Differential Geometry Chin-Lung Wang

Differential Geometry

Chin-Lung Wang

ABSTRACT. This is the preliminary version of my course notes in the fall term of 2006 at NCU and 2012 at NTU. The aim is to pro- vide basic concepts in differential geometry for first year graduate students as well as advanced undergraduate students.

Contents

Chapter 1. Differentiable Manifolds 5

1. The category of C^k manifolds 5

2. Cut offs and the partition of 1 8

3. Tangent spaces 12

4. Tangent maps 15

5. Sub-manifolds and the Whitney imbedding theorem 18

6. Submersions and Sard’s theorem 23

7. Vector fields, flows, Lie derivatives and the Frobenius

integrability theorem 27

8. Existence, Uniqueness and Smoothness Dependence of

ODE 33

9. Exercises 35

Chapter 2. Tensors and Differential Forms 41

1. The Tensor Algebra 41

2. The Exterior Algebra 45

3. Cartan’s operator d 46

4. Lie derivatives on tensors 48

5. Cartan’s homotopy formula 49

6. Integration on forms 52

7. Manifold with boundary and Stokes’ theorem 53 8. De Rham cohomology and the De Rham Theorem 56

9. Exercises 63

Chapter 3. Riemannian manifolds 69

1. Riemannian structure 69

2. Covariant Differentiation and Levi-Civita Connection 72 3. Geodesic, Exponential Map and Riemann Normal

Coordinate 76

3

4. Riemann Curvature Tensor 82

5. Variation of Geodesics 84

6. Jacobi Fields 90

7. Space Forms 94

8. The Second Fundamental Form 96

9. Variation of Higher Dimensional Submanifolds 98

10. Exercises 103

Chapter 4. Hodge Theorem 115

1. Harmonic Forms 115

2. Hodge Decomposition Theorem 116

3. Bochner Principle 119

4. Fourier Transform 121

5. Sobolev spaces 122

6. Elliptic Operators and Garding’s Inequality 125 7. Proof of Compactness and Regularity Theorem 128

8. Exercises 130

Chapter 5. Basic Lie Theory 137

1. Categories of Lie groups and Lie algebras 137

2. Exponential map 141

3. Adjoint representation 143

4. Differential geometry on Lie groups 147

5. Homogeneous spaces 150

6. Symmetric spaces 154

7. Curvature for symmetric spaces 159

8. Topology of Lie groups and symmetric spaces 161

9. Exercises 164

Index 166

Chapter 1

D

M

1. The category of C^kmanifolds

Definition 1.1. A (topological) manifold M is a topological space which is (1) locally Euclidean (2) Hausdorff and (3) Second countable.

Here are some explanations of these concepts:

(1) M is locally Euclidean if for each point p ∈ M there is a open neighborhood U 3 p which is homeomorphic to an open set in R^d for some d∈ _{N. Let}

ϕ: U → ϕ(U) ⊂ _R^d

be such a homeomorphism. The components xⁱ : U → _{R of ϕ are} called the coordinate functions and the pair(U, ϕ) is called a (coor- dinate) chart of M at p. It is customary to identify φ with the (column vector) coordinate function

x= (x¹,· · · , x^d)^t.

(2) M is Hausdorff if for any p 6= q in M there are neighborhood U 3 p, V 3 q such that U∩V =_∅.

(3) M is second countable if there is countable basis for its topology.

Recall that a basis is a collection of open subset such that any open set can be written as a (possibly infinite) union of certain constituents from that collection.

Exercise 1.1. Show thatR^d (with the standard Euclidean topology) is a manifold by finding an explicit countable basis.

It is not a priori clear why condition (3) should be there. A possi- ble reason goes as follows: If a topological space M is Hausdorff and

5

second countable, then any subset S ⊂ M with the induced topol- ogy is also Hausdorff and second countable. In particular any lo- cally Euclidean subset inR^dis a manifold. Conversely we will prove later that any manifold as defined above is indeed a subspace inR^d (the Whitney Imbedding Theorem). Hence the abstract definition of manifolds does not really lead to anything outside Euclidean spaces.

Given a manifold M and two charts(Ua, φa) inR^da and (U_b, φ_b) inR^dbwith Ua∩U_b 6=∅, we form the coordinate transition function

φ_ab:=φa◦φ_b⁻¹: φ_b(Ua∩U_b) →φ_b(Ua∩U_b)

which is a homeomorphism. It is intuitively clear that we should have da =d_b, which will be the dimension of M. However, the only known proofs are by no means elementary, except in one case:

Exercise 1.2. Let R^d1 ∼= R^d2 (homeomorphic). If d1 = 1 show that d2 = 1. Investigate the case d1 = 2 and reduce the problem to the Jordan Curve Theorem.

The general case will be outlined later (c.f. Exercise1.19) by means of certain approximation theorems and ideas in homotopy theory.

In this course we are mainly interested in differentiable manifolds instead of general topological manifolds. We call a collection of charts {(Ua, φa)}_a∈A a C^k atlas of M if (1) the transition functions φ_ab’s are all C^kmappings for some fixed k ∈_N∪ {_∞}and (2)^S_a∈AUa = M.

Exercise 1.3. For a manifold with a C^k atlas, k ≥ 1, show that the dimension d =dim M is well defined on each connected component of M.

When a manifold M is equi-dimensional of dimension d, we usu- ally denote it by M^d, if no confusion with the cartesian product M×

· · · ×M is likely to occur.

