2 WARM-UP CONCEPTS ON OPEN SETS

Differential GeometryMath 230a - Fall 2015

George Torres

Lecture - 9/3

1 Course information

Instructors: Hiro Tanaka (hirohirohiro@gmail.com), Phil Tynan (ptynam@math).Office Hours: Hiro: Tuesday 1:30-2:30, Wednesday 2-3, SC 341

Phil: Thursday 2-3 SC 536.Grading: Weekly problem sets, take-home final exam.

2 Warm-up concepts on open sets

To start, we introduce some key concepts on open sets of Rn, to be generalized to manifolds.

Definition: Let U ⊂ Rn be open. Then a function f : U → Rm is called:

• C0 if it is continuous;

• C1 if it is differentiable (i.e. has partial derivatives);

• Cr if it has all partial derivatives of order ≤ r, and they are continuous;

• C∞ if it is Cr for all r ∈ Z+.

Definition: A smooth map f : U → Rn is an immersion if dfx is injective for all x ∈ U .

Definition: A map f : U → V on open sets is a diffeomorphism if it is smooth, bijective and has smoothinverse.

Definition (naıve): Let U ⊂ Rn be open. A Riemannian metric on U is a smooth function g : U →Mn×n(R)such that g(x) is a symmetric, non-degenerate, positive definite matrix for all x.1.

Note: Any Riemannian metric g induces an inner product on the tangent space TxX:

g(u, v) = 〈u, v〉 = u>g(x)v

We speak of both the inner product and the matrix interchangeably.

Example(s):

Let U = Rn, and set g(x) = I. This is called the standard Riemannian metric on Rn. How do we makemore? Fix a smooth map f : U → Rm. Since f is C1, it induces a map from TxU → Tf(x)Rn, called thederivative dfx. The derivative can be thought of as the matrix of partial derivatives, or the Jacobian. Ifthere is a Riemannian metric h on Rm, it induces a bilinear product on TxU given by:

〈u, v〉 := h(dfx(u), dfx(v))

1We recognize Mn×n(R) ∼= Rn2

1

2.1 Curves and curvature 2 WARM-UP CONCEPTS ON OPEN SETS

Note that this is only non-degenerate when f is an immersion (i.e. dfx is injective).

This notion is called a pullback metric:

Definition: Let h be a Riemannian metric on V ⊂ Rm and let f : U → V be a smooth immersion. Then theinduced Riemannian metric is defined to be the pullback f∗h:

f∗h(u, v) = h(dfx(u), dfx(v))

Definition: Let g be a Riemannian metric on U ⊂ Rn. Then the volume of (U, g) is:∫U

√det(g) dx1...dxn

In some sense, this is the volume of U where it has been stretched differently at every point by g.

Definition: Let (U, g) and (V, h) be sets with Riemannian metrics. An isometry of these is a diffeomorphism fsuch that f∗h = g.

2.1 Curves and curvature

Exercise:

Let γ : R → Rn be a in immersion. Show there exists a diffeomorphism φ : R → R such that γ ◦ φ is

parameterized by arclength (namely ‖d(γ◦φ)dt ‖ = 1).

Solution (fixed):

Let `(t) =∫ t

0|γ′(s)|ds be the arclength function, which is smooth and consider γ ◦ `−1. By the chain rule

we have:d(γ ◦ `−1)

dt=dγ(`−1)

d`−1· d`−1

dt= γ′(`−1(t)) · d`

−1

dt

Since d`−1

dt = 1d`/d`−1 , we have:

d(γ ◦ `−1)

dt= γ′(`−1(t)) · 1

d`/d`−1=

γ′(`−1(t)

|γ′(`−1(t))|

which is a unit vector,

Definition: the unit velocity on a in immersion γ : R→Rn is a vector function T : R→ Rn given by:

T (t) =γ(t)

|γ(t)|

Definition: The curvature of γ is defined to be the derivative when parameterized by arclength s:

κ :=dT

ds=dT/dt

ds/dt

This can be thought of as the reciprocal of the radius of the best approximating circle to the curve at that point.Note two properties about κ:

2

3 MANIFOLDS

1. κ ⊥ T . To see this, consider the function δ(t) = 〈T (t), T (t)〉. Clearly dδdt = 0, since |T (t)| = 1. However,

using the product rule:

dδ

dt=

⟨d

dtT (t), T (t)

⟩+

⟨T (t),

d

dtT (t)

⟩= 2

⟨d

dtT (t), T (t)

⟩= 0

Therefore ddtT (t) ⊥ T (t) as desired.

