Smooth Manifolds Let X be a nonempty set

JIA-MING (FRANK) LIOU

1. Smooth Manifolds

Let X be a nonempty set. The power set P (X) is the set of all subsets of X. A topology is a family T of elements of P (X) satisfying the following properties.

(1) ∅ and X belong to T .

(2) Any union of members of T is a member of T .

(3) Any finite intersection of membets of T is again a membet of T .

Elements of T are called open sets in X. The pair (X, T ) is called a topological space.

A topological space X is Hausdorff if for any x 6= y in X, there exist open set U, V with U ∩ V = ∅ in X so that x ∈ U and v ∈ V.

Definition 1.1. Let X, Y be topological spaces. A function f : X → Y is continuous if for every open set V in Y, f⁻¹(V ) is open in X.

A function f : X → Y between topological spaces is a homeomorphism if f is a bijection and both of f : X → Y and f⁻¹ : Y → X are continuous.

Definition 1.2. A family {U_α} of open sets in a topological space X is called an open cover ifS

αUα= X.

An n-dimensional smooth manifold is a Hausdorff space X such that there exist an open cover {U_α} together with a family of homeomorphisms ϕ_α : U_α → ϕ_α(U_α) ⊂ Rⁿ such that for any Uα∩ U_β 6= ∅, the map

ϕ_β◦ ϕ⁻¹_α : ϕα(Uα∩ U_β) → ϕ_β(Uα∩ U_β)

is a C^∞-diffeomorphism. The pair (U_α, ϕ_α) is called a local chart and the family of local charts {(Uα, ϕα)} is called an atlas on X.

Example 1.1. Every open subset of Rⁿ is an n-dimensional smooth manifold.

Let i : Rⁿ → Rⁿ be the identity map of Rⁿ. For any open set U, we denote iU the restriction of i to U. Then i_U is a global chart for U. Hence U is a smooth manifold.

Example 1.2. (The real projective space RPⁿ. ) On Rⁿ⁺¹\ {0}, we define an equivalence relation as follows. We say that x ∼ y if x = λy for some λ ∈ R. The set (Rⁿ⁺¹\ {0})/ ∼of equivalence classes RPⁿis a smooth manifold of dimension n. It is called the n-dimensional real projective space.

Let X be a topological space and R be an equivalence relation on X. The set of all equivalence classes is denoted by X/R. There is a natural map π : X → X/R defined by x 7→ [x]. We say that U is an open set in X/R if π⁻¹(U ) is open in X. The set Y with this topology is a topological space called the quotient space of X modulo R whose topology is called the quotient topology.

1

We equip RPⁿ the quotient topology. Then RPⁿ is a Hausdorff space. We denote π(x₁, · · · , x_n+1) by (x₁: · · · : x_n+1) for every (x₁, · · · , x_n+1) ∈ Rⁿ⁺¹\{0}. Let U_ibe the set of points (x1: · · · : xn+1) so that xi 6= 0. Then {U_i: 1 ≤ i ≤ n+1} forms an open cover of RPⁿ. Let ϕ_i: U → Rⁿdefined by ϕ_i(x₁ : · · · : x_n+1) = (x₁/x_i, · · · , x_i−1/x_i, x_i+1/x_i, · · · , x_n+1/x_i).

Then we can check that {(Ui, ϕi)} is an atlas for RPⁿ⁺¹. 2. Sheaf of Smooth Functions

Let U be an open set in X. A continuous map f : U → R is said to be smooth at a point p ∈ U if there exists a local chart (V, ϕ) around p such that V ⊂ U and f ◦ ϕ⁻¹: ϕ(U ) → R is smooth at ϕ(p). We say that f is smooth on U if it is smooth at every point of U. The set of all smooth functions on U is denoted by C_X^∞(U ) or C^∞(U ).

Proposition 2.1. Let U be an open set in X. The set C_X^∞(U ) forms an algebra over R.

For every pair of open sets V ⊂ U, we have a natural restriction map rU,V : C_X^∞(U ) → C_X^∞(V ) such that for any triple of open sets W ⊂ V ⊂ U, we have r_W,V ◦ r_V,U = r_W,U. If {U_i} is an open cover for an open set U, and if f, g ∈ C_X^∞(U ) so that f |_Ui = g|_Ui, then f = g. If {Ui} is an open cover of an open set U, and for each i, there exists f_i ∈ C_X^∞(Ui) so that f_i|_Ui∩Uj = f_j|_Ui∩Uj, then there exists f ∈ C_X^∞(U ) so that f |_Ui = f_i. This observation motivates the definition of sheaves on a topological space.