Given a C^k atlas {(Ua, φa)}_a∈A on M, a chart (U, φ) is C^k related to it if both the transition functions φ◦φ⁻_a¹and φ⁻_a¹◦φare C^kfor all a∈ A. It is convenient to add all C^krelated charts into a given atlas.

Exercise 1.4. Show that the enlarged collection of charts{Uα, φα)}_α∈A

also forms a C^katlas. Moreover, it is a maximal atlas in the sense that any chart which is C^k related to it is already contained in it.

1. THE CATEGORY OF C^kMANIFOLDS 7

Definition 1.2. A C^k(differentiable) structure on M is a maximal atlas of C^k charts. A C^k manifold is a manifold together with a C^k struc- ture. When a C^k manifold is given, the term charts of it will always mean C^kcharts.

Formally the case k = 0 is simply a topological manifold. From the definition it is an immediately question whether it is possible to select from all charts a sub-collection which defines a C¹structure or even a C^kstructure for higher k. These are important and highly non- trivial problems in manifold theory. There are C⁰ manifolds which admit no C¹ structures. In later chapters we will address on some of these questions. For the moment, we will only remark that (1) A famous theorem Whitney says that any C¹ manifold indeed ad- mits (contains) C^∞ structures, (2) The C²condition is the minimum requirement to define the notion of curvature, a concept introduced by Gauss and Riemann which lead to the birth of modern differen- tial geometry, and will be vastly studied in this course. Thus in this course, differentiable manifolds will always mean C^∞ manifolds.

A function f : M^d →_{R is C}^kat p∈ M if f ◦x⁻¹is C^kat x(p) ∈ _R^d for one chart(U, x)which contains p. Since

f ◦x⁻¹

β = f ◦x⁻_α¹◦ (x_α◦x⁻¹

β ),

by the definition of C^k structure the notion of C^k is independent of the choice of charts. Denote by C^k(U)the space of functions that are C^kat all points in U.

Likewise a function f : M^m → Nⁿ between two C^k manifolds is called C^k if

y◦ f ◦x⁻¹ : x(f⁻¹(V) ∩U) ⊂_R^m →y(V) ⊂_Rⁿ

is C^k for any choice of charts (U, x) on M and (V, y) on N. It is enough to check it for any two special atlas. Denote by C^k(M, N)_the space of all such C^kfunctions.

A mapping f : M → N between two C^kmanifolds is a diffeomor- phism if f⁻¹ is well defined and both f and f⁻¹ are C^k. This is the notion of isomorphisms in the category of C^kmanifolds.

Exercise 1.5. For any C^k manifold M^d and p ∈ M, show that there are charts with x(U) = B0(r), the open ball of radius r inR^d, as well as charts with x(U) = _R^d.

Exercise 1.6. Consider M=R with one chart given by(_{R, φ})where φ(t) = t³. Show that this defines a C^∞ structure on M. Is M diffeo- morphic toR with the standard C^∞ structure(_{R, id})?

There could be many C^k structures on a manifold, but it is hard to find non-diffeomorphic ones. The set of equivalence classes of differentiable structures up to diffeomorphism is a delicate object for study, which again will be briefly discussed in later chapters.

2. Cut offs and the partition of 1

Are there any C^∞ functions on a C^∞ manifold M besides the con- stants? For each charts (U, x) the coordinate functions xⁱ’s are by definition C^∞ on U but it may not be possible to extend xⁱ to a C^∞ function on M.

One of the basic principles in differential geometry is try to (1) compute things locally via differential calculus and (2) find a way to patch local information together to get global results. This section establishes the existence of partition of unity which is the simplest tool in this regard.

Recall that a topological space M is paracompact if every open cover{U_α}_α∈A of it has a locally finite open refinement{V_β}_{β∈B, in} the sense that

(1) Local finiteness: for each p∈ M, there is a neighborhood U 3 p such that V_β∩U=∅ except possibly for a finite number of Uβ’s.

(2) Refinement: There is a map ρ : B → A such that V_β ⊂U_ρ(_β) for all β ∈ B. The map ρ may not be injective nor surjective.

A manifold is more than paracompact. In fact we have an easy but important

Lemma 1.3. Let M be a locally compact topological space which is Haus- dorff and second countable (e.g. a manifold), then M is σ compact. Namely,

2. CUT OFFS AND THE PARTITION OF 1 9

there is a countable sequence of increasing open sets{G_i}_i∈Nwith ¯G_icom- pact, ¯G_i ⊂G_i+1and M =^S∞_i=1G_i.

P^ROOF. Let{W_i}_i∈Nbe any given countable basis.

Exercise 1.7. Show that by removing those Wiwith noncompact clo- sure ¯Wi we still get a basis. (Notice that the Hausdorff condition is needed.)

Thus we may assume that ¯W_iis compact for all i.

Let G₁ =W₁. The set G_i is constructed inductively: Suppose that G_iis constructed. Since ¯G_iis compact and covered by Wj’s, there is a smallest j(_i) ∈_{N so that}

G¯_i ⊂W₁∪ · · · ∪W_j(i).

We then define Gi+1 =W₁∪ · · · ∪W_j(i). It remains to show that ¯G_i+1

is compact. This follows from

G¯_i+1 ⊂W^¯₁∪ · · · ∪W^¯_j(i)

since a closed set in a (finite union of) compact set is compact.  Lemma 1.4. Let M = ^S∞_i=1Gi be σ compact. Then every open cover {Uα}_α∈A has a countable locally finite refinement {Vj}_j∈N with ¯Vj being compact.

