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) =

f(p)vp(g)+ g(p)vp(f). Let (U, φ) a local coordinate for X at p, let Rm = Rmu1,...,um and write

φ = (x1, . . . , xm). Then we have special tangent vectors {∂

∂xk|p, 1 ≤ k ≤ m} (called the

partial derivatives)∂

∂xk|p: C

∞(X) → R

defined by∂

∂xk|p (f) =

∂(f ◦ φ−1)

∂uk|φ(p .

Then {∂

∂xk|p, 1 ≤ k ≤ m} forms a basis for TpX, moreover(from the proof), for every vp ∈

TpX, we can write

vp =m∑

k=1

vp(xk)

∂

∂xk|p .

1.2. Construction of the tangent bundle TX:

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Let X be a smooth differential manifold of dimension m. Let

TX = ∪p∈XTpX = {(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

the fiber π−1({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

ψ̃U : π−1(U) → Rm ×Rm

by

ψ̃U : (p,m∑

k=1

ak

∂

∂xk|p) 7→ (x(p), (a1, . . . , am)).

Then it is easy to check that ψ̃U is one-to one and ψ̃U(π−1(U)) is an open set in Rm × Rm.

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

ψ̃V ◦ (ψ̃U)−1 : ψ̃U(π−1(U ∩ V )) → Rm ×Rm

is given by

(a, b) 7→ (y ◦ x−1(a),m∑

k=1

∂y1

∂xk|x−1(a) bk, . . . ,

m∑

k=1

∂ym

∂xk|x−1(a) bk).

Since X is a smooth manifold, y ◦ x−1 is smooth, hence ψ̃V ◦ (ψ̃U)−1 is also smooth. Let

A∗ = {(π−1(U), ψ̃U ) | (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.

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1.3. The tangent bundle, cotangent bundle and the definition of general vector

bundle.

For each point p ∈ X the fiber π−1({p}) is the tangent space TpX of X at p hence an m-

dimensional vector space. For a chart x : U → Rm is A, we define ψU : π−1(U) → U ×Rm

by

ψU : (p,m∑

k=1

ak

∂

∂xk|p) 7→ (p, (a1, . . . , am)).

Obviously ψU is a diffeomorphism. Further more, it has the following important property:

the restriction of ψU to the tangent space TpX, i.e. ψp = ψU |TpX : TpX → {p}×Rm is given

by

ψp :m∑

k=1

ak

∂

∂xk|p 7→ (a1, . . . , am),

so it is a vector space isomorphism. The map ψU : π−1(U) → U×Rm 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:

(i) For each p ∈ X, the fiber π−1({p}) = Tp(X) is an m-dimensional vector space.

(ii) For each p ∈ X there exists a bundle chart (π−1(U), ψU ) (some book called it

trivialization) such that ψU : π−1(U) → U × Rm is a smooth diffeomorphism and for all

q ∈ U , the map ψq = ψU |Tq(X) : Tq(X) → {q} × Rm 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:

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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

(i) For each p ∈ X, the fiber Ep = π−1({p}) is a k-dimensional vector space.

(ii) For each p ∈ X there exists a bundle chart (π−1(U), ψU ) (some book called it

trivialization) such that ψU : π−1(U) → U × Rk is a smooth diffeomorphism and for all

q ∈ U , the map ψq = ψU |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 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

covering of X and let ψα : π−1(Uα) → Uα ×Rk be the trivialization. Then, on Uα ∩ Uβ 6= ∅,

the composition map

ψα ◦ ψ−1β : (Uα ∩ Uβ) × Rk → (Uα ∩ Uβ) ×Rk

is of the form, for every p ∈ Uα ∩ Uβ and b ∈ Rk,

ψα ◦ ψ−1β (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.

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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∈XTp(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 dimX = m. Let

(U, φU) be a chart of X with coordinate functions x1, . . . , xm. Then it defines a trivialization

ψU : π−1(U) → U × Rm by

ψU : (p,m∑

k=1

ak

∂

∂xk|p) 7→ (p, (a1, . . . , am)).

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

b = (b1, . . . , bm) ∈ Rm, and p ∈ U ∩ V ,

ψ−1V (p, b) = (p,

m∑

i=1

bi∂

∂yi|p).

