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 TX for a differential manifold X. Intuitively this is the object we get by gluing at each point p ∈ X the corresponding tangent space TpX. The differentiable structure on X induces a differentiable structure on TX making it into a differentiable manifold of dimension 2 dim(X). The tangent bundle TX is the most important example of what is called a vector bundle over X(see the definition below). 1.1. Review of the tangent space TpX: Let X be a smooth differential manifold of dimension m and let p ∈ X. The tangent space TpX 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(fg) = m m f(p)vp(g)+ g(p)vp(f). Let (U, φ) a local coordinate for X at p, let R = Ru1,...,um and write ∂ φ = (x1,...,xm). Then we have special tangent vectors { | , 1 ≤ k ≤ m} (called the ∂xk p partial derivatives) ∂ | : C∞(X) → R ∂xk p defined by ∂ ∂(f ◦ φ−1) | (f)= | . ∂xk p ∂uk φ(p ∂ Then { | , 1 ≤ k ≤ m} forms a basis for T X, moreover(from the proof), for every v ∈ ∂xk p p p TpX, we can write m ∂ v = v (xk) | . p p ∂xk p kX=1 1.2. Construction of the tangent bundle TX: 1 Let X be a smooth differential manifold of dimension m. Let TX = ∪p∈X TpX = {(p, v) | p ∈ X, v ∈ TpX}. Let π : TX → X be the natural projection map with π :(p, v) 7→ p. For a given point p ∈ X −1 the fiber π ({p}) of π is the m-dimensional tangent space TpX at p. The triple (TX,X,π) is called the tangent bundle of X. We can put a differentiable structure on TX 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 −1 m m ψ˜U : π (U) → R × R by m ∂ ψ˜ : (p, a | ) 7→ (x(p), (a ,...,a )). U k ∂xk p 1 m kX=1 −1 m m Then it is easy to check that ψ˜U is one-to one and ψ˜U (π (U)) is an open set in R × R . 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 −1 −1 m m ψ˜V ◦ (ψ˜U ) : ψ˜U (π (U ∩ V )) → R × R is given by m m −1 ∂y1 ∂ym (a, b) 7→ (y ◦ x (a), | −1 b ,..., | −1 b ). ∂xk x (a) k ∂xk x (a) k kX=1 kX=1 −1 −1 Since X is a smooth manifold, y ◦ x is smooth, hence ψ˜V ◦ (ψ˜U ) is also smooth. Let ∗ −1 A = {(π (U), ψ˜U ) | (U, x) ∈ A}, then A∗ is a C∞ atlas. So TX is an 2m smooth manifold. It is trivial that the projection map π : TX → X is also smooth. 2 1.3. The tangent bundle, cotangent bundle and the definition of general vector bundle. −1 For each point p ∈ X the fiber π ({p}) is the tangent space TpX of X at p hence an m- m −1 m dimensional vector space. For a chart x : U → R is A, we define ψU : π (U) → U × R by m ∂ ψ : (p, a | ) 7→ (p, (a ,...,a )). U k ∂xk p 1 m kX=1 Obviously ψU is a diffeomorphism. Further more, it has the following important property: m the restriction of ψU to the tangent space TpX, i.e. ψp = ψU |TpX : TpX →{p} × R is given by m ∂ ψ : a | 7→ (a ,...,a ), p k ∂xk p 1 m kX=1 −1 m so it is a vector space isomorphism. The map ψU : π (U) → U ×R is called a bundle chart. In summary: For a smooth manifold X, we get a triple (TX,X,π), which is called the tangent bundle of X, where π is a continuous surjective map(natural projection), TX is a smooth differential manifold of dimension 2 dim(X). Further, it satisfies the following property: −1 (i) For each p ∈ X, the fiber π ({p})= Tp(X) is an m-dimensional vector space. −1 (ii) For each p ∈ X there exists