PROOF. For each i ∈ N, consider the open annulus Si:= G_i+1\G^¯_i−2. (We put G_i = _{∅ for i} ≤ 0.) Then ¯G_i\G_i−1is compact and contained in Si. It is covered by{U_α∩S_i}_α∈Ahence is covered by a finite num- ber of them. By putting together all these finite open sets we get a countable sequence{Vj}_j∈_N. Each Vjis of the form Uα∩Si, so

V¯_j =Uα∩S_i ⊂S^¯_i ⊂G^¯_i+1

is closed in a compact set. Hence ¯Vjis itself compact.

Finally, {Vj} is locally finite since if p ∈ Si then only those Vj’s constructed from Si−1, Siand Si+1may possibly intersect Sinontriv-

ially. 

Now we discuss cut off (or bump) functions . Let

f(t) =







e^−1/t for t>0, 0 for t≤_0.

Exercise 1.8. Show that f ∈ C^∞(_R)and f⁽ⁿ⁾(0) =0 for all n∈ _N.

The function

g(t) = ^f(t)

f(t) + f(1−t) = ¹ 1+e^1t−1−t1

is then C^∞ and non-decreasing with g(t) =0 for t ≤0 and g(t) = 1 for t≥1.

The function h(t) = g(₂+t)g(₂−t) is a cut off function with h =1 on[−1, 1]and h=0 outside(−2, 2). For a higher dimensional version we consider

ψ(x¹,· · · , x^d) =

∏

h(xⁱ) ∈C^∞(_R^d).

Then ψ=1 on[−1, 1]^dand ψ=0 outside(−2, 2)^d. Alternatively we may consider the radially symmetric function

ψ(x) = h(|x|) ∈C^∞(_R) which has ψ|_B

0(1) =1 and ψ|_Rd\B₀(2) =0.

In general for a continuous function f on a topological space M its support is defined to be

supp f = {p ∈ M | f(p) 6=0}.

For a closed set B ⊂ M, a cut off function for B is a non-negative continuous function f such that supp f = B. The functions ψ above are special C^∞ cut off functions of standard cube and closed balls.

Definition 1.5. Given an open cover {Uα}_α∈A of a C^k manifold M, a partition of unity subordinate to{U_α}is a countable collection of C^k functions{_ψj}_j∈N on M such that

(1) 0≤ψj ≤1 for all j.

(2){supp ψj}is a locally finite (closed) refinement of{Uα}. (3)∑j∈Nψj(p) =1 for all p ∈ M.

2. CUT OFFS AND THE PARTITION OF 1 11

There will be no convergence issue in (3) since by (2) the sum will be a finite sum over a neighborhood of any point p.

Theorem 1.6(Existence of Partition of Unity). Let M be a C^kmanifold with{Uα}_α∈Aan open cover. Then there is a C^kpartition of unity{ψ_i}_j∈N subordinate to {Uα} with supp ψ_j being compact. Without the compact support requirement we may label the partition of unity by the same set A with ψ_α 6≡ 0 for at most a countable subset of A.

P^ROOF. Let M=^S∞_i=1G_ias given by Lemma1.3. We will modify the proof of Lemma1.4to construct{_ψj}_.

For each i∈ N, ¯Gi\Gi−1is compact and contained in Si =Gi+1\G^¯i−2. For each p ∈ G^¯i\Gi−1, let (Wp, x) be a chart at p such that Wp ⊂ Uα∩Si for some α ∈ A and x(Wp) = B0(3). Let Vp = x⁻¹(B^¯0(2)). Define a C^kcut off function for ¯Vp ⊂Wp ⊂Uαby

ψ¯p =







ψ◦x on Wp, 0 on M\Wp.

There is a finite subcover of the open cover {Vp} of ¯Gi\Gi−1. By putting together all such finite open sets for all i ∈ N, we get the desired locally finite refinement{V_j}_j∈Nas in Lemma1.4.

Let ¯ψ_jbe the corresponding cut off function for V_j. For each p ∈ M, there is a (finite number of) ¯ψ_j with ¯ψ_j(p) 6= 0, hence we may define

ψ_j = ^ψ¯j

∑iψ¯i

∈ C^∞(M),

which clearly satisfies∑jψ_j = 1 with supp ψ_j =supp ¯ψ_j = V^¯_j being compact.

For the last statement, for each α∈ A, we may simply let ψα =

∑

V¯_j⊂U_α

ψj.

Here ψα ≡0 if no such i exists. The proof is complete.  Exercise 1.9. Investigate the theorem for the case when M =_{R with} the open cover given by a single set U = M=_R.

Exercise 1.10. Let A (resp. U) be a closed (resp. open) set in a C^k manifold M with ¯A ⊂ U. Show that there exists f ∈ _C^k(M) such that f|_A ≡1 and f|_M\U ≡0. Is that possible to make supp f = A?

3. Tangent spaces

It is a priori not obvious how to generalized the concept of tan- gent vectors to manifolds. In fact this is a challenge problem for C⁰ manifolds. We will give two definitions of it for C^k manifolds with k ∈_N∪ {_∞}.

Let p ∈ _R^d_{and X} ∈ _R^dbe a vector. For a C¹ function f defined near p, the directional derivative

X f :=D_Xf(p) = ^d

dtf(p+tX)

t=0=lim

t→0

f(p+tX) − f(p) t

is defined, which is a derivation (first order differential operator) in the sense that

(1) Linearity: X(a f +bg) = aX f +bXg and (2) Lebnitz rule: X(f g) = (X f)g(p) + f(p)Xg.

