fjv ≥ 0 for every j ∈ J. Let us set carr fj = {x ∈ M | fj (x) 6= 0} and supp fj the closure of the set carr fj . We have carr fjv = {u ∈ E | fj (p(u) 6= 0} ⊂ p−1 (supp fj ) and hence supp fjv ⊂ p−1 (supp fj ), because supp fj is v a closed set. But {supp fj }j∈J is a locally finite P v covering. P Hence {supp fj }j∈J is a locally finite covering, too. Finally, we have fj (u) = fj (p(u)) = 1. q.e.d. From the geometric viewpoint, the assumption that M is paracompact is natural and we shall accept it in what follows. Here is an important example of vector bundle. Example 1.3.1 The tangent bundle of a differentiable manifold. Let M be a differentiable manifold [ and Tx M the tangent vectol space in the point x of M , x ∈ M. Let be T M = Tx M and we define the application τ : T M → M by τ (Xx ) = x, x∈M

where Xx ∈ Tx M. Then we consider the vectorial charts (Ui , ϕ0i , IRn ), where {(Ui , ϕi )} is a differentiable atlas on Mand ϕ0i : τ −1 (Ui ) → Ui × IRn are taken as ϕ0i (Xx ) = ∂ · The application ϕ0i,x is, obviously, bijective. (x, Xx1 , ..., Xxn ), if Xx = Xxi ∂xi x

Fibred Spaces

9

Let be now another vectorial chart (Uj , ϕ0j , IRn ), with Ui ∩ Uj 6= ∅. Let us eluci    0 0 ∂ −1 −1 1 n 0 i 0 0 date the application ϕj ◦ ϕi . We have ϕj ϕi (x, Xx , ..., Xx ) = ϕj Xx = ∂xi x ! !   0 0 ∂ ∂xi ∂xi i X = Xxi . ϕ0j 0 x ∂xi ∂xi x ∂xi ϕi (x) ! 0 ∂xi It is obvious that the structure functions x → g ij (x) = ∈ GL(n, IR) ∂xi ϕi (x)

are differentiable. We have used the notation from Section 1.2. Hence (T M, τ, M ) is a vector bundle. Its structure functions are given by the Jacobian matrix of the transformation of coordinates (2.2). Examining the Example 1.2.2, one establishes that the structure functions of the principal fibre bundle of the frames of a manifold are defined by the same matrix. This property suggests the existence of a link between these two fibred spaces. Indeed, the fibred space associated with L(M ), according to the method indicated at the end of Section 1.2, where F = IRn is exactly the vector bundle (T M, τ, M ), if we identify the equivalence class [(X )x , (z i )] with the vector a   ∂ , (one sums over a and i from 1 to n), where (X )x is a frame in the z a X ix ∂xi x a a point x ∈ M and (z i ) ∈ IRn . We end this section with the remark that we can consider also the product or trivial vector bundle, that is (M × IRm , pr1 , M ).

1.4

Vector bundles morphisms

The notion of vector bundle morphisms is interesting for the applications of the vector bundles in Finsler geometry. We shall show that, in natural geometric conditions, for a pair of vector bundles there exists a morphism from one to another. Definition 1.4.1 Let (E, p, M ) and (E 0 , p0 , M 0 ) be two vector bundles and f0 : M → M 0 a differentiable application. We call f0 –morphism from (E, p, M ) to (E 0 , p0 , M 0 ) an application f : E → E 0 with the property that for every point x ∈ M , there exists a vectorial chart (U, ϕ, F ) of (E, p, M ) in x, a vectorial chart (U 0 , ϕ0 , F 0 ) of (E 0 , p0 , M 0 ) in the point f0 (x) and a differential application hU 0 U : U ∩ f0−1 (U ) → L(F, F 0 ) such 0 −1 −1 0 that fx ◦ ϕ−1 x = ϕf0 (x) ◦ hU 0 U (x), for every x ∈ U ∩ f0 (U ), where fx is the restriction of the application f to Ex . From the above definition, it results immediately that fx : Ex → Ef0 0 (x) is a linear application and that hU 0 U (x) = ϕ0f0 (x) ◦ fx ◦ ϕ−1 x . It is easy to verify also that the application f is differentiable. Let now f 0 : E 0 → E 00 a f00 –morphism from the vector bundle (E 0 , p0 , M 0 ) to the vector bundle (E 00 , p00 , M 00 ). The application f 0 ◦ f : E → E 00 is a f00 ◦ f0 –morphism from the vector bundle (E, p, M ) to the vector bundle (E 00 , p00 , M 00 ). We observe then that id|E : E → E is an id|M –morphism. Consequently, the vector bundles and the morphisms of vector bundles provide a category denoted by F V. Definition 1.4.2 A f0 –morphism of vector bundles is called vector bundle morphism if M 0 coincides with M and f0 is the identity on M .

10

Chapter 1

A morphism of vector bundles is called monomorphism (epimorphism), if fx is monomorphism (epimorphism) of linear spaces for every point x from the base manifold. If f is a morphism from (E, p, M ) to (E 0 , p0 , M 0 ) which is also a bijection, then the applications fx , x ∈ M , are isomorphisms from Ex to Ex0 and f −1 : E 0 → E is a vector bundle morphism having the associated functions (hU 0 U (x))−1 . In this case, one says that f is an isomorphism of vector bundles and the corresponding vector bundles are called isomorphic. Let us consider a differentiable application h : M 0 → M and (E, p, M ) a vector bundle. The set h∗ E = {(x0 , u) ∈ M 0 × E | h(x0 ) = p(u)} has a vector bundle structure, with the base M 0 and the projection h∗ p(x0 , u) = x0 . The vector bundle (h∗ E, h∗ p, M 0 ) is called the vector bundle induced by the application h or the inverse image of the vector bundle (E, p, M ) by h. Any chart (U, ϕ, F ) of the vector bundle (E, p, M ) determines a chart (h−1 (U ), ϕ0 , F ) of the induced vector bundle (h∗ E, h∗ p, M 0 ), where ϕ0 : (h∗p )−1 (h−1 (Ui ) → h−1 (Ui ) × F, ϕ0 (x0 , u) = (x0 , ϕh(x) (u)). One easily establishes that the structure functions of the vector bundle induced by the application h are g ∗ ij (x) = g ji (h(x)). Let us add also that its local fibre is (h∗ E)x0 = {x0 } × Eh(x) and that there exists an h–morphism f : h∗ E → E, defined by f (x0 , u) = u. This morphism makes commutative the following diagram: f

∗ h E −→ E  ∗ y yp h p h

M 0 −→

M

Let g : E → E 0 be a vector bundle morphism, from (E, p, M ) to (E 0 , p0 , M 0 ). If h : M 0 → M is a differentiable application, then there exists a unique morphism of vector bundles h∗ g : h∗ E → h∗ E 0 , so that f 0 ◦ h∗ g = g ◦ f, where f : h∗ E → E and f 0 : h∗ E 0 → E 0 are the h–morphisms defined above. The morphism h∗ g is defined by h∗ g(x0 , u) = (x0 , g(u)). If M 0 is a submanifold of M , the considerations just made may be applied to the inclusion i : M 0 → M. The inverse image (i∗ E, i∗ p, M 0 ) of the vector bundle (E, p, M ) by i is called the restriction of this vector bundle to M 0 and it is denoted by (E|M 0 , p, M 0 ). Finally, let us observe that if h is the identity map, the vector bundle induced by h coincides with the initial one. If h is a constant application, then the vector bundle induced by h is the trivial vector bundle. Given the vector bundles (E, p, M ) and (E 0 , p0 , M 0 ) and two f0 –morphisms of them, f1 , f2 : E → E 0 , it has sense to consider the sum f1 +f2 , defined by (f1 +f2 )x = f1,x +f2,x , for every x from M , and the application af1 , too, where a is a real function on M , defined by (af1 )x = a(x)f1,x , x ∈ M. The following proposition is been directly proved, by using the Definition 1.4.1. Proposition 1.4.1 The sum f1 + f2 of two f0 –morphisms of vector bundles and the product af1 of an f0 –morphism of vector bundles with a real function are, each of them, an f0 –morphism.