a bundle chart (π (U), ψU ) (some book called it −1 m trivialization) such that ψU : π (U) → U × R is a smooth diffeomorphism and for all m q ∈ U, the map ψq = ψU |Tq(X) : Tq(X) →{q} × R is a vector space isomorphism. A smooth map v : X → TX is called a smooth vector field (or smooth section) if π ◦ 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 π : E → X be a smooth surjective map. The triple (E,X,π) is called a (smooth) vector bundle of rank k over X if −1 (i) For each p ∈ X, the fiber Ep = π ({p}) is a k-dimensional vector space. −1 (ii) For each p ∈ X there exists a bundle chart (π (U), ψU ) (some book called it −1 k trivialization) such that ψU : π (U) → U × R is a smooth diffeomorphism and for all k q ∈ U, the map ψq = ψU |Eq : Eq(X) →{q} × R 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 ψ : E → X × Rk. Definition 2: Let (E,X,π) be a vector bundle over X. A smooth map σ : X → E is said to be a smooth section of the bundle (E,X,π) if π ◦ σ(p)= p for every p ∈ X. The set of all smooth sections is denoted by Γ(X, E) or just Γ(E). Definition 3: Let (E,X,π) be a vector bundle of rank k over X. Let {Uα} be an open −1 k covering of X and let ψα : π (Uα) → Uα × R be the trivialization. Then, on Uα ∩ Uβ =6 ∅, the composition map −1 k k ψα ◦ ψβ :(Uα ∩ Uβ) × R → (Uα ∩ Uβ) × R k is of the form, for every p ∈ Uα ∩ Uβ and b ∈ R , −1 ψα ◦ ψβ (p, b)=(p, gαβ(p)(b)) for some smooth map gαβ : Uα ∩ Uβ → GL(k, R) where GL(k, R) is the set of k × k non- singular matrices. The smooth GL(k, R)-valued maps {gαβ} 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. This bundle is called the trivial bundle. 2. Let E = TX = ∪p∈X Tp(X). It is called the tangent bundle, denoted by TX. The rank of this bundle is m (the dimension of TX as a manifold is 2m), where dim X = m. Let 1 m (U, φU ) be a chart of X with coordinate functions x ,...,x . Then it defines a trivialization −1 m ψU : π (U) → U × R by m ∂ ψ : (p, a | ) 7→ (p, (a ,...,a )). U k ∂xk p 1 m kX=1 We now calculate the transition functions. Let(U, φU ), (V,φV ) two charts on X, with coordinate functions x1,...,xm and y1,...,ym respectively, where U ∩ V =6 ∅. For every m b =(b1,...,bm) ∈ R , and p ∈ U ∩ V , m −1 ∂ ψV (p, b)=(p, bi i |p). i ∂y X=1 Since, on U ∩ V , ∂ m ∂xj ∂ i |p= i j |p, ∂y j ∂y ∂x X=1 we conclude that, on U ∩ V , m m m j −1 ∂ ∂x ∂ ψV (p, b)=(p, bi i |p)=(p, bi i j |p). i ∂y j i ∂y ∂x X=1 X=1 X=1 Hence, m 1 m m −1 ∂x ∂x ψU ◦ ψV (p, b)=(p, bi i ,..., bi i ). i ∂y i ∂y X=1 X=1 This means the transition map gUV is, for every p ∈ U ∩ V , ∂xi gUV (p)= j |φV (p). ∂y !1≤i,j≤m 5 3. Besides the tangent bundle TX 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 TpX, ∗ m p ∈ X, is called the cotangent space to X at p, denoted by Tp X. Suppose that x : U → R ∂ be a local coordinates for X at p, then { | , 1 ≤ k ≤ m} forms a basis for T X, i.e. ∂xk p p ∂ for every v ∈ T X. The dual basis to { | , 1 ≤ k ≤ m} is traditionally denoted by p p ∂xk p k {dx |p, 1 ≤ k ≤ m}. (Sometimes, I may drop the subscript p from the notation.) Thus an m ∗ k arbitrary element of Tp X is expressed as akdx |p for some ak ∈ R. The disjoint union k=1 of all the cotangent space X ∗ ∗ T X = ∪p∈X Tp X is called the cotangent bundle of X.