1 The Tangent bundle and vector bundle - UH

1 Tangent bundle , vector bundles and vector fields by Min Ru 1 The Tangent bundle and vector bundle The aim of this section is to introduce the Tangent bundle T X for a differential manifold X. Intuitively this is the object we get by gluing at each point p X the corresponding Tangent space Tp X. The differentiable structure on X induces a differentiable structure on T X making it into a differentiable manifold of dimension 2 dim(X). The Tangent bundle T X. is the most important example of what is called a vector bundle over X(see the definition below). Review of the Tangent space Tp X: Let X be a smooth differential manifold of dimension m and let p X.

2 The Tangent space Tp X is a collection of Tangent vectors vp to X at the point p. A Tangent vector vp is a map vp : C (X) R such that (i) vp (af + bg) = avp (f ) + bvp (g), (ii) vp (f g) =. f (p)vp (g) + g(p)vp (f ). Let (U, ) a local coordinate for X at p, let Rm = Rm u1 ,..,um and write . = (x1 , .. , xm ). Then we have special Tangent vectors { k |p , 1 k m} (called the x partial derivatives).. k |p : C (X) R. x defined by (f 1 ). | p (f ) = | (p . xk uk . Then { |p , 1 k m} forms a basis for Tp X, moreover(from the proof), for every vp . xk Tp X, we can write m . vp (xk ) k |p . X. vp =. k=1 x Construction of the Tangent bundle T X: 1.)

3 Let X be a smooth differential manifold of dimension m. Let T X = p X Tp X = {(p, v) | p X, v Tp X}. Let : T X X be the natural projection map with : (p, v) 7 p. For a given point p X. the fiber 1 ({p}) of is the m-dimensional Tangent space Tp X at p. The triple (T X, X, ). is called the Tangent bundle of X. We can put a differentiable structure on T X making it into a differentiable manifold of dimension 2 dim(X) as follows: Let X be a differential manifold with maximal atlas A. Let x : U Rm in A be a chart for X and define U : 1 (U) Rm Rm by m X . U : (p, ak |p ) 7 (x(p), (a1 , .. , am )). k=1 xk Then it is easy to check that U is one-to one and U ( 1 (U)) is an open set in Rm Rm.

4 We now check that overlap(transition) maps are smooth maps. In fact, Let (U, x) and (V, y). be two charts in A such that p U V . Then the overlap(transition) map V ( U ) 1 : U ( 1 (U V )) Rm Rm is given by m m X y1 X ym (a, b) 7 (y x 1 (a), k | b x (a) k 1 , .. , | 1 b ). k x (a) k k=1 x k=1 x Since X is a smooth manifold, y x 1 is smooth, hence V ( U ) 1 is also smooth. Let A = {( 1 (U), U ) | (U, x) A}, then A is a C atlas. So T X is an 2m smooth manifold. It is trivial that the projection map : T X X is also smooth. 2. The Tangent bundle , cotangent bundle and the definition of general vector bundle . For each point p X the fiber 1 ({p}) is the Tangent space Tp X of X at p hence an m- dimensional vector space.

5 For a chart x : U Rm is A, we define U : 1 (U) U Rm by m X . U : (p, ak |p ) 7 (p, (a1 , .. , am )). k=1 xk Obviously U is a diffeomorphism. Further more, it has the following important property: the restriction of U to the Tangent space Tp X, p = U |Tp X : Tp X {p} Rm is given by m X . p : ak |p 7 (a1 , .. , am ), k=1. xk so it is a vector space isomorphism. The map U : 1 (U) U Rm is called a bundle chart. In summary: For a smooth manifold X, we get a triple (T X, X, ), which is called the Tangent bundle of X, where is a continuous surjective map(natural projection), T X is a smooth differential manifold of dimension 2 dim(X).

6 Further, it satisfies the following property: (i) For each p X, the fiber 1 ({p}) = Tp (X) is an m-dimensional vector space. (ii) For each p X there exists a bundle chart ( 1 (U), U ) (some book called it trivialization) such that U : 1 (U) U Rm is a smooth diffeomorphism and for all q U, the map q = U |Tq (X) : Tq (X) {q} Rm is a vector space isomorphism. A smooth map v : X T X is called a smooth vector field (or smooth section) if v(p) = p for each p X. Finally, motivated by the above construction, we introduce the following general defini- tion: 3. Definition 1: Let E and X be smooth manifolds and : E X be a smooth surjective map.

7 The triple (E, X, ) is called a (smooth) vector bundle of rank k over X if (i) For each p X, the fiber Ep = 1 ({p}) is a k-dimensional vector space. (ii) For each p X there exists a bundle chart ( 1 (U), U ) (some book called it trivialization) such that U : 1 (U) U Rk is a smooth diffeomorphism and for all q U, the map q = U |Eq : Eq (X) {q} Rk is a vector space isomorphism. vector bundles of rank 1 is also called the line bundle . The vector bundle of rank r over X is said to be trivial if there exists a global bundle chart : E X Rk . Definition 2: Let (E, X, ) be a vector bundle over X. A smooth map : X E is said to be a smooth section of the bundle (E, X, ) if (p) = p for every p X.

8 The set of all smooth sections is denoted by (X, E) or just (E). Definition 3: Let (E, X, ) be a vector bundle of rank k over X. Let {U } be an open covering of X and let : 1 (U ) U Rk be the trivialization. Then, on U U 6= , the composition map 1 : (U U ) Rk (U U ) Rk is of the form, for every p U U and b Rk , 1 (p, b) = (p, g (p)(b)). for some smooth map g : U U GL(k, R) where GL(k, R) is the set of k k non- singular matrices. The smooth GL(k, R)-valued maps {g } are called the transition func- tions for a vector bundle E. 4. Examples 1. Let E = X Rk . Then E is a vector bundle of rank k. In this case, the trivialization map is an identity map.

9 This bundle is called the trivial bundle . 2. Let E = T X = p X Tp (X). It is called the Tangent bundle , denoted by T X. The rank of this bundle is m (the dimension of T X as a manifold is 2m), where dim X = m. Let (U, U ) be a chart of X with coordinate functions x1 , .. , xm . Then it defines a trivialization U : 1 (U) U Rm by m X . U : (p, ak |p ) 7 (p, (a1 , .. , am )). k=1 xk We now calculate the transition functions. Let(U, U ), (V, V ) two charts on X, with coordinate functions x1 , .. , xm and y 1 , .. , y m respectively, where U V 6= . For every b = (b1 , .. , bm ) Rm , and p U V , m X . V 1 (p, b) = (p, bi |p ). i=1 y i Since, on U V , m X xj.

10 | p = | , j p y i i j=1 y x we conclude that, on U V , m m X m X X xj . V 1 (p, b) = (p, bi | p ) = (p, bi |p ). i=1 y i j=1 i=1 y i xj Hence, m m X x1 X xm U V 1 (p, b) = (p, bi , .. , bi ). i=1 y i i=1 y i This means the transition map gU V is, for every p U V , xi ! gU V (p) = | V (p) . y j 1 i,j m 5. 3. Besides the Tangent bundle T X above, we also have the cotangent bundle T X as follows: Consider a smooth manifold X of dimension m. The dual space to the Tangent space Tp X, p X, is called the cotangent space to X at p, denoted by Tp X. Suppose that x : U Rm . be a local coordinates for X at p, then { k |p , 1 k m} forms a basis for Tp X, x.