Since, on U ∩ V ,∂

∂yi|p=

m∑

j=1

∂xj

∂yi

∂

∂xj|p,

we conclude that, on U ∩ V ,

ψ−1V (p, b) = (p,

m∑

i=1

bi∂

∂yi|p) = (p,

m∑

j=1

m∑

i=1

bi∂xj

∂yi

∂

∂xj|p).

Hence,

ψU ◦ ψ−1V (p, b) = (p,

m∑

i=1

bi∂x1

∂yi, . . . ,

m∑

i=1

bi∂xm

∂yi).

This means the transition map gUV is, for every p ∈ U ∩ V ,

gUV (p) =

(

∂xi

∂yj

)

1≤i,j≤m

|φV (p).

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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,

p ∈ X, is called the cotangent space toX at p, denoted by T ∗pX. Suppose that x : U → Rm

be a local coordinates for X at p, then {∂

∂xk|p, 1 ≤ k ≤ m} forms a basis for TpX, i.e.

for every vp ∈ TpX. The dual basis to {∂

∂xk|p, 1 ≤ k ≤ m} is traditionally denoted by

{dxk |p, 1 ≤ k ≤ m}. (Sometimes, I may drop the subscript p from the notation.) Thus an

arbitrary element of T ∗pX is expressed as

m∑

k=1

akdxk |p for some ak ∈ R. The disjoint union

of all the cotangent space

T ∗X = ∪p∈XT∗pX

is called the cotangent bundle ofX. The cotangent bundle can be given a smooth structure

making it into a manifold of dimension 2 dim(X) by an argument very similar to the one for

the tangent bundle as above). It is easy to verify that the transition functions for T ∗X is

gUV (p) =

(

∂yi

∂xj

)

1≤i,j≤m

|φU (p).

4. Tensor Bundles. Consider the (r, s)−type tensor space

T rs (p) = Tp(x) ⊗ Tp(X) ⊗⊗T ∗

p (X) ⊗ · · · ⊗ Tp(X)∗,

where the first products for V is taken r times, and the second products for V ∗ is taken s

times. Let

T rs = ∪p∈XT

rs (p).

Then, similar above, we can show that T rs is a vector bundle, which is called an (r, s)-type

tensor bundle on X.

5. Bundle Operations. Let (E,X, π) be a vector bundle, we can define, in an obvious

way, the dual bundle (E∗, X, π), which is called the dual bundle. Similarly, let E,E ′ be two

vector bundles, we can define E ⊕E ′ and E ⊗E ′ the direct sum and the the tensor product

bundles.

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2 Smooth Vector fields

Let X be a smooth manifold of dimension m. A vector field v on X is a section of the

tangent bundle TX, ie v : X → TX such that π ◦ v(p) = p for every p ∈ X. In other words.

A vector field on X is a map v which assigns to each point p ∈ X a tangent vector

v(p) = vp ∈ Tp(X).

Let x : U → Rm be a local chart of X, and p ∈ U , then

v(p) =m∑

i=1

vp(xi)∂

∂xi|p .

The real-valued functions vi : U → R defined by vi(p) = vp(xi), 1 ≤ i ≤ m, are called the

components of v related to the local chart x : U → Rm.

A vector field is said to be smooth if v is a smooth section tangent bundle TX, i.e. v

is smooth as a map. It can be checked (we omit it here) that v is smooth if and only if its

components are smooth for all charts in some atlas for X. Denote by Γ(X) the set of all

smooth vector fields on X. We have the following algebra structure on Γ(X):

For v, w ∈ Γ(X), a ∈ R and f ∈ C∞(X):

(i) v + w ∈ Γ(X), i.e. (v + w)(p) = v(p) + w(p);

(ii) av ∈ Γ(X), i.e. (av)(p) = av(p);

(iii) fv ∈ Γ(X), i.e. (fv)(p) = f(p)v(p).

There is another way of thinking about vector fields: Recall that, for every p ∈ X, the

tangent vector vp map vp : C∞(X) → R such that (i) vp(af + bg) = avp(f) + bvp(g), (ii)

vp(fg) = f(p)vp(g) + g(p)vp(f). Since v assigns at each point p ∈ X a tangent vector vp, we

can define, for each f ∈ C∞(X), v(f) as a function on X by v(f)(p) = vp(f). If the vector

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field is smooth, then v(f) is also a smooth function on X for every f ∈ C∞(X). Hence, we

can think of a smooth vector field v ∈ Γ(X) as a map

v : C∞(X) → C∞(X)

by

f 7→ v(f),

where (vf)(p) = vp(f).