2. If γ has image a circle of radius R in R2, then |κ| = 1R . Without loss of generality, assume γ(t) =

R(cos t, sin t). Then T (t) = (− sin t, cos t). We can then calculate κ:

κ =dT/dt

ds/dt=

(− cos t, sin t)

R

and so we indeed have |κ| = 1R .

Lecture - 9/8

3 Manifolds

To generalize the above notions, we must define the concept of a manifold.

3.1 Topological Manifolds

Definition: A topological space X is locally Euclidean if ∀x ∈ X there is an integer d ≥ 0 and some open setU ⊂ Rd such that U is homeomorphic to a neighborhood of x.

Definition: A space X is second countable if X admits a countable basis of open sets.

Remark: If X is a topological manifold, every connected component of X will be:

• locally Euclidean

• Hausdorff

• second countable

Definition: An open cover Uα = U of a space X is locally finite if ∀x, there is an open set W ⊂ X containingx such that W ∩ Uα 6= ∅ for only finitely many α.

Definition: X is paracompact if every open cover admits a locally finite refinement.

Definition: A topological manifold a space that is2:

• locally Euclidean (to do calculus)

• Hausdorff (to do geometry)

• paracompact (for partitions of unity)

2Some definitions of a topological manifold replace paracompact with second countable

3

3.2 Smooth structures 3 MANIFOLDS

Remark: Even though we have defined a topological manifold to have the properties we need, we still cannotdo calculus on these objects without a smooth structure (or atlas). To motivate this, consider a topologicalmanifold X with two homeomorphisms of neighborhoods of x ∈ X, shown in the figure below. Then, for calculusto even make sense on X, we must have derivatives agree on both parameterizations. In particular, we needψ ◦ φ−1 to be smooth.

This motivates a few definitions:

Definition: If X is a topological manifold, a chart on X is a pair (U, φU ), where U ⊂ X is open and φu is ahomeomorphism to an open set in Rn.

Definition: A Cr atlas on X is a collection of charts {(Uα, φα)}α∈A such that:

• {Uα} cover X

• ∀α, β ∈ A, the function φβφ−1α is Cr (where defined).

3.2 Smooth structures

We now have the tools to define a manifold with a smooth structure:

Definition: A Cr manifold is a pair (X,A), where X is a topological manifold and A is a Cr atlas. Usually welet r =∞ (in which case this is a called a smooth manifold).

Definition: Let (X,A) and (Y,AY ) be two Cr manifolds. Then a continuous function f : X → Y is Cr if∀x ∈ X, there are charts (U, φ) ∈ A, (V, ψ) ∈ AY such that x ∈ U, f(x) ∈ V and ψ ◦ f ◦ φ−1 is Cr.

Definition: A function f : (X,A)→ (Y,AY ) is a Cr diffeomorphism if:

• f is a bijection

• f and f−1 are Cr

Note that we can place an equivalence relation on smooth structures. Namely, two atlases A,A′ are equivalentif A ∪ A′ is also a smooth atlas3. Further, if A is an equivalence class of atlases, there is a natural choice of“maximal atlas” as a representative: ⋃

A∈A

A (maximal atlas)

3One can check that this is equivalent to saying there exists a diffeomorphism between (X,A) and (X,A′).

4

4 PARTITIONS OF UNITY

For certain topological manifolds, the equivalence classes A are nontrivial; i.e. there are multiple smoothstructures that aren’t equivalent. The following are three results that demonstrate the non-triviality of smoothstrutures:

Theorem: Not every topological manifold admits a C∞ atlas.Theorem (Milner): The seven sphere S7 admits non-diffeomorphic smooth structures.Theorem (Donaldson, Freedman): Euclidean 4 space R4 has uncountably many non-diffeomorphic smoothstructures.

Lecture - 9/10

4 Partitions of Unity

Exercise:

Let j : R2 → R3 be defined by j(x, y) = (x, cos(y), sin(y)). Compute j∗i, where i is the standard metricon R3.

Solution:

We compute the derivative:

dj =

1 00 − sin y0 cos y

By definition of the pullback:

f∗i = dj>idj = dj>dj =

1 00 − sin y0 cos y

(1 0 00 − sin y cos y

)=

(1 00 1

)———————

Notation: We refer to both charts and parameterizations between neighborhoods of a manifold and a neighbor-hood of Rn. It is a chart when it starts from a manifold, and it is a parameterization when it starts from Rn.Namely, if U ⊂ X, then

U Rnchart

param.

Partitions of unity are a necessary tool used on manifolds to turn locally supported functions into globallysupported ones.

Definition: Let X be a smooth manifold and fix an open cover U = {Uα}, α ∈ B. A partition of unitysubordinate to U is a collection of smooth functions {fβ : X → R≥0}β , β ∈ B such that:

1.∑β∈B fβ(x) = 1 for all x ∈ X.

2. ∀β we have supp(fβ) = {x | fβ(x) 6= 0} ⊂ Uβ .

3. The collection {supp(fβ)} is locally finite.

Theorem: Let X be a Cr manifold. Then for every cover {Uα}, there exists a Cr partition of unity subordinateto {Uβ}.

5

5 SUBMERSIONS

Remark : We especially like the above result when r =∞.

To prove this, we need two lemmas:

Lemma 1: Let U ⊂ Rn be open, and K ⊂ U be compact. Then ∃ a smooth function f : U → R such thatf(K) ⊂ R>0 and supp(f) ⊂ U .

Lemma 2: Let {Cγ} be a collection of closed subsets of a topological manifold X. If {Cγ} is locally finite, then⋃Cγ is also closed.

Proof (of theorem):

Let {Uβ} be an open cover of of X. The proof will proceed in three steps:

A. If {Wε} is a refinement of the given cover {Uβ}, and {Wε} has a partition of unity, then so does{Uβ}.Proof (of A):

Let κ : {ε} → {β} be a function such that Wε ⊂ Uβ . Then, if {fε} is a partition of unitysubordinate to {Wε}, define:

fβ =∑

ε∈κ−1(β)

fε

We claim this is a partition of unity subordinate to Uβ . It is easy to see that it satisfied parts1 and 3 of our definition because {fε} does. To show it satisfies condition 2, we use Lemma 2:

supp(fβ) =⋃

supp(fε)

=⋃

supp(fε) (Lemma 2)

⊂Wε ⊂ Uβ

B. We can always choose a refinement {Wε} of {Uβ} so that each Wε is compact (proof on Hw 1).

C. Fixing such a refinement {Wε}, we can find a locally finite refinement {Yε} of {Wε} such that Yε ⊂Wε.Now, from Lemma 1, we have functions fε such that f(Yε) ⊂ R>0 and supp(fε) ⊂ U . Then we define:

gε =fε∑fε

We see that, by construction, this satisfies the conditions of a partition of unity.

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5 Submersions

Definition: Let f : U → V be a smooth map, with U ⊂ Rn, V ⊂ Rm open. Then f is a submersion at x ∈ Uiff dfx is surjective. We call f a submersion if it is a submersion at every point.

Remarks:

• This is only possible when n ≥ m

• f : U → V given by inclusion is a submersion.

6

6 TANGENT SPACES

• The projection f : Rn → Rm given by (x1, ..., xn) 7→ (x1, ..., xm) is a submersion.

Submersions are a handy tool for constructing submanifolds:

Definition: A subset X ⊂ U , with U open in Rn is a smooth submanifold of U if ∀x ∈ X there is an open setW ⊂ U and a diffeomorphism φ : Rn →W such that φ(Ri) = X ∩W . Here, Ri is defined as:

Ri := {x ∈ Rn | x = (x1, ..., xi, 0, ..., 0)}

Note that a smooth submanifold is itself a smooth manifold.

Definition: A continuous map f : X → Y between topological spaces is proper if the preimage of any compactset is compact.

Theorem (Sumbersion): Let f : U → V be a submersion. Then for all y ∈ V , the preimage f−1(g) ⊂ U is asmooth submanifold of U .

Example:

Consider the map f : R2 → R given by x 7→ |x|2. This is a submersion, and therefore by the Submersiontheorem:

M = f−1(1) = S1

is a smooth manifold.

Lecture - 9/15

6 Tangent Spaces

Observation: Tangent spaces give us a way to take derivatives. We would like a definition that is coordinatefree.

Definition: Let X be a smooth manifold. Then the set of C∞ functions f : X → R is denoted by C∞(X) orC∞(X,R).

We will define the tangent space using the algebraic notion of a derivation:

Definition: Let A,B be commutative algebras over R, and fix an R-algebra homomorphism e : A → B. Thenan e−derivation, or simply a derivation, is a function d : A→ B satisfying two conditions:

(1) d(αf + g) = ad(f) + d(g)

(2) d(fg) = d(f)e(g) + e(g)d(f)

From here out, we take A = C∞(X), B = R and ex given by evaluation at x for some fixed x (f 7→ f(x)).

Definition: Let M be a smooth manifold. Then the tangent space of M at x, denoted Tx(M), is defined tobe:

Tx(M) := {ex-derivations Xx : C∞(M)→ R}

One can check that Tx(M) is a vector space over R.

In order for this to be a reasonable definition of the tangent space, we should verify a few things:

• T0(Rn) ∼= Rn

• Chain rule

7

6 TANGENT SPACES

Proposition: Let x ∈M be contained in an open set U ⊂M , and let f, g be C∞ functions on M . If f = g onU , then ∀Xx ∈ Tx(M), we have Xx(f) = Xx(g).

Proof: Choose some compact ball B so that x ∈ int(B) and B ⊂ U . Fix a function h : M → R so that h|B = 1and supp(h) ⊂ U4. Given a derivation Xx ∈ Tx(M), we can evaluate it at h · (f − g). Since f = g on U , we haveh · (f − g) = 0, so:

0 = Xx(h · (f − g))

=���

�����:0

Xx(h)(f − g)(x) +���*1

h(x)Xx(f − g)

= Xx(f − g)

So by linearity Xx(f) = Xx(g).

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This proposition tells us that the tangent space, while defined using global functions, is actually just a localproperty.

Proposition: Let j : N → M be a smooth map of manifolds, and let x ∈ N . Then there exists a linear mapdjx : TxN → Tj(x)M defined by:

Xx 7→(f 7→ Xx(f ◦ j)

)Proof: Exercise for the reader.

Proposition: Let Nj−→M

h−→ L be smooth functions between manifolds. Then the chain rule for the derivativeis satisfied:

d(h ◦ j)x = dhj(x) ◦ djxProof: Let f ∈ C∞(L). Then:

d(h ◦ j)x(Xx)(f) = Xx(f ◦ (h ◦ j))= Xx((f ◦ h) ◦ j)= djx(Xx)(f ◦ h)

= dhj(x) ◦ djx(Xx)(f)

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Proposition: T0(Rn) ∼= Rn as vector spaces.

Proof: Not that the assignment ∂∂xi

∣∣∣0

: C∞ → R given by f 7→ ∂f∂xi

∣∣∣0

is a derivation. We claim that the set:

D =

{∂

∂x1

∣∣∣0, ...,

∂

∂xn

∣∣∣0

}Forms a basis for T0(Rn). To see this, we use Taylor’s Theorem, which asserts that any C∞ function can bewritten as:

f(x) = f(0) +∑

xigi(x)

4We can do this because of problem 6 on homework 1

8

7 SUBMERSION THEOREM (REVISITED)

where the gi are smooth with the property gi(0) = ∂f∂xi

∣∣∣0. Given a derivation X0, let us evaluate it at f ∈ C∞(M).

We first note that f(0) can be thought of as a constant function and therefore X0(f(0)) = 0. Further:

X0(f) = X0(f(0)) +∑

X0(xigi(x))

= 0 +∑

X0(xi)gi(0) +∑���*

0xi(0)X0(gi(x)) (product rule)

=∑

X0(xi)∂f

∂xi

∣∣∣0

Therefore X0 =∑X0(xi)

∂∂xi

∣∣∣0. This confirms that D is a spanning set. These are also linearly independent

because ∂∂xi

xj = δij . Thus T0(Rn) is an n dimensional vector space over R, as desired.

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Three remarks to wrap up this section:

1. If j : Rm → Rn is a smooth function, then

dj0

(∂

∂xi

∣∣∣0

)=∑

(dj0)ij∂

∂xi

∣∣∣0

Where dj0 on the right is the ex-derivation definition of the derivative, and the dj0 is the matrix definitionof the derivative.

2. For all y ∈ Rn, consider the diffeomorphism Ty : Rn → Rn given by x 7→ x+ y. Then

∂

∂xi

∣∣∣y

= d(Ty)

(∂

∂xi

∣∣∣0

)3. By the chain rule, any diffeomorphism of manifolds j : M → N induces a linear isomorphism dfx : TxM →Tj(x)N .

7 Submersion Theorem (revisited)

Definition: Let f : M → N be smooth. Apoint y ∈ N is a regular value if ∀x ∈ f−1(y), the derivative dfx is asurjection.

Definition: A subset Z ⊂M of a smooth manifold is a smooth submanifold if ∀z ∈ Z, there is an open set U ⊂ Zand a diffeomorphism h : V → U , where V ⊂ Rn is open, such that h(Rm) = U ∩ Z, where Rm ⊂ Rn.

Theorem (Submersions): Let M,N be smooth manifolds and f : M → N be a smmooth function. Then forevery regular value y ∈ N , the preimage f−1(y) ⊂M is a smooth submanifold.

Proof: Next lecture

Lecture - 9/17

Proof (Submersion Theorem): Choose coordinates (φ,U) and (ψ, V ) around y ∈ f−1(x) and x ∈M respectivelyso that:

U V

φ(U) ψ(V )

f

φ ψ

9

9 TANGENT BUNDLES

Note that, since f is regular, we have φ(U) ⊂ Rn and ψ(V ) ⊂ Rm for n ≥ m. We can ensure that both φ(U)and ψ(V ) contain the origin, and so we can let g = ψ ◦ f ◦ φ−1 be a diffeomorphism between them. Further,we may assume that φ and ψ are chosen so that g is of the form (x1, ..., xn) 7→ (x1, ..., xm) (the cononicalsubmersion)5. Then we have g−1(0) = (0, ..., 0, xm+1, ..., xn). Now we have φ−1(Rn−m) = f−1(y) ∩ U , and sowe are done.

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8 Lie Brackets

Definition: For a manifold M , define

Γ(TM) := {R− linear derivations from C∞(M) to itself with respect to e = id}= {X : C∞(M)→ C∞(M) | X(af + g) = aX(f) +X(g) and X(fg) = X(f)g +X(g)f}

An element X ∈ Γ(TM) is called a vector field

Remark: For every x ∈ M , consider the function Γ(TM) → Tx(M) given by X 7→ evx ◦ X := Xx. This is aderivation because evx is a ring homomorphism. We think of the Xx as the associated “vector“ for the vectorfield at x. Since vector fields are homomorphisms between C∞ and itself, we can ask about composing them.Specifically, a useful object is the Lie Bracket:

Proposition: Let X,Y ∈ Γ(TM) be vector fields. Define [X,Y ] := X ◦ Y − Y ◦X. Then:

1. [·, ·] : Γ(TM)× Γ(TM)→ Γ(TM)

2. [·, ·] is bilinear over R

3. [X,Y ] = −[Y,X] for any X,Y .

4. [·, ·] satisfies the Jacobi identity:

[X, [Y, Z]] = [[X,Y ], Z] + [Y, [X,Z]]

Equivalently: for any X ∈ Γ(TM), the map DX = [X, ·] is a derivation with respect to [·, ·]. This meansDX [Y,Z] = [DX(Y ), Z] + [Y,DX(Z)]. Thinking of these vector fields inducing flows on manifolds, we seethe Lie bracket as a measure of how non-commutative they are.

Definition: Let V be an R vector space. Any bilinear map V × V → V is a lie bracket if it satisfies conditions3 and 4 above. The pair (V, [·, ·]) is called a Lie Algebra.

Remark: For all commutative rings A, the set Der(A,A) is a Lie algebra under [X,Y ] = X ◦ Y − Y ◦X.

9 Tangent Bundles

We seek to construct the tangent bundle TM of a manifold M :

Definition: Given a smooth manifold M , define:

TM :=∐

Tx(M)

We wish to give this a topology and a smooth structure. More specifically, we want several things from ourconstruction:

5This assumption isn’t trivial, but it is in fact true

10

9 TANGENT BUNDLES

1. A smooth structure on TM together with a smooth function

TMπ−→M

(x, y) 7→ y

2. For all x, the set π−1(x) is a vector space over R

3. We have a local trivializations. That is, for x ∈M , we have U ⊂M open containing x and a diffeomorphismΦ making the following commute:

U × Rk TU TM

U M

pr

Φ

π

i

π

i

Where i is inclusion and pr(y, v) = y. Further, we require Φ(y, ·) : Ty(U) → {y} × Rk to be a linearisomorphism for all y ∈ U .

Definition: Fix a manifold M and let E be a smooth manifold with a map π : E → M . If π−1(x) is a an Rvector space for any x and we have local triviality as defined in 3. above, the pair (E, π) is called a rank k vectorbundle over M .

Constructing the Tangent Bundle

To achieve the desired properties of TM , we use the following steps:

• Take a sufficeint open cover U = {Uα} of TM

• identify TUα ∼= Uα × Rk, so TUα inherits a smooth structure

• Set an equivalence relation on∐α TUα so that TM =

∐α TUα/ ∼, where this relation tells us when

v ∈ TUα and v′ ∈ TUβ are from the same tangent vector on M .

Here goes:

Let A = {(Uα, φα)} be a smooth atlas for M . For all x ∈ Uα, the derivative of φα gives us an isomorphismTxUα → Tφα(x)Rn. Over all x ∈ TxUα this gives a map:∐

x∈Uα

TxUα →∐

Tφα(x)Rn

We’ve arlready establised that:

Tφα(x)Rn =

⟨∂

∂x1

∣∣∣φα(x)

, ...,∂

∂xn

∣∣∣φα(x)

⟩So for each x we have the isomorphism

Tφα(x)Rn ∼= {φα(x)} × Rn

x 7→ (φα(x), a1, ..., an)

with x =∑ai

∂∂xi|φα(x). We can now rewrite our original map as:∐

x∈Uα

TxUα → {φα(Uα)} × Rn

11

9 TANGENT BUNDLES

Denote TUα =∐x∈Uα TxUα. We see this is a diffeomorphism, giving TUα a smooth structure. Now we set an

equivlaence relation on∐α TUα. Let Uα, Uβ be two elements of the cover and consider the diagram below:

T (Uα ∩ Uβ)

φα(Uα)× Rk φβ(Uβ)× Rk

(Uα ∩ Uβ)× Rk (Uα ∩ Uβ)× Rk

φ−1α × id φ−1

β × id

Γαβ

Where Γαβ sends (x, v) 7→ (x, d(φβ ◦φ−1α (v)). One can check that Γαα = id and Γδβ ◦Γβα = Γδα, so we may use

this as an equivalence relation. Namely (x, v) ∼ (y, w) if Γαβ(v) = w. Then, we can check that is satisfies ourinitial requirements for defining: ∐

α

TUα/ ∼ = TM

12