A presheaf of abelian groups (rings, algebras, vector spaces) on a topological space X is an assignment F such that

(1) For each open set U of X, there corresponds to a abelian group (rings, algebras, vector spaces) F (U ),

(2) For every inclusion of open sets V ⊂ U, there is a homomorphism rU,V : F (U ) → F (V ), subject to the following two conditions:

(a) r_U,U is the identity morphism on F (U ),

(b) for every triple of open sets W ⊂ V ⊂ U, we have rW,V ◦ r_V,U = rW,U.

If V ⊂ U is an inclusion of open set, for each s ∈ F (U ), we denote r_U,V(s) by s|_V. If F is a presheaf on X, elements of F (U ) are called sections of F over U for each open set U.

A presheaf F on X is a sheaf if

(1) (Locality) If {U_i} is an open cover for an open set U, and if s, t ∈ F (U ) so that s|Ui = t|Ui, then s = t.

(2) (Gluing) If {U_i} is an open cover of an open set U, and for each i, there exists si∈ F (U_i) so that si|_Ui∩Uj = sj|_Ui∩Uj, then there exists s ∈ F (U ) so that s|Ui = fi. Corollary 2.1. Let X be a smooth manifold. The assignment U 7→ C_X^∞(U ) is a sheaf of algebras over X.

This leads to the notion of ringed spaces.

Definition 2.1. A ringed space is a pair (X, O_X) where X is a topological space and O_X is a sheaf of rings on X.

Let U and V be open sets in X. Suppose f : U → R and g : V → R are smooth functions.

We say that (f, U ) and (g, V ) have the same germ at x ∈ U ∩ V if there is an open set W ⊂ U ∩ V containing x so that f = g on W. An equivalent class [(f, U )] is called a germ at x. The set of all germs at x forms an algebra C_X,x^∞ called the stalk of the sheaf C_X^∞at x.

For simplicty, we denote (f, V ) by f.

Let F be a sheaf on a topological space X. Let U, V be open neighborhoods of x in X.

Suppose that s ∈ F (U ) and t ∈ F (V ). We say that (s, U ) is equivalent to (t, V ) if there exists an open set W ⊂ U ∩ V with x ∈ W so that s|W = t|W. The set Fx of all equivalent classes is called the stalk of F at x.

3. Tangent bundles, Cotangent bundles, Vector bundles A tangent vector to X at p ∈ X is a linear functional δ : C_X,p^∞ → R such that

δ(f g) = δ(f )g(p) + f (p)δ(g).

The set of all tangent vectors to X at p forms a vector space denoted by TpX; it is called the tangent space to X at x. The cotangent space T_p^∗X to X at p is defined to be the dual space of TpX.

Proposition 3.1. Let X be an n-dimensional smooth manifold and p ∈ X. The tangent space TpX is an n-dimensional vector space.

Proof. Let (ϕ, U ) be a local coordinate system around x. Assume that ϕ(q) = (x¹(q), · · · , xⁿ(q)) for each q ∈ U. For each f ∈ C_X,p^∞ , we define

∂

∂xⁱ(p)[f ] = ∂(f ◦ ϕ⁻¹)

∂xⁱ (ϕ(p)).

Then we can check that

 ∂

∂xⁱ(p) : 1 ≤ i ≤ n



forms a basis for TpX. 

Corollary 3.1. The cotangent space T_p^∗X is an n-dimensional real vector space.

Proof. The cotangent space T_p^∗X has a natural basis {(dx_i)_p: 1 ≤ i ≤ n} dual to

 ∂

∂xⁱ(p) : 1 ≤ i ≤ n

 .

 The union T X =S

p∈XT_pX and T^∗X =S

p∈XT_p^∗X are called tangent bundle and cotan- gent bundle of X respectively. They are examples of vector bundles on X.

Let us study the tangent bundle for a moment. Denote the canonical projection T X → X by π. Since X is a smooth manifold, we can choose an atlas {(Uα, ϕα)} for X. We denote ϕ_α= (x¹_α, · · · , xⁿ_α). Hence on U_α, we define h_α: U_α× Rⁿ→ π⁻¹(U_α) by

h_α(p, a₁, · · · , a_n) =

n

X

i=1

a_i ∂

∂xⁱ_α(p).

Then {π⁻¹(U_α)} forms an open cover for T X and {(π⁻¹(U_α), h⁻¹_α )} is an atlas for T X.

Therefore T X is a 2n-dimensional smooth manifold. In fact, T X is an example of vector bundle over X.