Conversely, it is interesting to see whether a derivation deter- mines a vector. We will see shortly that this is indeed the case for derivations on C^∞ functions.

For a C^kmanifold, denote by C^k_pthe space of germs of C^kfunctions at p. It consists of functions which are defined on some neighbor- hood of p and two functions f , g are identified if f|_U =g|_{U for some} U 3 p.

Definition 1.7. Let M be a C^k manifold and p ∈ M. The Zariski tan- gent space DpM is the vector space which consists of all derivations X : C^k_p →_R.

For any chart(U, x)at p, partial derivatives ∂/∂xⁱ|_pare examples of tangent vectors: For f ∈C^k_p,

∂

∂xⁱ  

pf := ^∂(f ◦x⁻¹)

∂xⁱ (x(p)).

The following lemma is a useful substitute of the Taylor expan- sion especially for functions that has only limited differentiability.

3. TANGENT SPACES 13

Lemma 1.8. Let f ∈ C^k(B₀(r)). Then f(x¹,· · · , x^d) =

∑

xⁱg(x¹,· · · , x^d) with gi ∈ _C^k−1(B0(r))and gi(0) = ∂ f/∂xⁱ(0).

P^ROOF. By the Fundamental Theorem of Calculus, f(x) − f(0) =

Z ₁

0

d

dtf(tx)dt=

∑

Z ₁

0

∂ f

∂xⁱ(tx)xⁱdt

=

∑

xⁱ Z ₁

0

∂ f

∂xⁱ(tx)dt =

∑

xⁱg(x),

where g_i(x) = Z ₁

0

∂ f

∂xⁱ(tx)dt ∈ C^k−1(B₀(r))and g(0) = ^{∂ f}

∂xⁱ(0)as ex-

pected. 

Theorem 1.9. For a C^∞ manifold with (U, x) a chart at p, the partial derivatives form a basis of DpM. Indeed for any X ∈ DpM,

X =

∑

X(xⁱ) ^∂

∂xⁱ   p.

PROOF. Since X(1) = X(1·1) = X(1) ·1+1· X(1) = 2X(1), we have X(1) = 0, hence X(a) = aX(1) = 0 for any constant a.

For simplicity of notations we assume that x(p) = 0. Then for any f ∈ C^∞_p,

X f =X(f − f(p)) =X(

∑

=

∑

=

∑

=

∑

This proves the theorem. 

The proof (and the theorem) fails for C^k manifolds if k < _{∞ be-} cause then gi ∈ _C^k−1only and the second line about the Lebnitz rule does not make sense. This can be analyzed in purely algebraic terms:

Proposition 1.10. For any C^kmanifold M, k ∈ {0, 1, 2,· · · ,∞}, DpM ∼= (mp/m²_p)^∗.

Where mp = {f ∈ C^k_p | f(p) = 0} is the maximal ideal at p and m²_p consists of all finite sum of products∑ fig_i with f_i, g_i ∈ mp.

PROOF. Let X ∈ DpM. X is by definition a linear functional X : mp → R. To show that X induces a map mp/m²_p → R we need to verify that X|_m2

p = 0. But this follows from the Lebnitz rule readily:

For f_i, g_i ∈mp,

X(

∑

∑

Conversely given ψ : mp/m²_p → R we claim that Xψf := ψ(f − f(p))defines a derivation Xψ on C^k_p. Indeed,

Xψ(_{f g}) = _ψ(_{f g}− _f(_p)_g(_p))

=ψ((f − f(p))(g−g(p)) + (f − f(p))g(p) + f(p)(g−g(p))

= (Xψf)g(p) + f(p)Xψg

where we use the fact that(f − f(p))(g−g(p)) ∈m²_p.  Exercise 1.11. Let M be a C^k manifold. Show that

dim DpM=dim mp/m²_p=







dim M if k =_∞,

∞ if k <_∞.

(Hint: For k=1, study functions f(x) = (x¹)^afor 1 <a<2.) Let(U, x)and(V, y)be two charts at p, then for any f ∈ C^∞_p,

∂

∂xⁱ  

pf = ^∂(f ◦x⁻¹)

∂xⁱ (x(p)) = ^∂(f ◦y⁻¹◦y◦x⁻¹)

∂xⁱ (x(p))

=

∑

j

∂(f ◦y⁻¹)

∂y^j

(y(p))^∂y

j

∂xⁱ(x(p)) =

∑

j

∂y^j

∂xⁱ(x(p)) ^∂

∂y^j   pf .

4. TANGENT MAPS 15

Thus two vector representations

∑

∑

charts correspond to the same vector precisely when their coefficients satisfy the transformation rule

b^j=

∑

i

aⁱ∂y^j

∂xⁱ(x(p)).

This condition only requires C¹structure, so we make the

Definition 1.11. The tangent space TpM for a C^k manifold M with k ≥ 1 consists of compatible systems of vectors in each chart at p which satisfy the transformation rule.

4. Tangent maps

From now on we work in the C^∞ category. In particular TpM = DpM and tangent vectors are precisely derivations. Given f : M → N a C^∞ map and p ∈ M, we define the tangent map d fp : TpM → T_f(p)N via

(d fpX)h=X(h◦ f), where X ∈ TpM and h∈ C^∞_f(p).

The tangent map is indeed the generalization of derivative of a map in Calculus and is also denoted by D fp, D f(p), d f(p), f∗p and perhaps the most commonly used f⁰(p). It is the linearization (first order approximation) of the original map.