The smooth vector field

v : C∞(X) → C∞(X)

also satisfies the following properties: For every f, g ∈ C∞(X),

(i) v(af + bg) = av(f) + bv(g),

(ii) v(fg) = fv(g) + gv(f).

An exterior differential p-form is a section of∧p T ∗(X). In a local chart (U, (x1, . . . , xm)),

an exterior differential p-form

ω =∑

1≤j1<···<jp≤m

aj1···<jpdxj1 ∧ · · · ∧ dxjp,

where aj1···<jpare real valued functions on U .

Λp(X) will denote the vector space of exterior differential p-forms.

Γ(X) will denote the vector space of (smooth) vector fields.

T sr (X) denote the vector space of (r, s)-tensor fields.

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3 Lie bracket of smooth vector fields

Let v, w be two smooth vector fields (thought as maps from C∞(X) → C∞(X)). Define the

Lie bracket of v and w, denoted by

[v, w] : C∞(X) → C∞(X),

by

[v, w](f) = v(w(f)) − w(v(f)).

The Lie bracket has the following properties:

(i) (R-linearity): [au+ bw, v] = a[u, v] + b[w, v];

(ii) (skew-symmetry): [v, w] = −[w, v];

(iii) (Jacobi-identity) [u, [v, w]] + [w, [u, v]] + [v, [w, u]] = 0;

(iv) for every f, g ∈ C∞(X), [fv, w] = f [v, w] − (w(f))v, [v, gw] = g[v, w] + (v(g))w;

(v) In local coordinates, if v =∑m

i=1 vi ∂∂xi , and w =

∑mi=1w

i ∂∂xi , then

[v, w] =m∑

i,j=1

(

vj ∂wi

∂xj− wj ∂v

i

∂xj

)

∂

∂xi.

4 Exterior Differentials

Recall that an exterior differential p-form is a section of∧p T ∗(X). In a local chart (U, (x1, . . . , xm)),

an exterior differential p-form

ω =∑

1≤j1<···<jp≤m

aj1···<jpdxj1 ∧ · · · ∧ dxjp,

where aj1···<jpare real valued functions on U .

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Let

Λ(X) =m⊕

p=0

Λp(X).

It is called the algebra of exterior differential forms. Note that it has the structure of an

algebra with respect to wedge product “∧”.

We now define the exterior differential operator d : Λp(X) → Λp+1(X) as follows, for

every ω ∈ Λp(X), and for every X1, . . . , Xp+1 ∈ Γ(X),

dω(X1, . . . , Xp+1) =p+1∑

i=1

Xi(ω(X1, . . . , X̂i, . . . , Xp+1)

+∑

i<j

(−1)i+jω([Xi, Xj], X1, . . . , X̂i, . . . , X̂j, . . . , Xp+1).

We also have a local expression, a local chart (U, (x1, . . . , xm)), i.e. let

ω =∑

1≤j1<···<jp≤m

aj1···<jp∧ dxj1 ∧ · · · ∧ dxjp.

Then

dω =∑

1≤j1<···<jp≤m

daj1···<jp∧ dxj1 ∧ · · · ∧ dxjp.

For this expression we see that, for any η ∈ Λp(X),

d(η ∧ ξ) = dη ∧ ξ + (−1)pη ∧ dξ.

According to the definition, for f ∈ Λ0(X), we have d(f)(X) = X(f), and for ω ∈ Λ1(X),

we have

dω(X, Y ) = X(ω(Y )) − Y (ω(X)) − ω([X, Y ]).

Now let F : M → N be a smooth map, then we have d(F ∗ω) = F ∗(dω) for any differential

form ω on N .

Note that we can define the integration of a k-form with compact support on a k-

dimensional (orientable) manifold, and have the following statement of Stokes’ formula: Let

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M be a smooth compact manifold of dimension n with boundary, and ω is a (n− 1)-form on

M . Then∫

Mdω =

∫

∂Mω.

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