Definition 3.1. A smooth family of real (complex) vector spaces over X is a smooth manifold E together with a smooth map π : E → X such that for each x ∈ X, the set π⁻¹(x) has a structure of real (complex) vector space. The set π⁻¹(x) is called the fiber of π over x, and X is called the base, and E is called the total space.

Example 3.1. Let X be a smooth manifold. The product space E = X ×Rⁿhas a structure of a smooth manifold. Let π : E → X defined by π(x, v) = x. This family is called a trivial family.

A smooth family of vector spaces is denoted by (E, π, X). A morphism from (E, π, X) to (E⁰, π⁰, X) is a smooth map ϕ : E → E⁰ such that π⁰◦ ϕ = π. An isomorphism between smooth family of vector spaces is defined in the obvious way, i.e. a morphism ϕ : (E, π, X) → (E⁰, π⁰, X) is an isomorphism if there is a morphism ψ : (E⁰, π⁰, X) → (E, π, X) so that ψϕ = 1_E and ϕψ = 1_E⁰.

Definition 3.2. A smooth family of vector spaces (E, π, X) is said to be trivial if it is isomorphic to a trivial family. A smooth family of vector spaces (E, π, X) is said to be a vector bundle if for each x ∈ X, there exists an open neighborhood U of x so that π⁻¹(U ) is trivial. A vector bundle (E, π, X) is said to have rank r if there exists an open cover {U_α} of X so that π⁻¹(Uα) is isomorphic to the trivial family Uα× R^r for all α.

Let π : E → X be a vector bundle on X. If U is an open set of X so that π⁻¹(U ) is trivial, then any choice of isomorphism from π⁻¹(U ) to U × R^r is called a local trivialization.

Example 3.2. The tangent bundle T X and cotangent bundle T^∗X are vector bundles of rank n = dim X on X.

A (smooth) section s of a vector bundle (smooth family of vector spaces) E over U is a smooth map s : U → E so that π ◦ s(x) = x for all x ∈ U. The set of all sections of E over U is denoted by Γ(U, E). On Γ(U, E), we define

(s + t)(x) = s(x) + t(x), (f s)(x) = f (x)s(x),

where s, t ∈ Γ(U, E) and f ∈ C_X^∞(U ). Elements of Γ(X, E) are called global sections of E.

Proposition 3.2. The set Γ(U, E) is a C_X^∞(U )-module.

Proof. We leave it to the reader as an exercise. 

This oberservation also leads to the notion of sheaf of modules on a ringed space:

Let (X, OX) be a ringed space. A sheaf of OX-module on X is a sheaf F on X such that for each open set U of X, the abelian group F (U ) is a O_X(U )-module, and for each inclusion of open set V ⊂ U, the restriction rU,V : F (U ) → F (V ) is compatible with the module structures via the ring homomorphism O_X(U ) → O_X(V ).

Definition 3.3. Let X be an n-dimensional smooth manifold, and 0 ≤ k ≤ n. A smooth k-form on an open set U of X is a section of the exterior bundle Λ^kT^∗X. The space Γ(U, Λ^kT^∗X) of k-forms on U is also denoted by Ω^k_X(U ). The space global sections of Λ^kT^∗X is simply denoted by Ω^k_X.

Corollary 3.2. Let X be a smooth manifold. The assignment U 7→ Ω^k_X(U )

is a sheaf of C_X^∞-module on X.

Let ω be a one-form on X. Then ω is a section of the cotangent bundle π : T^∗X → X on X. Let (x¹, · · · , xⁿ; U ) be a local coordinate system on X. Then we know that the map h : U × Rⁿ→ π⁻¹(U ) given by

h(p, a₁, · · · , a_n) =

n

X

i=1

a_i(dxⁱ)_p

is a local trivialization of T^∗X over U. Let ωU be the restriction of ω to U. Then ωU : U → π⁻¹(U ). We see that

h⁻¹◦ ω_U(p) = (p, ω₁(p), · · · , ω_n(p)), for some smooth functions ω₁, · · · , ω_n on U. Hence ω_U =P_n

i=1ω_i(p)dxⁱ. The one form ω_U on U is called the local representation of ω on U.

In general, let η be a k-form on X. Suppose that (x¹, · · · , xⁿ; U ) is a local coordinate system on X. Let η_U = η|_U. Then η_U has the formal expression

ηU = X

1≤i1<···<ik≤n

ηi1···ikdxⁱ¹ ∧ · · · ∧ dx^ik called the local representation of η on U.