As in Calculus, d fp is linear and satisfies the chain rule. Namely for g : N →S be another C^∞ map we have

d(g◦ f)_p =dg_f(p)◦d fp.

The proof in Calculus is a bit tricky, but the proof now is completely formal: For X ∈ TpM and h∈ C_{g f}^∞_(p),

(d(g◦ f)_pX)h= X(h◦g◦ f) = (d fpX)(h◦g) = dg_f(p)(d fpX)h.

Exercise 1.12. Show that in charts (U, x) at p ∈ M and (V, y) at f(p) ∈ N, d fp is represented by d ˜f_x(p) with Jacobian matrix

∂ ˜f^j

∂xⁱ



where ˜f=y◦ f ◦x⁻¹: x(U∩ f⁻¹(V)) → _R^{dim N}_{. Namely} d fp

 ∂

∂xⁱ   p



=

∑

j

∂ ˜f^j

∂xⁱ(x(p)) ^∂

∂y^j   p.

Two special cases with one of the manifolds beingR are particu- larly interesting.

Example 1.12. The first is the total differential d f of a function f : M→ R. Let y be the coordinate of R, then d fpX = a∂/∂y|_p for some a.

By plug in h =y in the definition of d fp we get a = X f . Since there is only one basis ∂/∂y|_p for TpR, following the usual convention we denote a vector inR simply by its coefficient. Also we drop the point p if no confusion is likely to occur. Then

d f(X) = X f .

Each coordinate xⁱ is a C^∞ function at p and we compute dxⁱ ∂

∂x^j



= ^∂x

i

∂x^j =δⁱ_j.

That is, the differentials dxⁱ’s form a dual basis of the cotangent space T_p^∗M := (TpM)^∗ =Hom(TpM,R)

with respect to the basis ∂/∂xⁱ’s. Moreover, d f =

∑

i

∂(f ◦x⁻¹)

∂xⁱ dxⁱ.

This follows by looking at the values of both sides on the basis vec- tors ∂/∂xⁱ’s.

Example 1.13. The second example is the tangent vector γ⁰(t) = dγt

∂

∂t



of a parameterized C¹curve γ : (a, b) ⊂ R→ M with parameter t ∈ (a, b). Here, following the usual convention, we identify γ⁰(t) ≡ dγt

as its image vector since there is one basis vector ∂/∂t on Tt(a, b). In local chart(U, x), the curve is represented by

t7→ γ˜(t) :=x◦γ(t) = (x¹(t),· · · , x^m(t))^t

4. TANGENT MAPS 17

and

γ⁰(t) =

∑

i

(xⁱ)⁰(t) ^∂

∂xⁱ   γ(t).

The way d fp approximates f is best explained through the in- verse/implicit function theorem. We start with the simplest notions, namely the injectivity and surjectivity of f .

Definition 1.14. A map f ∈ C^∞(M, N)is an immersion at p∈ M if the linear map d fpis injective, it is a submersion at p if d fpis surjective.

Lemma 1.15. Let f : M^m → Nⁿ be C^∞.

(1) If f is an immersion at p (so m ≤n), then there are charts(U, x) at p and (V, y) at f(p) such that f|_U is represented by yⁱ =

˜fⁱ(x) = xⁱ for i =1, . . . , m and yⁱ =0 for i≥m+1. That is, U is a coordinate slice of V.

(2) If f is a submersion at p (so m≥ n), then there are charts(U, x) at p and (V, y) at f(p) such that f|_U is represented by yⁱ =

˜fⁱ(x) = xⁱ for i = 1, . . . , n. That is, ˜f is a coordinate projection from U to V.

P^ROOF. For (1), start with any charts such that f(U) ⊂ V. Since d ˜f_x(p) is injective, it has rank m. By reordering of coordinates yⁱ’s’

we may assume that the first m×m square matrix

∂ ˜fⁱ

∂x^j(x(p))

^m

i,j=1

is invertible. Denote by y=^{ y}

1

y²



= ˜f(x) = ^{ f}

1(x) f²(x)



underRⁿ =_R^m×_R^n−m, then by the inverse function theorem y1 = f¹(x)is invertible over some x(p) ∈W ⊂x(U). By using(x⁻¹(W), y¹= f¹◦x)as a new chart at p and let g = f²◦ (f¹)⁻¹, the map f becomes a graph of g:

y=

 y¹ g(y¹)

 . By a simple change of coordinates

z =^{ z}

1

z²



=

 y¹ y²−g(y¹)



near f(p) ∈ N, we get the desired coordinate charts.

For (2), again we start with any charts with f(U) ⊂ V. Since d ˜f_x(p) is surjective, it has rank n. By reordering of coordinates xⁱ’s we may assume that the first n×n square matrix

∂ ˜fⁱ

∂x^j

(x(p))

n i,j=1

is invertible. Denote by x = (x¹, x²)^t under R^m = _Rⁿ×_R^m−n and consider the map F : x(U) → y(V) ×_R^m−n defined by

h y x²

i

:= F(x¹, x²) =

 ˜f(x¹, x²) x²

 . Since

dF_x(p) =

"

D1˜f D2 ˜f 0 idm−n

#

is invertible, the inverse G= F⁻¹exists over a smaller neighborhood W 3 x(p).

The result follows by using(y, x²)^t as the new coordinate system

at p. 

Exercise 1.13. Show that f ∈ C^∞(M, N) can be locally represented by

˜f(x) = (x¹,· · · , x^k, 0,· · · , 0)^t

for some k ≤ m if and only if that d fp has constant rank k for all p ∈ M.