Fibred Spaces

11

Theorem 1.4.1 If (E, p, M ) and (E 0 , p0 , M ) are vector bundles with the paracompact base M , then there exists a vector bundle morphism f : E → E 0 . Proof. Let {ai } be a differentiable partition of unity which is subordinated to the covering {Ui } of the base M , Ui being domains of vectorial charts of (E, p, M ). For any fixed Ui , any differentiable application Ui → L(F, F 0 ) defines, by means of the local charts, a morphism of vector bundles fi : E|Ui → E 0 |Ui . It results that ai fi : E|Ui → E 0 |Ui are morphisms of vector bundles. Each ai fi may be prolonged to E, considering (ai fiP )x = 0, if x ∈ / Ui , because supp ai ⊂ Ui . The application f : E → E 0 defined by fx = ai (x)fi,x is a vector bundle morphism. q.e.d.

1.5

Operations with vector bundles

Let be ξ = (E, p, M ) and ξ 0 = (E 0 , p0 , M 0 ) two vector bundles. Then (E × E 0 , p × p0 , M × M 0 ) is a vector bundle. It will be denoted by ξ × ξ 0 and will be called the product of the vector bundles ξ and ξ 0 . 0 0 [ Assuming that ξ and ξ have the same base M , we consider the set E ⊕ E = Ex × Ex0 and the maping p ⊗ p0 which carries (u, u0 ) ∈ Ex × Ex0 to x ∈ M. Then x∈M

(E ⊕ E 0 , p ⊕ p0 , M ) is a vector bundle with the type fibre F × F 0 . It is called the Whitney sum of ξ and ξ 0 and is denoted by ξ ⊕ ξ 0 . Its structure functions are given by the matrices



g ji (x) 0

,

0

0 g ji (x) where g ji and g 0 ji are the structure functions of ξ and ξ 0 , respectively. It is obvious that when M = M 0 , ξ×ξ 0 is isomorphic to ξ×ξ 0 . As the local fibres Ex and Ex0 of the vector bundles ξ and resp. ξ 0 are linear spaces, we may consider 0 0 0 the [ linear space L(Ex , Ex ) of all linear maps from Ex to Ex . Then (L(E, E ) = L(Ex , Ex0 ), ρ, M ), where ρ maps z ∈ L(Ex , Ex0 ) to x, is a vector bundle. It will x∈M

be denoted by L(ξ, ξ 0 ). Its type fibre is L(F, F 0 ) and its structure functions in a point x ∈ Ui ∩ Uj have the values given by the mappings h → g 0 ji (x) ◦ h ◦ g ji (x), where h ∈ L(F, F 0 ). More general, given q vector bundles ξi = (Ei , pi , M ), i = 1, 2, ..., q, one may construct the vector bundle L(ξ1 , ..., ξq ; ξ 0 ) = (L(E1 , ..., Eq ; E 0 ), ρq , M ), where L(E1 , ..., Eq ; E 0 ) = ∪L(E1,x , ..., Eq,x ; Ex0 ) and ρq is similar to ρ. Some notable vector bundles are getting when in the above construction ξ 0 is replaced by (M × IR, pr1 , M ). For E1 = E2 = · · · = Eq we set L(E1 , ..., Eq , M × IR) = Lq (E, IR) and L1 (E, IR) will be denoted by E ∗ or ξ ∗ . The vector bundle ξ ∗ = (E ∗ , ρ1 , M ) is called the dual of ξ. Its type fibre is the dual F ∗ of F . the values of its structure functions in a point x ∈ Ui ∩ Uj are given by the mappings θ → θ ◦ g ij (x), θ ∈ F ∗ . A remarkable subspace of the space Lq (F, F 0 ) is given by the skew–symmetric multilinear mappings. It will be denoted by Aq (F, F 0 ). Considering Aq (E, E 0 ) = ∪Aq (Ex , Ex0 ), the restriction of ρ to it and M one gets a new vector bundle Aq (ξ, ξ 0 ). 0 0 0 0 The tensor product [ of the vector bundles ξ = (E, p, M ) and ξ = (E , p , M ) is ξ ⊗ ξ 0 = (E ⊗ E 0 = Ex ⊗ Ex0 , p ⊗ p0 , M ), where p ⊗ p0 maps every u ⊗ u; ∈ Ex ⊗ Ex0 x∈M

12

Chapter 1

to x ∈ M. It is obvious that the tensor product of a finite number of vector bundles could be defined. Vector subbundles. Let ξ = (E, p, M ) be a vector bundle and E 0 ⊂ E with the property that for every x ∈ M there exists a vectorial chart (U, ϕ, F ) and a linear subspace F 0 ⊂ F such that ϕ(p−1 (U )∩E 0 ) = U ×F 0 . It follows that ξ 0 = (E 0 , p E 0 , M ) is a vector bundle with type fibre F 0 . It will be called a vector subbundle of ξ. It is easy to see that E 0 is a submanifold of E. Moreover, the inclusion map i : E 0 → E is a vector bundle morphism. In order to obtain a first example of vector subbundle, let us consider the monomorphism f : E 0 → E from ξ 0 = (E 0 , p0 , M 0 ) to ξ = (E, p, M ). Then f (E 0 ) is the total space of a vector subbundle of ξ, which is isomorphic to ξ 0 . It will be denoted by Im f and will be called the image of the monomorphism f . A second example of vector subbundles is provided by a vector bundle [ epimor0 phism. If f : E → E is a vector bundle epimorphism, the set Ker f = Ker fx , x∈M

where Ker fx is the kernel of fx , is the total space of a vector subbundle of ξ 0 denoted by Ker f and called the kernel of the epimorphism f. To the pair (ξ, ξ 0 ), where ξ 0 is a vector subbundle of ξ, one may associate a new vector bundle called the quotient of ξ by ξ 0 . Namely, if Ex and Ex0 are the local fibres in [ x of ξ and resp. ξ 0 , one considers the quotient space Ex /Ex0 . The set E/E 0 = Ex /Ex0 has a vector bundle structure over M with type fibre F/F 0 . We x∈M

denote this vector bundle by ξ/ξ 0 or E/E 0 . The sequence of vector bundles over M : (5.1)

f

g

0 → ξ 0 −→ ξ −→ ξ 00 → 0

is said to be exact if f is a monomorphism, g is an epimorphism and Im f = Ker g. If the sequence (5.1) is exact, it is clear that one may identify ξ 00 to ξ/ξ 0 . One says that the vector subundle ξ 0 = (E 0 , p0 , M ) of ξ = (E, p, M ) splits ξ if there exists a vector subundle ξ 00 = (E 00 , p00 , M ) of ξ such that ξ = ξ 0 ⊕ ξ 00 . Using a partition of unity on M one proves Theorem 1.5.1 Let ξ 0 be a vector subbundle of ξ = (E, p, M ) with paracompact base M . There exists an epimorphism f : E → E 0 such that f ◦ i = id E , where i : E 0 → E is the inclusion map. By this theorem, E 00 = Ker f is a vector subbundle of ξ and E = E 0 ⊕ E 00 . Therefore, we have Corollary 1.5.1 Any vector subbundle of a vector bundle ξ with paracompact base M, splits it. Principal fibre bundle associated to a vector bundle. Let F be the type fibre of the vector bundle ξ = (E, p, M ) and the trivial vector bundle (M ×[ F, pr1 , M ). We consider the vector bundle (L(M × F, E), ρ, M ) and P = {u | ρ(u) = x∈M

x, u : F → Ex a linear isomorphism} as subset of L(M × F, E).