Definition 3.4. A vector field V on a smooth manifold X is a section of the tangent bundle T X on X. If V, W are vector fields on X, we define [V, W ] = V W − W V.

If f is a mooth function on X, and V be a vector field on X, we define V (f ) to be the function p 7→ V (p)(f ).

Proposition 3.3. Let V, W, Z be vector fields on X, a, b ∈ R, and f, g be smooth functions on X. Then

(1) [V, W ] = −[W, V ],

(2) [aV + bW, Z] = a[V, Z] + b[V, Z],

(3) [[V, W ], Z] + [[W, Z], V ] + [[Z, V ], W ] = 0, (4) [f V, gW ] = f g[V, W ] + f V (g)W − gW (f )V.

Let V, W be vector fields on X, and ω be an one-form on X. Define dω by (dω)(V, W ) = V (ω(W )) − W (ω(V )) − ω([V, W ]).

Then we see that dω is a two-form on X.

Proposition 3.4. Let ω be an one-form on X. Suppose that (x¹, · · · , xⁿ, U ) is a local coordinate system on U and ωU =P

iωidxⁱ is a local representation of ω on U. Then dω_U =

n

X

i=1

dω_i∧ dxⁱ.

Proof. We leave it to the reader as an exercise. 

In general, suppose that η is a smooth k-form on X. Define the k + 1-form dη by dη(V1, · · · , Vk+1) =

n

X

i=1

(−1)ⁱ⁺¹Vi(η(V1, · · · , ˆVi, · · · , Vk+1))

=X

i<j

(−1)^i+jη([V_i, V_j], V₁, · · · , ˆV_i, · · · , ˆV_j, · · · , V_k+1).

Similarly, one can replace one-forms by any k-forms in Proposition 3.4 and prove similar results for k-forms. Therefore all the properties of differential forms on Rⁿ also work for differential forms on smooth manifolds.

Definition 3.5. Let R be a ring. A sequence of R-modules {Cⁱ} together with a family of R- linear maps dⁱ : Cⁱ→ Cⁱ⁺¹is called a cochain complex if dⁱ⁺¹◦ dⁱ= 0. The i-th cohomology module Hⁱ(C^•, d^•) of the cochain complex is the quotient R-module ker dⁱ/ Im dⁱ⁻¹.

Then we obtain a cochain complex

(3.1) Ω⁰_X −−−−→ Ω^{d 1}_X −−−−→ · · ·^d −−−−→ Ω^{d k}_X −−−−→ Ω^{d k+1}_X −−−−→ · · · .^d

The i-th cohomology group (space) H_dRⁱ (X) of this cochain complex is called the de Rham cohomology of the manifold X.

One of our goal in this class is to show the following theorem.

Theorem 3.1. Let X be any compact oriented connected Riemannian manifold. Then H_dRⁱ (X) are all finite dimensional real vector spaces.

Let X be an n-dimensional compact oriented Riemannian manifold. The i-th Betti number b_i(X) is the dimension of the de Rham cohomology space H_dRⁱ (X), i.e. b_i(X) = dim_RH_dRⁱ (X). The Euler characteristic of X is defined to be

χ(X) =

n

X

i=0

(−1)ⁱbi(X).

Let X be a compact oriented connected Riemannian surface. Let K be the curvature function on X. We will also discuss the famous Gauss-Bonnet theorem:

Z

X

Kdµ = 2πχ(X), where dµ is the volume form on X.

4. Integration on Manifolds

Let X be an n-dimensional manifold and ω be a smooth k-form on X. The support of ω is the closure of the set of points p ∈ X such that ω(p) 6= 0. The space of smooth k-forms with compact support is denoted by Ω^k_c(X).

Let U be an open set in Rⁿ. Now, assume that ω = f (x)dx¹ ∧ · · · ∧ dxⁿ is a smooth n-form on Rⁿso that the support K of ω is compact and contained in U. We define

Z

U

f (x)dx¹∧ · · · ∧ dxⁿ= Z

K

f (x)dx¹· · · dxⁿ, where the right hand side is the Riemann integral of the function f.

Let ϕ : U ⊂ Rⁿ→ Rⁿbe a smooth diffeomorphism from an open set U into the open set V = ϕ(U ) of Rⁿ. Suppose y = ϕ(x) for x ∈ U. The Jacobian of ϕ at p ∈ U is the following determinant:

J (ϕ)(p) = det ∂y^j

∂xⁱ(p)



1≤i,j≤n

.

Suppose ω is a one-form on a smooth manifold X. Let (U, ϕ) be a local coordinate system on X and ϕ = (x¹, · · · , xⁿ) on U. Suppose the support of ω is compact and contained in U.

Denote ωU = f (x)dx¹∧ · · · ∧ dxⁿ. We might define Z

U

ω = Z

ϕ(U )

f (x)dx¹∧ · · · ∧ dxⁿ.

Now suppose (V, ψ) is another local coordinate on X with ψ = (y¹, · · · , yⁿ) such that the support of ω is contained in V. Assume that ω_V = g(y)dy¹∧ · · · ∧ dyⁿ. To have a well-defined value forR

Uω, we need to show that Z

ϕ(U )

f (x)dx¹∧ · · · ∧ dxⁿ= Z

ψ(V )

g(y)dy¹∧ · · · ∧ dyⁿ.

In other words, we have to show that the integral of n-forms on a local piece of a manifold does not depend on the choice of local charts. To do this, we need a further assumption.

We need to assume that J (ψ ◦ ϕ⁻¹) > 0. We know that

f (x) = (g ◦ ψ ◦ ϕ⁻¹)(x)J (ψ ◦ ϕ⁻¹)(x).

From elementary calculus, Z

ψ(V )

g(y)dy¹· · · dyⁿ= Z

ϕ(U )

f (x)|J (ψ ◦ ϕ⁻¹)|dx¹· · · dxⁿ. If J (ψ ◦ ϕ⁻¹) > 0, then |J (ψ ◦ ϕ⁻¹)| = J (ψ ◦ ϕ⁻¹). Then

Z

ψ(V )

g(y)dy¹∧ · · · ∧ dyⁿ= Z

ψ(K)

g(y)dy¹· · · dyⁿ

= Z

ϕ(K)

(g ◦ ψ ◦ ϕ⁻¹)(x)|J (ψ ◦ ϕ⁻¹)(x)|dx¹· · · dxⁿ

= Z

ϕ(K)

(g ◦ ψ ◦ ϕ⁻¹)(x)J (ψ ◦ ϕ⁻¹)(x)dx¹· · · dxⁿ

= Z

ϕ(K)

f (x)dx¹· · · dxⁿ

= Z

ϕ(U )

f (x)dx¹∧ · · · ∧ dxⁿ.

In order to have a well-defined theory of integration of differential forms on manifolds, we need the notion of oriented manifolds.

Definition 4.1. We say that a smooth manifold X is orientable if there exists an atlas {(U_α, ϕ_α)} such that

J (ϕ_β◦ ϕ⁻¹_α ) > 0

for all α, β. If X is orientable, a (maximal) choice of such an atlas is called an orientation.

A manifold with an orientation is said to be oriented.

Let X be an n-dimensional oriented manifold. If ω is an n-form with compact support, the support of ω may not be contained in a local chart. Hence we do not know how to make sense of the integration of ω from the previous definition. To remedy this situation, we introduce the notion of partition of unity. Moreover, we assume that X is compact.

Let {Vα} be an open covering of a compact manifold. A finite family {ρ_i : 1 ≤ i ≤ m} of smooth functions is called a partition of unity subordinate to the covering {Vα} if

(1) Pm

i=1ρ_i= 1

(2) 0 ≤ ρi ≤ 1, and the support of ρ_i is contained in Vα for some α.

Proposition 4.1. Let X be a compact manifold and {V_α} be a covering of X. A partition of unity subordinate to {Vα} always exists.

Remark. From now on, we assume that X is a compact oriented n-dimensional smooth manifold.

Let ω be a smooth n-form on X. Let {(Uα, ϕα)} be an orientation of X. Using Proposition 4.1, we can choose a partition of unity {ρ_i : 1 ≤ i ≤ m} subordinate to the open covering {U_α}. Since the support of ρ_i is contained in some Vi = Vαi, the support of ρiω is contained in V_i. Since X is compact and oriented, the support of ρ_iω is also compact. Hence the integral R

Uiρiω is well-defined. We define Z

X

ω =

m

X

i=1

Z

Ui

ρiω.

Proposition 4.2. Let ω be an n-form on X. The integralR

Xω is independent of the choice of partition of unit.

Proof. The proof can be found in do Carmo’s book. We leave it to the reader.  One can also introduce the notion of manifolds with boundary. If X is an n-dimensional manifold with boundary ∂X, then ∂X is an n − 1-dimensional smooth manifold. When X is orientable, so is ∂X. Let i : ∂X → X be the inclusion map.

Theorem 4.1. (Stoke’s Theorem) Let ω be an n − 1 form on X. Then Z

∂X

i^∗ω = Z

X

dω.