5. Sub-manifolds and the Whitney imbedding theorem

There is a well defined notion of sub-objects in a reasonably given category.

Definition 1.16. For a manifold N, a topological subspace M ⊂ N is a submanifold if there is an atlas {(Uα, xα)}_α∈A on N such that the restriction

{(Uα∩ M, xα|_Uα∩M)}_α∈A

also form an atlas on M. This definition applies to any C^k manifolds.

5. SUB-MANIFOLDS AND THE WHITNEY IMBEDDING THEOREM 19

Let f : M → N be an immersion. By lemma 1.15 (1), for any p ∈ M there is a chart U 3 p so that f|_U is injective and f(U) is a submanifold of N. However, f may not be injective globally, e.g.

parameterized plane curves with self-intersections.

Even if f is an injective immersion, the image may not be a mani- fold at all!

Example 1.17. Consider the plane curve in polar coordinates r = sin 2θ with θ ∈ (0, π). The parametrization γ :(0, π) → _R²given by

(x(θ), y(θ)) = (r cos θ, r sin θ) = (sin 2θ cos θ, sin 2θ sin θ) is an injective immersion of (_{0, π}) _{into R}². But the point (_{0, 0}) ∈ γ((0, π))does not have any locally Euclidean neighborhood, when the image γ((0, π))is equipped with the subspace topology inR².

Even if the image is a manifold, it may not be equipped with the induced subspace topology:

Exercise 1.14. Let a∈ _R\Q and consider the map f : R→ S¹×S¹: t7→ (e^it, e^iat)

where we identify S¹ as a subset in C. Show that f is an injective immersion and f(_R)is dense in S¹×S¹.

Definition 1.18. A C^∞ map f : M → N is an imbedding if it is an in- jective immersion which induces a homeomorphism f : M→^∼ f(M) with f(M) ⊂ N being equipped with the subpace topology.

Lemma 1.19. If f : M → N is a C^∞ imbedding then f(M) ⊂ N is a C^∞ submanifold and f : M→ f(M)is a diffeomorphism.

PROOF. The condition that f being a homeomorphism means that for U a open neighborhood at p, there are open set V ⊂ N at f(p) such that f⁻¹(V) = U and U is homeomorphic to f(U) = V∩ f(M) under f . By lemma1.15(1) we may select U to be a coordinate slice of V and hence f(M)is a C^∞ submanifold of N. f : M → f(M)is a diffeomorphism since they have identically the same atlas.  A continuous map f : M → N between topological spaces is called closed if the image of a closed set is closed. It is clear that an

injective continuous closed map induces a homeomorphism onto its image, so an injective closed immersion is an imbedding.

Similarly f is open if it send open sets to open sets. An injec- tive open immersion is also an imbedding. However, an imbedding needs not be closed nor open. E.g. an interval(a, b)along the x-axis inR².

A continuous map is proper if the inverse image of a compact set is compact.

Exercise 1.15. Let f ∈C(M, N)with N being Hausdorff. Show that:

(1) If M is compact then f is proper as well as closed. (2) If M, N are manifolds and f is proper then it is also closed.

Thus for compact domain manifolds there is no serious topo- logical issues to concern; the notion of immersions and imbeddings (=injective immersions here) are precise and convenient. For non- compact domain manifolds, extra information on f are usually cru- cial.

The Whitney imbedding theorem says that any manifold is noth- ing more than an imbedded submanifold in the Euclidean space. Be- fore we prove this fundamental result we need the

Definition 1.20. A set A⊂_R^dhas measure zero if for any e>0 there is a countable cover by balls B_i with ∑i|B_i| < e. A set A ⊂ M^din a C^k manifold (k ≥ 1) has measure zero if for any chart(U, x) the set x(A∩U)has measure zero inR^d.

By the standard diagonal argument we see that a countable union of measure zero sets also has measure zero. Also it is trivial that measure zero sets can not contain open sets. To see that the later definition makes sense we need “measure zero” to be independent of choice of coordinates. Indeed, more is true:

Exercise 1.16. (1) If f : U →_R^dis C¹and A ⊂U ⊂_R^dhas measure zero then f(A)also has measure zero. (2) If f : M^m → Nⁿ is C¹ and m <n, then f(M)has measure zero, in particular it is not surjective.

5. SUB-MANIFOLDS AND THE WHITNEY IMBEDDING THEOREM 21

Theorem 1.21(Whitney Imbedding Theorem, 1936). Every C^∞ mani- fold M^dadmits a C^∞ closed imbedding intoR^2d+1and a C^∞closed immer- sion inR^2d.

PROOF. Here we will only give the proof for the (simpler) case that M is compact.

Step 1: Construct an imbedding f : M → _R^N for some large N ∈N. (This step requires only the C¹structure.)

For any p ∈ M, consider a chart (Up, xp) with xp(Up) = B0(2). The open cover {U⁰_p := x⁻_p¹(B0(1))}_p∈M admits a finite subcover indexed by 1, . . . , k. Consider cut off functions{ψ_i}^k_i=1 with ψ_i ≡ 1 on U_i⁰and supp ψ_i ⊂U_i. Define a C^∞map

f :=

∏

(_ψixp_i, ψi)

≡ (_ψ1xp₁, ψ1,· · · _{, ψk}xp_k, ψk)_{: M}→ (_R^d+1)^k =_R^k(d+1)_. To see that f is an immersion, we notice that

d fp = (d(_ψ1xp₁), dψ₁,· · · , d(_ψkxp_k), dψ_k) : TpM →T_f(p)R^k(d+1). Let p ∈ U_i⁰. Since ψ_i|_U⁰

i ≡ 1, we get d(_ψixp_i)_p = d(xp_i)_p. This is the identification map TpM ∼= T_xpi(p)R^d hence in particular that d fp is injective.