Fibred Spaces

13

If {Ui }i∈I are domains [ of the vectorial charts of a vector bundle atlas of (L(M × F, E), ρ, M ), then P = (ρ−1 (Ui ) ∩ P ). The set ρ−1 (Ui ∩ P ) is diffeomorphic to i∈I

Ui × Aut (F ) and is open since the set Ui × Aut (F ) is so. Thus P is a submanifold of L(M × F, E). We denote by π the restriction of ρ to P . The group Aut (F ) acts on P at the right according to the law Rg : P → P, Rg (u)f = u(g(f )), g ∈ Aut (F ), f ∈ F. The diffeomorphisms ρ−1 (Ui ) ∩ P → Ui × Aut (F ) are used for verifying the condition (b) from the Definition 1.2.3. It follows that (P, π, M, Aut (F )) is a principal fibre bundle. It is called the principal bundle of frames associated to the vector bundle ξ = (E, p, M ). An element u ∈ P , being a linear isomorphism u : F → Fx , it can be identified to a basis in Ex , the image by u of a fixed basis in F . For F = IRn with the canonical basis ei = (0, ..., 1i , 0, ..., 0), u can be identified to (u(ei ), a basis of Ex . This identification explains the term of frame in x for u and the term of principal bundle of frames for (P, π, M, Aut (F )). The preceding construction can be applied, in particular, to the tangent bundle (T M, τ, M ) of M . The principal fibre bundle obtained on such a way is isomorphic to the frame bundle L(M ). The isomorphism is getting by taking ϕ : Aut (F ) → Aut (F ) as identity map and f : P → L(M ) the mapping which associates to the linear isomorphism u : IRn → Tx M the frame u(ei ) in Tx M. We shall identify those two principal fibre bundles by using the isomorphism just described.

1.6

Sections in vector bundles

Let be the vector bundle ξ = (E, p, M ) and U an open subset of M . A differentiable mapping s : U → E with the property p ◦ s = id U , where id means the identity, is called a section of ξ over U . We shall denote by S(U, ξ) or S(U, E) the set of all sections of ξ over U . We define the sum of two sections and the product of a section by a real function on U as follows: (s + s0 )(x) = s(x) + s0 (x), (as)(x) = a(x)s(x), s, s0 ∈ S(U, E), a ∈ F(U ).

(6.1)

The set S(U, ξ) with the operations (6.1) is an F(U )-module. The existence of sections on open sets in M one establishes by using the vectorial charts of ξ. If (U, ϕ, F ) is a vectorial chart of ξ, the mapping s : U → E, s(x) = ϕ−1 (x, f ) for a fixed elementf ∈ F , is a section of ξ over U . Let us notice that taking f as fα , α = 1, ..., m, the elements of a basis in F , one obtains the sections sα (x) = ϕ−1 (x, fα ) which are linearly independent. Conversely, given m linear independent sections ! over X U , we get a vectorial chart setting ϕ : p−1 (U ) → U × F , ϕ aα (x)sα (x) = α ! X x, aα (x)fα . α

14

Chapter 1

The sections of ξ = (E, p, M ) over M are called global sections. The existence of a global section is assured by the hypothesis that M is paracompact. It is getting by gluing together some local sections with the help of a partition of unity on M . If h : M 0 → M is a differentiable mapping and s is a section over U in the vector bundle ξ = (E, p, M ) then h∗ s : h−1 (U ) → h∗ E, (h∗ s)(x0 ) = (x0 , s(h(x0 ))) is a section over h−1 (U ) in the pull–back of ξ by h. Every morphism f : E → E 0 induces an F(M )-linear mapping F : S(M, E) → S(M, E 0 ), (F (s))(x) = f (s(x)). Conversely, every F(M )-linear mapping F : S(M, E) → S(M, E 0 ) defines a vector bundle morphism f : E → E 0 . The sections of the tangent bundle (T M, τ, M ) are called vector fields on M . We end this Section with some comments on G-structures. According to the remarks in the end of Section 3 and to the Theorem 1.2.2, a G-structure on M , that is a reduction of GL(n, IR), to G exists if and only if the tangent bundle (T M, τ, M ) admits a vectorial atlas whose structure functions are G-valued. In terms of local sections, this condition is equivalent with the existence of an open covering {Ui } of M and, for every open set Ui , the existence of a set of vector fields (Xa ), a = 1, ..., n, such that i

1◦ (Xa (x)) is a frame in x ∈ Ui , i

2◦ Xb (x) = g ji (x)ab Xa (one sums over a), x ∈ Ui ∩ Uj , j

i

◦

3 the matrix (g ji (x)ab ) is an element of G. The frames (Xa (x)) are called adapted frames to the considered G-structure. i   ∂ are Definition 1.6.1 A G-structure on M is called integrable if the frames ∂xi x adapted to it for every x ∈ M. Definition 1.6.2 Let ξ = (E, p, M ) be a vector bundle with the type fibre a linear space F of dimension m and G a Lie subgroup of GL(m, IR). One says that ξ admits a G-structure if there exists a reduction to G of the principal fibre bundle of frames of ξ. In other words, the vector bundle ξ admits a G-structure if there exists a vectorial atlas of it whose structure functions to be G-valued. In terms of local sections, the existence of a G-structure for ξ is equivalent to the existence o an open covering {Ui } of M and, for any open set Ui , to exist a set of sections si,a ∈ S(Ui , ξ), a = 1, ..., m, such that 1◦ si,a (x) is a frame in Ex , x ∈ Ui , 2◦ sj,b (x) = g ji (x)ab si,a (one sums over a), x ∈ Ui ∩ Uj , 3◦ the matrix (g ji (x)ab ) is an element of G. The existence of a G-structure for (E, p, M ) does not depend on the existence of a G-structure on M .

Chapter 2

Connections in Vector Bundles The theory of connections is an important field of differential geometry. In this chapter we present from this theory only the results which are useful in the geometry of vector bundles. We shall begin with the general notion of nonlinear connection in a vector bundle specializing it in view of applications to the theory of Finsler, Lagrange and generalized Lagrange spaces. The linear connections will appear as particular cases of nonlinear connections. One distinguishes several modalities of defining these connections, as well as some remarkable geometric objects associated to them.

2.1

Nonlinear connections in vector bundles

The notion of nonlinear connection has appeared in the study of Finsler spaces. The first global formulation of it is due to W. Barthel [20]. Then it was studied by J. Vilms [147], J. Grifone [42], T.V. Duc [33], etc. We present the concept of nonlinear connection and its properties on a way suggested by the afore mentioned papers. Let ξ = (E, p, M ) be a vector bundle with the type fibre F = IRm and pT : T E → T M the differential of mapping p, called the tangent mapping of p. The mapping pT is a p-morphism from tangent bundle of E, (T E, τE , E) to tangent bundle of M , (T M, τ, M ). The kernel of this p-morphism is a vector subbundle of the bundle T (T E, τE , E). It will be denoted by (V E, τV , E) [= Ker p and it will be called the vertical subbundle. Its total space is V E = Vu , where Vu = Ker pT u, u ∈ E. u∈E

Let i : V E → T E be the inclusion mapping. In the following constructions the local structures of the manifolds T M, E, T E, V E are involved. In order to examine these local structures, let us fix a local chart (U, ϕ) on M , such that U to be a domain of a vectorial chart (U, ϕ0 , IRm ) for ξ. 15

16

Chapter 2

We have the diffeomorphisms: ∼ ∼ T M U −→ ϕ(U ) × IRn , E U −→ ϕ(U ) × IRm , (1.1) ∼ T E U −→ ϕ(U ) × IRm × IRn × IRm . If x ∈ U we set x ¯ = ϕ(x) = ϕ(x), x ¯ = (xi ), i = 1, ..., n. A vector Xx tangent in the point x to M will be locally represented by the pair (x, y), where y = (y i ) is given by ∂ Xx = y i i · ∂x Let (V, ψ) be another local chart on M with the property U ∩ V 6= ∅ and (V, ψ 0 , IRm ) be a vectorial chart for ξ. We express the mapping ψ ◦ ϕ−1 in the form: (1.2)

0

0

0

xi = xi (x1 , ..., xn ), rank (∂xi /∂xj ) = n.

The components of the tangent vector Xx in the chart (V, ψ), given by Xx = y i

0



∂ ∂xi0



are expressed as functions of y i by 0

0

yi =

(1.3)

∂xi i y, ∂xi

because of 0

∂ ∂xi ∂ · = ∂xi ∂xi ∂xi0

(1.4)

Therefore, the transformations of coordinates on the differentiable manifold T M are of the form  0 i i0 1 n i0 i   x = x (x , ..., x ), rank (∂x /∂x ) = n, (1.5) i0   y i0 = ∂x y i , ∂xi The Jacobian matrix of the mapping (1.5) is

0

∂xi

∂xi

0

∂ 2 xi j

y ∂xi ∂xj

(1.6)

0

∂xi ∂xi



·



A point u ∈ E, E being the total space of a vector bundle ξ = (E, p, M ), is locally represented by the pair (xi , y a ) ∈ IRn × IRm , where xi are xi ◦ p, interpreting the coordinates (xi ) as real functions on M . According to the formulae (3.2)–(3.4) from Chapter 1, the transformations of coordinates on the differentiable manifold E are of the form: 0

0

0

xi = xi (x1 , ..., xn ), rank (∂xi /∂xi ) = n, (1.7)

0

0

y a = Maa (x)y a ,

0

rank (Maa (x)) = m.

,

Connections in Vector Bundles

17

The Jacobian matrix of these transformations is

0 0

∂xi

∂xi (1.8)

∂M a0 0

a y a Maa

∂xi





·



0

and the nonsingular matrices (Maa (x)) satisfy the relations: 0

0

ϕ−1 (x) ∈ U = V

Maa (x) = δaa ,

(1.9)

00

0

00

Maa0 (x0 )Maa (x) = Maa (x),

ϕ−1 (x) = ψ −1 (x) ∈ U ∩ V ∩ W.

Now, let us consider the vertical subbundle (V E, τV , E). A vector Xu tangent to E in the point u is locally represented by (xi , y a , X i , Aa ), where the elements (X i ) ∈ IRn ∂ ∂ and (Aa ) ∈ IRm are defined by Xu = X i i + Aa a · The tangent mapping pT ∂x ∂y has the local expression pT (x, y, X, A) = (x, X). So the local fibres of the bundle  ∂ ∼ (T E, pT , T M ) are isomorphic to {x}×IRm ×{X}×IRm −→ IR2m . Since pT = 0, ∂y a ∂ it follows that , (a = 1, ..., m) determine a local basis for Vu . Consequently, the ∂y a vertical distribution {u → Vu | u ∈ E} is integrable. The submanifold V E contains the elements which are locally represented by (x, y, 0, A). Therefore, the fibres of the vertical subbundle are isomorphic to IRm . Let p∗ T M be the pullback by p : E → M of the tangent bundle and let be p! : T E → p∗ T M , p!(Xx ) = (u, pTu (Xu )). This mapping is a morphism of vector bundles. It quickly follows that the mapping p! is onto and that its kernel satisfies the relations Ker p! = Ker pT = V E. Consequently, we have: Proposition 2.1.1 The following sequence of vector bundles over E, is exact: (1.10)

i

p!

0 −→ V E −→ T E −→ p∗ T M −→ 0.

Now we can give the definition of the notion of nonlinear connection. Definition 2.1.1 A nonlinear connection in the vector bundle ξ = (E, p, M ) is a left splitting of the exact sequence (1.10). According to this definition a nonlinear connection in the bundle ξ is a vector bundle morphism C : T E→V E such that C ◦ i = id V E . The kernel of the morphism C is a subbundle of the vector bundle (T E, τE , E), which will be denoted by (HE, τ , E) E HE

or, briefly, by HE and will be called the horizontal subbundle. It follows: Proposition 2.1.2 In the presence of a nonlinear connection the vector bundle (T E, τE , E) is the Whitney sum of the horizontal subbundle HE and of the vertical subbundle V E.

18

Chapter 2

Conversely, the existence of a subbundle HE of T E satisfying the condition T E = HE ⊕ V E implies the existence of a left splitting of the sequence (1.10), that is the existence of a nonlinear connection in the vector bundle ξ. Indeed, we can define in this case C on fibres, as the projection on the first term of the direct sum Hu E ⊕ Vu E, u ∈ E. Consequently, we have: Theorem 2.1.1 A nonlinear connection in the vector bundle ξ is characterized by the existence of a subbundle HE of the tangent bundle to E, (T E, τE , E) with the property T E = HE ⊕ V E. Corollary 2.1.1 A nonlinear connection in the vector bundle ξ is a distribution {u ∈ E → Hu E} on E, such that Tu E = Hu E ⊕ Vu E. Let us consider a nonlinear connection in the vector bundle ξ. Evidently, the restriction p! HE of the mapping p! to HE is an isomorphism of vector bundles. The component pT : HE → T M of the mapping p! HE is a p-morphism whose restrictions to fibres are isomorphisms. Hence, for any vector field X on M , there exists an horizontal vector field X h on E, that is a section in the horizontal subbundle HE, such that pT (X h ) = X. Definition 2.1.2 The vector field X h is called the horizontal lift of the vector field X. Using the inverse of the isomorphism p! HE we define the morphism of vector bundles D : p∗ T M → T E which, evidently, will satisfy the condition p!◦D = id p∗ T M . In other words, D is a right splitting in the exact sequence (1.10). It is clear that the bundle Im D coincides to the horizontal subbundle HE. The tangent bundle T E will decompose as Whitney sum of horizontal and vertical subbundles, too. We define the morphism C : T E → V E on fibres as being the identity on the vertical vectors (elements from Vu E) and zero on the horizontal vectors. It follows that C is a left splitting of the exact sequence (1.10). In addition, the mappings C and D satisfy the equality i ◦ C + D ◦ p! = id T E . These considerations show that we have: Theorem 2.1.2 A nonlinear connection in the vector bundle ξ is characterized by a right splitting of the exact sequence (1.10), D : p∗ T M → T E, p! ◦ D = id p∗ T M . The existence of the nonlinear connections is assured under natural geometric conditions. Theorem 2.1.3 In any vector bundle ξ = (E, p, M ), with the paracompact base M there exist nonlinear connections. Proof. Assuming M paracompact, from Theorem 1.3.1, it follows that the total space E is paracompact, too. Applying the Theorem 1.5.1, to the vector bundle (T E, τE , E) and to the subbundle V E we obtain the existence of a left splitting C in the exact sequence (1.10). On the same way, using Corollary 1.5.1, we obtain the existence of a horizontal subbundle suplementary to the vertical subbundle and, again we have assured the existence of a nonlinear connection in (E, p, M ).

Connections in Vector Bundles

2.2

19

Connection map associated to a nonlinear connection

As we have seen above, the local fibres of the vertical subbundles are isomorphic to IRm . And the local fibres of the vector bundle ξ = (E, p, M ) are isomorphic to IRm , as well. By composing these two isomorphisms one gets an isomorphism rn : Vu E → Ep(u) . The set of mappings ru defines a canonical isomorphism r between the vertical subbundle and the vector bundle p∗ E. According to the definition of the bundle p∗ E the following diagram is commutative: p∗ E ↓ p1 E

p2

→ p

→

E ↓p M

Let us consider a nonlinear connection on E, defined by a left splitting C of the exact sequence (1.10). Definition 2.2.1 The map K : T E → E, K = p2 ◦ r ◦ C is called the connection map associated to the nonlinear connection C. From this definition it follows that the connection map K is a p-morphism of vector bundles, whose kernel is the horizontal subbundle HE. Generally, the map K is nonlinear on the fibres of the bundle (T E, pT , T M ). Definition 2.2.2 The connection C is called homogeneous (resp. linear) if the connection map K associated to C is homogeneous (resp. linear) on the fibres of the bundle (T E, pT , T M ). We continue the considerations of local nature made in the previous section. Using (1.1) the exact sequence (1.10) takes the form: i

0 −→ ϕ(U ) × IRm × {0} × IRm −→ ϕ(U ) × IRm × IRn × (2.1.)

p!

×IRm −→ ϕ(U ) × IRm × IRn −→ 0. The map i has the local form (x, y, 0, A) → (x, y, 0, A) and the map p! has the local expression (x, y, X, A) → (x, y, X). The differentiable manifold p∗ E is locally diffeomorphic to ϕ(U ) × IRm × IRm and the isomorphism r one expresses by (x, y, 0, A) → (x, y, A). The map C has the local form (x, y, X, A) → (x, y, 0, Cϕ (x, y, X, A)), where Cϕ : ϕ(U ) × IRm × IRn × IRm → IRm is a linear mapping in the variables X and A. From the condition C ◦ i = id V E we get Cϕ (x, y, 0, A) = A. The last relation shows that the functions Cϕ0 (x, y, X, A) = Cϕ (x, y, X, A) − A do not depend on A, when we combine it with Cϕ0 (x, y, X, A) − Cϕ0 (x, y, X, A0 ) = Cϕ0 (x, y, 0, A − A0 ). The mapping K has the local representation Kϕ of the form: Kϕ (x, y, X, A)

= (p2 ◦ r)(x, y, 0, Cϕ (x, y, X, A)) = = p2 (x, y, Cϕ (x, y, X, A)) = (x, Cϕ (x, y, X, A)).

20

Chapter 2

But Cϕ (x, y, X, A) = A + Cϕ0 (x, y, X), where Cϕ0 does not depend on A and is linear with respect to X. Therefore, we can write: Kϕ (x, y, X, A) = (x, Aa + Nia (x, y)X i ).

(2.2)

The differentiable functions Nia (x, y), i = 1, ..., n; a = 1, ..., m defined on E are called coefficients of the nonlinear connection C. Theorem 2.2.1 A necessary and sufficient condition that a p-morphism K : T E→E to be a connection map associated to a nonlinear connection C is that its local expression be given by (2.2). Proof. The necessity of the condition was proved above. We now prove the sufficiency. Evidently, the mapping K locally given by (2.2) is linear on the fibres of the vector bundle (T E, τE , E) and preserves these fibres. Its differentiability follows from the differentiability of the functions Nia (x, y). According to the definition of the mapping p2 ◦ r we can write K = p2 ◦ r ◦ C, where C : T E → V E is locally defined by (x, y, X, A) → (x, y, 0, Aa + Nia (x, y)X i ). It quickly follows that (C ◦ i)ϕ (x, y, 0, A) = (x, y, 0, A), that is C ◦ i = id V E . Hence C is a left splitting of the exact sequence (1.10). q.e.d. Corollary 2.2.1 The connection C in vector bundle (E, p, M ) is homogeneous (resp. linear) if and only if its connection map K has the local expression (2.2), where the functions Nia (x, y) are homogeneous (resp. linear) in the variables y a . Proof. Taking into account the Definition 2.2.2, Theorem 2.2.1 and the local structure of the fibres of (T E, τE , E) the conclusion is immediate. In the hypothesis that the connection C is linear, the connection map associated to it takes the local form (2.3)

a Kϕ (x, y, X, A) = (x, Aa + Kbi (x)X i y b ), i = 1, ..., n0 a, b = 1, ..., m).

a The functions Kbi (x) defined on M will be called the Christoffel coefficients of the linear connection C. We seek for the rule of transformation of the set of functions Nia (x, y) to a change of local charts on M and on E. Using the matrix (1.8) we have the following transformations of local coordinates on the manifold T E: 0

0

(2.4)

0

0

0

a

0

xi = xi (x1 , ..., xn ), y a = Maa (x, y) , X i = 0

0

Aa =

0 ∂Maa a i y X + Maa Aa . i ∂x

The inverse of the mapping (1.7) can be written as follows: (2.5)

0

0

0

xi = xi (x1 , ..., xn ), y b = Mab0 (x0 )y a .

∂xi i X , ∂xi

Connections in Vector Bundles

21

Kϕ

Thus, we have (x, y, X, A) −→ (x, Aa + Nia (x, y)X i ). On the other hand, ! a0 i0 (2.4) Kε i ∂Ma (x) a i a0 a i0 a0 a ∂x −→ X , y X + Ma (x)A (x, y, X, A) −→ x , Ma y , ∂xi ∂xi ! i0 (2.5) i i0 a0 a i a0 a a0 0 0 ∂x x , Ma y X + Ma (x)A + Ni0 (x , y ) i X −→ ∂x ! ! 0 i0 ∂Maa (x) a a0 ∂x i b b 0 x , A + Ma0 (x ) y + Ni0 Xi . ∂xi ∂xi ( ) 0 i0 ∂Maa (x) a a0 ∂x b 0 It follows: Ma0 (x ) y + Ni0 = Nib . Therefore, we have: ∂xi ∂xi Proposition 2.2.1 The transformations of local coordinates in the vector bundle have as effect the following transformation of the coefficients of the nonlinear connection C: 0

(2.6)

0

∂Maa (x) a ∂xi a0 a = M y . (x)N − a i ∂xi ∂xi Now, we can formulate a new characterization of the nonlinear connection. 0

Nia0

Theorem 2.2.2 The existence of a nonlinear connection in ξ is characterized by the existence of a set of real functions (Nia ) i = 1, ..., n; a = 1, ..., m, defined on every domain of local chart on E, which on the intersections of the domains of local charts satisfy the equations (2.6). Proof. Indeed, we have seen before that a nonlinear connection C in the vector bundle ξ determines a set of functions (Nia ) with the required properties. Conversely, if we have the set of functions (Nia ) with the mentioned properties, we can define Kϕ by the formula (2.2) and the connection map K as the application which has the local representation Kϕ . The equations (2.6) assure the correctness of this definition of the application K. The splitting of the exact sequence (1.10) one obtains in the same manner as in the proof of Theorem 2.2.1. q.e.d. Proposition 2.2.2 Any homogeneous connection in the vector bundle ξ is a linear connection. Proof. According to the Theorem 2.2.2 and Corollary 2.2.1 an homogeneous connection in the vector bundle ξ is characterized by the set of functions (Nia (x, y)), differentiable and homogeneous with respect to y a . But, any real function, which is differentiable in 0 and homogeneous, is linear. Hence, the functions Nia (x, y) are linear with respect to y a , that is the considered connection is linear. Remark 2.2.1 The proof of the previous proposition shows that in order to have the concept of homogeneous nonlinear connection, which is essential in Finsler geometry it is necessary to avoid the null section in the vector bundle ξ. In other words, we need to consider the nonlinear connection in the bundle (E − 0, p E−0 , M ), where E − 0 is the differentiable manifold E from which we have deleted the image of the null section of ξ.

22

2.3

Chapter 2

Other characterizations of the nonlinear connections

Let X (M ) be the module of vector fields on M and X (E) be the module of vector fields of the total space of a vector bundle ξ = (E, p, M ). Assuming that ξ is endowed with a nonlinear connection C : T E→V E, C ◦ i = id V E , the morphism C induces a morphism C : X (E) → S(V E). We always denote by the same letter the morphisms between the spaces of sections induced by the morphisms between the vector bundles. We define a morphism v : X (E) → X (E) by v(X) = C(X) if X is vertical and v(X) = 0 if X is horizontal. It follows that v has the properties: 1◦ v(X (E)) ⊂ S(V E);

2◦ {v(X) = X} ⇐⇒ X is vertical.

Conversely, a morphism v : X (E) → X (E) with the properties 1◦ and 2◦ defines a nonlinear connection in the vector bundle ξ, because if we set S(HE) = Ker v we can see that S(HE) is suplementary in S(V E). So, we have proved: Theorem 2.3.1 A nonlinear connection in the vector bundle (E, p, M ) is characterized by an endomorphism v : X (E) → X (E) with the properties: 1◦ v(X (E)) ⊂ S(V E);

2◦ {v(X) = X} ⇐⇒ {X ∈ S(V E)}.

From the definition of the operator v we can see that v 2 = v, that is v is a projector, called the vertical projector associated to the nonlinear connection C. We notice that v coincides with the application C considered as morphism between modules of sections. Theorem 2.3.2 A nonlinear connection in the vector bundle (E, p, M ) is characterized by an endomorphism h : X (E) → X (E) with the properties: 1◦ h2 = h;

2◦ Ker h = S(V E).

Proof. If C is a nonlinear connection in the vector bundle ξ we set h = I − v, where v is the vertical projector of C and I is the identity on X (E). It follows h2 = h and Ker h = {X | h(X) = 0} = {X | X = v(X)} = S(V E). Conversely, given g with the properties 1◦ and 2◦ we put v = I − h. if X ∈ X (E), then the vector field v(X) = X − h(X) is vertical and {v(X) = X} ⇐⇒ {h(X) = 0} ⇐⇒ {X ∈ S(V E)}. Consequently, v has the properties from Theorem 2.3.1 and so it determines a nonlinear connection. It follows that h, having the properties 1◦ and 2◦ , determines a nonlinear connection. q.e.d. Corollary 2.3.1 The endomorphism h has the properties: 3◦ h(X (E)) ⊂ S(HE);

4◦ {h(X) = X} ⇐⇒ {X ∈ S(HE)}.

The map h is called the horizontal projector associated to the nonlinear connection C. The existence of a nonlinear connection C leads to the decomposition X (E) = S(HE) ⊕ S(V E). We set X = hX + vX, X ∈ X (E), preserving the notations X h and X v for the horizontal lift and vertical lift of X ∈ X (M ), respectively.

Connections in Vector Bundles

23

Theorem 2.3.3 A nonlinear connection in the vector bundle (E, p, M ) is characterized by an almost product structure P on the total space E whose distribution of the eigenspaces corresponding to the eigenvalue −1 coincides to the vertical distribution on E. Proof. Given a nonlinear connection C in the vector bundle (E, p, M ) we consider P = I −2v. It follows P 2 = I. Hence P is an almost product structure on E. Moreover, we have {P (X) = −X} ⇐⇒ {X ∈ S(V E)}. Conversely, an almost product structure P on E with the property {P (X) = −X} ⇐⇒ {X ∈ S(V E)} be given, we set 1 v = (I − P ). One easily obtains that v satisfies the required conditions and so it 2 determines a nonlinear connection on E. q.e.d. Finally, the following equations hold: P − 2h − I, P = h − v.

(3.1)

Continuing the study of a nonlinear connection we associate to it some new operators. Definition 2.3.1 A law of covariant derivative associated to a nonlinear connection C is an application D : X (M ) × S(E) → S(E), which maps every pair (X, A) ∈ X (M ) × S(E) into the section K ◦ AT ◦ X ∈ S(E). Denoting D(X, A) = DX A we have DX A = K ◦ AT ◦ X and DX A will be called the covariant derivative of the section A with respect to the vector field X. Proposition 2.3.1 The covariant derivative DX A has the local expression   a ∂A a + N (x, A) sa , (3.2) DX A = X i i ∂xi where sa are local linear independent sections of ξ and A = Aa sa . Proof. The section A : M →  E is defined locally by  x → (x, Aa (x)). The application a ∂A AT has the form (xi , X i ) → xi , Aa (x), X i , X i . Hence K(AT (Xx )) is locally i ∂x   ∂Aa i a i represented by xi , X + N (x, A)X . q.e.d. i ∂xi From (3.2) we infer Proposition 2.3.2 The covariant derivative associated to a nonlinear connection has the properties: 1◦ DX1 +X2 A = DX1 A + DX2 A 2◦ Df X A = f DX A, X1 , X2 ∈ X (M ), A ∈ S(E). 3◦ DX (f A) = Xf A + f DX A 42 If A1 x = 0, x ∈ M, then DX (A1 + A2 ) x = DX A1 x + DX A2 x . Let c : I ⊆ R → M, t → c(t) be a differentiable curve on M and the mapping c˙ : DA = Dc˙ A I → T M defined by c(t) ˙ = cT (t, 1), the tangent field along on c. Putting dt

24

Chapter 2

DA = 0. According to (3.2) dt the section A = Aa sa is parallel along c, with c(I) ⊂ U if and only if the following equations hold: we shall say that the section A is parallel along c if

dAa dxi + Nia (x, A) = 0. dt dt

(3.3)

2.4

Linear connections in vector bundles

We have defined a linear connection in the vector bundle ξ = (E, p, M ) as a nonlinear connection whose connection map K is linear on the fibres of the vector bundle (T E, pT , T M ). A first characterization of the linear connection is given in Corollary 2.2.1. The local repreentation of the connection map K associated to a linear connection is given in the formula (2.3). Using the Proposition 2.2.1 it follows that the a Christoffel coefficients Kbi (x) transform with respect to a change of local charts by the rule: 0

0

∂Mba ∂xi a0 a = M K − · a bi ∂xi ∂xi We can formulate the following characterization of linear connections:

(4.1)

0

0

Kba0 i0 Mbb

Theorem 2.4.1 A linear connection in the fibre bundle (E, p, M ) is characterized by a a set of real differentiable functions Kbi (x), (i = 1, ..., n; a, b = 1, ..., m) defined on every domain of local chart of the base manifold M which on the intersections of local charts it satisfies the equations (4.1). The other characteristic properties of the linear connections one gets by considering for every t ∈ IR∗ the morphism ht : E → E, u → tu, u ∈ Ex , x ∈ M. Hence ht restricted to the fibre Ex is the homothety of factor t of Ex . The mapping ht , locally defined by (x, y) → (x, ty) is differentiable for any t 6= 0. Its differential hTt : T E → T E has the local expression (x, y, X, A) → (x, ty, X, tA). From the equation p ◦ ht = p, taking its differential, we obtain the equality pT ◦ hTt = pT , which implies: (4.2)

hTt,u (Vu E) = Vht (u) E.

Theorem 2.4.2 A nonlinear connection in the vector bundle (E, p, M ) is a linear connection if and only if the connection map K has the property (4.3)

ht ◦ K = K ◦ hTt , for any t ∈ IR∗ .

Proof. The local expression of the application K is Kϕ (x, y, X, A) = (x, Aa + N a (x, y)X i ). According to the Corollary 2.2.1, the considered connection is linear if and only if the differentiable functions Nia (x, y) are linear in the variables y a . Assuming that the functionsNia (x, y) are homogeneous only with respect to y a , one easily verifies the condition (4.3) by using the local expressions of the mappings ht and hTt . Conversely, the equality (4.3) implies the homogeneity of the functions Nia (x, y) in the arguments y a and, since these functions are differentiable in the points (x, 0), it follows their linearity with respect to y a . q.e.d.

Connections in Vector Bundles

25

Remark 2.4.1 In the case when the nonlinear connection is defined in the bundle (E − 0, p E−0 , M ), the Theorem 2.4.2 provides a characterization of homogeneous connections. That is, we have: Theorem 2.4.3 A nonlinear connection in the bundle (E − 0, p E−0 , M ) is an homogeneous connection if and only if the connection map satisfies (4.3). Theorem 2.4.4 A nonlinear connection in the vector bundle (E, p, M ) is linear if and only if we have: hTt,u (Hu E) = Hht (u) E.

(4.4)

Proof. From the definition of the connection map K it follows, as we have remarked, that Ker K = HE. If the connection is linear, it results (4.3) and its consequence (4.4). Conversely, we assume that (4.4) holds and let be Au ∈ Tu E. We can write Au = hAu +vAu . Hence ht (K(Au )) = ht (K(vAu )), while K(hTt (Au )) = K(hTt (vAu )). The equality ht (K(vAu )) = K(hTt (vAu )) is verified by using the local expressions of the considered maps and taking into account (4.2). Remark 2.4.2 The equality (4.4) characterizes the homogeneous connections in the bundle (E − 0, p E−0 , M ). Another properties of linear connections are given. Theorem 2.4.5 A nonlinear connection in the vector bundle (E, p, M ) is linear if and only if one from the following three conditions is satisfied : 1◦ hTt ◦ v = v ◦ hTt ;

2◦ hTt ◦ h = h ◦ hTt ;

3◦ hTt ◦ P = P ◦ hTt ,

where v, h and P are the vertical and horizontal projectors and the almost product structure associated to the considered connection. Proof. The equivalence of those thre conditions is evident. We prove that 1◦ is equivalent to the linearity of connection. The relation (4.4) assures that 1◦ is verified for X ∈ S(H). The equality (4.2) and the equivalence {v(X) = X} ⇐⇒ {X ∈ S(V )}, show that 1◦ is satisfied also for X ∈ S(V ). If 1◦ holds, taking into account that Ker v = HE we deduce (4.4). Hence the considered connection is linear. q.e.d.

2.5

Covariant derivative associated to a linear connection

The covariant derivative of a section, in the case of a linear connection has the local expression:   a ∂A a b i + Kbi (x)A sa . (5.1) DX A = X ∂xi

26

Chapter 2

Proposition 2.5.1 The law of covariant derivative D associated to a linear connection has the properties: 1◦ DX+Y A = DX A + DY A, 2◦ Df X A = f DX A, 3◦ DX f A = Xf · A + f DX A, 4◦ DX (A + B) = DX A + DX B, X, Y ∈ X (M ), A, B ∈ S(E). 5◦ The application D is local. Namely, if A vanishes on an open set U ⊂ M , then DX A vanishes on U , for any X ∈ X (M ). Also, we have: Theorem 2.5.1 A linear connection in the vector bundle (E, p, M ) is characterized by an application D : X (M ) × S(E) −→ S(E), (X, A) −→ DX A having the properties 1◦ − 5◦ from the Proposition 2.5.1. Proof. The law of covariant derivative associated to a linear connection has the properties 1◦ − 5◦ , by the previous Proposition. Conversely, an application D : X (M ) × S(E) → S(E) with the properties 1◦ − 5◦ determines a linear connection in the vector bundle (E, p, M ). For proving this fact, let U be the domain of a local chart on M and (sa ) be m linear independent section on U . If A is a section of the bundle (E, p, M ) its restriction to U can be given in the form A = Aa sa , where Aa are m real functions ∂ defined on U . Let X be a vector field expressed on U by X = X i i · By using the ∂x properties 1◦ − 5◦ from the Proposition 2.5.1 we get  a  ∂A a b DX A = X i + K A sa bi ∂xi where we set: (5.2)

b (x)sb . D ∂ sa = Kai ∂xi

b The functions Kai (x) defined on every local chart by (5.2) satisfy the equation (4.1) on intersections of domains of local charts. Hence a linear connection is defined by them. q.e.d.

Proposition 2.5.2 Let D and D0 be two laws of covariant derivatives in the vector bundle (E, p, M ). There exists a section d in the vector bundle (L(T M, E; E), ρ, M ) such that: (∗)

0 DX A = DX A + d(X, A), X ∈ X (M ), A ∈ S(E).

Proof. From the properties 3◦ and 4◦ of the laws of covariant derivative we have 0 0 0 (DX − DX )(A + B) = (DX − DX )A + (DX − DX )B and 0 0 (DX − DX )(f A) = f (DX − DX )A.

Connections in Vector Bundles

27

0 In other words, D0 − D is a X (M )-linear mapping. It follows that DX − DX is defined by its action on a basis of local sections. Hence it determines a section in the vector bundle (L(E; E), ρ, M ). Thanks to the linearity with respect to X, the application D0 − D defines a section d in (L(T M, E; E), ρ, M ) and, evidently, the equation (∗) holds.

Proposition 2.5.3 Let D be a law of covariant derivative in the vector bundle (E, p, M ) and d a section in the vector bundle (L(T M, E; E), ρ, M ). Then, the application D0 defined by the equality (∗) from the Proposition 2.5.2 is a law of covariant derivative in (E, p, M ). Proof. By a straightforward calculation the properties 1◦ − 5◦ for D0 are verified. q.e.d. Let ξi = (Ei , pi , M ), i = 1, ., , , k, be k vector bundles each of them endowed with a law of covariant derivative Di , i = 1, ..., k. We associate to the vector bundle ξ1 ⊕ · · · ⊕ ξk (Whitney’s sum) the mapping D defined by (5.3)

1 k DX (A1 + · · · + Ak ) = DX A1 + · · · + DX Ak , Ai ∈ S(E), X ∈ X (M ).

One proves that D is a law of covariant derivative. To the bundle given by tensorial product ξ1 ⊗· · ·⊗ξk we associate the application D defined by

(5.4)

DX (A1 ⊗ · · · ⊗ Ak ) =

k X

i A1 ⊗ · · · × DX Ai ⊗ · · · ⊗ Ak ,

i=1

Ai ∈ S 0 (Ei ), X ∈ X (M ). This application can be extended to any section in the bundle ξ1 ⊗ · · · ⊗ ξk by the condition (5.5)

DX (f A + gB) = Xf · A + f · DX A + Xg · B + gDX B, A, B ∈ S(ξ1 ⊗ · · · ⊗ ξk ).

By direct calculation we prove that D is a law of covariant derivative in ξ1 ⊗ · · · ⊗ ξk . In the case when ξ1 = · · · = ξk = ξ = (E, p, M ) the formulae (5.4) and (5.5) give us a law of covariant derivative in the tensor bundle ⊗k ξ. Using (5.3) we get a law of covariant derivative in the tensor algebra of the vector bundle ξ, induced by a linear connection in ξ. Let (E, p, M ) and (E 0 , p0 , M 0 ) be two vector bundles endowed with linear connections whose law of covariant derivatives are D and D0 , respectively. Let us consider ω : M → L(E, E 0 ) a section in the vector bundle of the linear applications from E to E 0 . For any X ∈ X (M ) we set (5.6)

0 ˜ X ω)(A) = DX (D (ω(A)) − ω(DX A), A ∈ S(E).

In the equality (5.6) ω is considered as a linear application S(E) → S(E 0 ). By a direct ˜ is a law of covariant derivative in the vector bundle calculation, one verifies that D 0 0 0 L(ξ, ξ ). In particular, if E = M × IR, then S(E 0 ) = F(M ) and consider DX = LX , where LX is the Lie derivative LX f = Xf. One easily verifies that LX is a law of

28

Chapter 2

derivative in the trivial vector bundle (M × IR, pr1 , M ). In this case the formula (5.6) gives us a law of covariant derivative in the vector bundle ξ ∗ , the dual of the vector bundle ξ. Then, for ω ∈ (S 0 (E))∗ , we have: (5.7)

∗ (DX ω)(A) = LX (ω(A)) − ω(DX A) = Xω(A) − ω(DX A).

Now it is clear how we can extend a law of covariant derivative, induced by a linear connection in the vector bundle ξ = (E, p, M ), to the vector bundles obtained as tensor products of ξ and ξ ∗ , and more general, how one gets a law of covariant derivative in the tensorial algebra of the bundle ξ. For example, if ω ⊗ A is a section in the vector bundle ξ ∗ ⊗ ξ, called the adjoint vector bundle, we have: (5.8)

ad ∗ DX (ω ⊗ A) = (DX ω) ⊗ A + ω ⊗ DX A = DX ◦ ω ⊗ A − ω ⊗ A ◦ DX .

In the second hand, we interpret ω ⊗ A as linear application S(E) → S(E). This is ∼ possible, due to the isomorphism ξ ∗ ⊗ ξ −→ L(ξ, ξ). The formula (5.8) one extends by ∗ linearity to the arbitrary sections in ξ ⊗ ξ (sections which locally one expresses by linear combinations of sections of the considered type). We mention that the laws of covariant derivative in the tensorial algebra of the vector bundle ξ, obtained in the above, commute with the operation of contraction in this algebra.

2.6

Curvature of a linear connection

Let us consider a vector bundle ξ = (E, p, M ) together with a linear connection whose law of covariant derivative is denoted by D. The curvature R of the linear connection D one defines by: (6.1)

R(X, Y )A = DX DY A − DY DX A − D[X,Y ] A, Y ∈ X (M ), A ∈ S(E).

From (6.1) it follows that R : X (M ) × X (M ) × S(E) → S(E) is an application F(M )-linear in all three arguments. An useful interpretation of R is as follows: Definition 2.6.1 One calls q-form with values in the vector bundle ξ = (E, p, M ) a q-X (M )-linear mapping ψ, skew symmetric, which associates to the vector fields X1 , ..., Xq a section ψ(X1 , ..., Xq ) in the bundle ξ. The q-forms with the values in the bundle ξ will be called vectorial, while the usual q-forms will be called scalar. Identifying the adjoint bundle ξ ∗ ⊗ ξ to the bundle L(ξ, ξ) we can interpret the curvature R of a linear connection D as a 2-form with values in the adjoint bundle. Let (U, ϕ) be a local chart on the base manifold M and (U, ϕ0 , IRm ) a vectorial chart for the bundle ξ = (E, p, M ). Denote by (sa ) the dual frame of the frame determined by m local sections (sa ), linearly independent. Consequently, sa (sb ) = δba . If we put:   ∂ , ∂ sb = Rb a ij sa , (6.2) R ∂xj ∂xi

Connections in Vector Bundles

29

from (5.2) and (6.1), we get: a a ∂Kbj ∂Kbi c a c a − + Kbi Kcj − Kbj Kci . ∂xj ∂xi Therefore, the curvature R of a linear connection with the law of covariant derivative D can be locally expressed by

(6.3)

Rb a ij =

(6.4)

R=

1 (Rb a ij dxi ∧ dxj )sb ⊗ sa , or by 2 R = Rba sb ⊗ sa ,

(6.5)

1 where Ra b = Rb a ij dxi ∧ dxj are scalar 2-forms. 2 We remark that the 1-forms a ωb a = Kbi dxi ,

(6.6)

give by means of (6.3) ”the structure equations” of the considered connection: Rba = dωba + ωca ∧ ωbc .

(6.7)

Let IR∗ be the curvature given by the formula (6.1) for a law of covariant derivative D∗ , induced by D, in the vector bundle ξ ∗ . We have: ∗ ∗ ∗ ∗ R∗ (X, Y )ω = DX DY∗ ω − DY∗ DX ω − D[X,Y ] ω, ∀X, Y ∈ X (M ), ω ∈ (S(E)) .

(6.8)

By using (5.7) and (6.8), we get: (R∗ (X, Y )ω)A = −ω(R(X, Y )A), A ∈ S(L(ξ, ξ)).

(6.9)

But, the last formula can be written in the following form: R∗ (X, Y ) = −t R(X, Y ), ∀X, Y ∈ X (M ),

(6.10)

where t R(X, Y ) is the transposed operator of R(X, Y ). That is, t

R(X, Y ) : (S(E))∗ −→ (S(E))∗ , (t R(X, Y )ω)(A) = ω(R(X, Y )A).

To a linear connection D in the vector bundle (E, p, M ) we can associate an operator of differentiation of q-forms with values in ξ = (E, p, M ), defined by

(6.11)

q X ˆ i , ..., Xq )+ dΦ(X0 , ..., Xq ) = (−1)i DXi (Φ(X0 , ..., X i=0 X ˆ i , ..., X ˆ j , ..., Xq ), + (−1)i+j Φ([Xi , Xj ], X0 , ..., X 0≤i