To see that f is injective, given p 6= p⁰ ∈ M, if there is an i such that p, p⁰ ∈ U_i⁰, then the component ψ_ix_pi = x_pi gives different coor- dinates for p and p⁰. Otherwise p ∈ U_i⁰ and p⁰ 6∈ U_i⁰ for some i and then ψ_i(p) =1>ψ_i(p⁰).

Step 2: Reduction of imbedding dimension N to 2d+1.

This step works for any given closed imbedding f : M → _R^N (we need only the C² condition on M and f , and M may be non- compact). The idea is find a direction v ∈ S^N−1 and compose f with the projection map πv : R^N → v^⊥ ∼= R^N−1. The new map fv := _πv◦ f is closed since projection maps are clearly closed maps and composition of closed maps are again closed.

To have fv being injective it is equivalent to require that for any p 6= q ∈ M, the vector−−−−−→

f(p)f(q) is not parallel to v. More precisely,

let∆ : M → M×M be the diagonal map∆(p) = (p, p)and consider the secant map

σ: M×M\_∆(M) → S^N−1/{±1} =: RP^N−1 defined by

σ((p, q)) = ± ^f(p) − f(q)

||f(p) − f(q)||^.

Since σ is a C^∞ map from a 2d manifold to an N −1 manifold, if 2d < N−1 (that is, N >2d+1) then σ can not be surjective; in fact Im σ has measure zero. Thus fvis injective if we select v 6∈im σ.

To have fv being an immersion it is equivalent to require that d(πv)x =πvis injective on Txf(M)for all x ∈ _f(M). Without loss of generality we identify M as its image f(M)inR^N. Then the tangent bundle TM ⊂ M×_R^N and the unit sphere bundle

S(TM) ⊂ M×S^N−1

as a C^∞ manifold of dimension d+ (d−1) = 2d−1 is defined. The map

T : S(TM) → S^N−1/{±₁}_: (p, v) 7→ ±v

is a C^∞map from a 2d−1 manifold to an N−1 manifold. If 2d−₁<

N−1 (that is, N > 2d) then im T has measure zero and so T is not surjective. Thus fvis an immersion if we select v 6∈imT.

When N > 2d+1, im σ∪im T ⊂ S^N−1 also has measure zero hence the desired projection direction v can be selected. This com-

pletes the proof. 

Exercise 1.17. For a C^k manifold M, show that TM is a C^k−1 mani- fold by constructing an atlas on it and computing the transition func- tions.

A similar idea leads to applications to homotopy theory:

Theorem 1.22. If f : S^k → Sⁿ is continuous with k < n, then f is homotopic to a constant map. That is, π_k(Sⁿ) =0 for k <n.

6. SUBMERSIONS AND SARD’S THEOREM 23

PROOF. It is trivial if f is not onto since Sⁿ \ {p} ∼= Rⁿ is con- tractible. This is indeed the case if f is C¹by Exercise1.16.

Now the idea is simply to approximate f by a C¹ (in fact C^∞) function ˜f : S^k →Sⁿ within a δ-error with δ<_π.

Exercise 1.18. Prove the C^∞ approximation for f ∈ C(S^k, Sⁿ)within any δ>0. In fact show that the C¹approximation is always possible for any f ∈C(M, N)where M, N are both compact C¹manifolds.

With this done, then for each x ∈ S^k, the two vectors f(x)_{, ˜f}(x) span a two dimensional plane Vx ⊂ _Rⁿ⁺¹ and there is a unique homotopy F(x, t) from f(x) to ˜f(x) through the shorter great cir- cle Vx∩Sⁿ. F(x, t) is clearly continuous, hence f is homotopic to ˜f, which is C¹and hence homotopic to a constant map.  Exercise 1.19 (Invariance of dimension). As a corollary, show that Rⁿ ∼=R^m (homeomorphic)⇐⇒ n=m.

6. Submersions and Sard’s theorem

After discussing submanifolds induced from immersions, we now consider those induced from submersions, i.e. d f is surjective.

N

M f⁻¹(q2)

f⁻¹(q1)

q2

q1

Definition 1.23. Let f : M → N be smooth. A point q ∈ N is called a regular value of f if d fp is surjective for all p ∈ _f⁻¹(q). Otherwise q is called a singular value . When q is a regular value, the preimage f⁻¹(q)is a, possibly non-connected, submanifold of M and f can be parametrized as a coordinate projection locally at p by Lemma 1.15.

Any point p∈ f⁻¹(q), i.e. d fp surjective, is referred as a regular point .

Definition 1.24. A point p ∈ M is called a critical point of f if p is not regular, i.e. d fp is not surjective. If N = R, this means d fp = 0. We denote by C(f)the set of all critical points.

Intuitively, the map f establishes a kind of nice local fiber space structure on M outside the singular values f(C(f)). Thus it is impor- tant to know more about the properties of f(C(f)).

Theorem 1.25(Sard’s Theorem). f(C(f))has measure 0 in N.

PROOF. Only have to prove the case of charts(U, x)with bounded U.

Consider f : U ⊂_R^m →_Rⁿ_{. Let}

C_i = {x ∈U | D^αf(x) = _0, ∀_α, |_α| ≤i} and C =C(f)_{. So C} ⊃C₁ ⊃C₂⊃ · · ·_.

This proof consists three steps:

(1) f(C\C1)has measure 0.

(2) f(Ci\Ci+1)has measure 0.

(3) f(C_k)has measure 0 for some k large enough.

And we use induction on the dimensions m+n.

If m+n = 1, C = C₁ and f(C) consist of only one point. The theorem is trivial in this case. So we assume that m+n ≥2.

Let p∈ C\C₁, say ∂ f¹(p)/∂x¹ 6=0. Through a coordinate change h : U → _R^m,

x 7→ (f¹(x), x²,· · · , x^m)^t, we have

dhp =







∂1f¹(p) ∗ 0 I_m−1







which has non-zero Jacobian. By the Inverse Function Theorem, there is a neighborhood V ⊂U such that h⁻¹exists on ˜V :=h(V).

6. SUBMERSIONS AND SARD’S THEOREM 25

U V

p

h(p) V˜

f

h⁻¹ h

R^m

˜f= f ◦_h⁻¹

Let ˜f= f ◦h⁻¹: ˜V →_R^m. We have ˜f¹(t,· · · ) =t and q ∈C(˜f) ∩V^˜ ⇐⇒ h⁻¹(q) ∈C(f) ∩V.

Then ˜f : ˜V →_Rⁿ splits into

˜f_t : ({t} ×_R^m−1) ∩V^˜ → {t} ×_Rⁿ⁻¹, and

d ˜f=







1 0

∗ d ft





 , q = (t, r) ∈C(˜f) ⇐⇒ r ∈ C(˜ft).

By the induction hypothesis, C(˜f_t) has measure zero in the hy- perplane{t} ×_Rⁿ⁻¹. Since C(˜f) \C₁ ⊂^S_tC(˜f_t), by Fubini theorem in Lebesgue integral,

|f(C(˜f) \C₁)| ≤ Z

t

 ˜ft(C(˜ft))dt =0.

Remark 1.26. This argument does not really need the full power of Lebesgue theory. We only need the theory of measure 0 for the proof.

Consider the same argument on all p ∈ C\C1. Since C\C1 can be covered by countable union of such (C(˜f) \C1)’s, we conclude that f(C\C1)has measure 0.

Secondly, for any p ∈ C_i\C_i+1, we may assume D^αf(p) = 0 for all|α| ≤i but D^βf¹(p) 6=0 for some β =α+ (1, 0,· · · , 0).

Write f^(α)(x) = D^αf¹(x). Again by changing coordinates, h : U → _R^m,

x 7→ (f^α(x), x²,· · · , x^m)^t,

dhp =







∂1f^(α)(_p) ∗ 0 I_m−1







is invertible and there is a neighborhood V ⊂U such that h⁻¹exists.

Also, h(C_i∩V) ⊂ {0} ×_R^m−1. Let ˜f= f ◦h⁻¹. We have h⁻¹(q) ∈ C_i(f) ⇐⇒ q ∈ C_i(˜f) = C_i(˜f₀) where ˜f0 : ({0} ×_R^m−1)^Th(V) →_Rⁿ.

By induction, f(C(˜f₀)) has measure 0, and so does f(Ci\Ci+₁) by the countable covering argument as before.

Thirdly, we claim that f(C_k)has measure 0 for large k through a volume estimate. By the Taylor expansion, there exists a constant A which depends only on k, m and n such that for p∈ C_k,

f(p+h) = f(p) +R(p, h), |R(p, h)| ≤ A|h|^k+1

for all|h| <δ. Let vn =vol(B0(1)). In each such ball Bp(δ)we have

|f(C_k∩Bp(_δ))| ≤vnAⁿδ^(k+1)n

Now we cover C_k by a finite number of such balls. We need at most (2d/δ)^m balls with d=diam U. Pick k satisfying k+1>m/n, then

|f(C_k)| ≤ vnAⁿ(2d)^mδ^(k+1)n−m.

Since A and d are independent of δ, by taking δ → 0 we get that f(C_k)has measure 0. The proof is complete.  Remark 1.27. Sard’s Theorem holds for C^k maps with k > max{m− n, 0}.

7. VECTOR FIELDS AND FROBENIUS THEOREM 27

7. Vector fields, flows, Lie derivatives and the Frobenius integrabil- ity theorem

Definition 1.28. Let U ⊂ M be open. X ∈ T(U)is called a vector field if for each p ∈U, we assign a vector X(p) ∈ TpM at p. We say X ∈ C^∞ if X f ∈ C^∞(U)for all f ∈ C^∞(U).

X

TM

M

•

p

•Xp

? π

We’ve known that TM is a smooth manifold where charts and coordinate functions on TM are of the form

(π⁻¹(U), x¹,· · · , x^m, dx¹,· · · , dx^m).

So we also have equivalent definitions for C^∞ vector fields:

Proposition 1.29. The followings are equivalent:

(1) X(f) ∈ C^∞(U)for any f ∈C^∞(U); (2) X =_{∑ a}ⁱ(∂/∂xⁱ), aⁱ ∈ C^∞(U); (3) X : U →TM is C^∞.

In Euclidean space, for a C^k vector field F : U ⊂ _R^m → _R^m we can find a solution to the initial value problem







∂x

∂t = F(x(t)), x = (x¹(t),· · · , x^m(t)) x(_{0, x0}) =x₀

.

This is the existence and uniqueness theorems of ODE.

Theorem 1.30. There exists a unique solution x(t, x₀)which is C^k in both t and x0. (In fact, C^k+1in t.)

We also have a manifold version: