Differential geometry notes

John Kerl

November 1, 2005

Abstract

The following are notes to help me prepare for the University of Arizona math department’s geometry- topology qualifier in 2006. The content is distilled from Spivak’s A Comprehensive Introduction to Differential Geometry, Spivak’s Calculus on Manifolds, Lang’s Algebra, and other sources. The list of topics is based on, but presented in a different order from, Prof. Doug Pickrell’s fall 2005 differential geometry course (Math 534A). My intention is to gather together precise definitions and examples. This document is a work in progress. Please do not base any important decisions on its contents; for definitive information, please consult the cited references.

1 Contents

1 Categories 3 1.1 Functors ...... 4 1.2 Hom functors ...... 4 1.3 Pushouts and pullbacks ...... 4

2 Linear algebra and tensors 5 2.1 Basis and dual space ...... 5 2.2 Algebras ...... 6 2.3 Tensors ...... 6 2.4 Pushouts and pullbacks ...... 7 2.5 Tensor algebras ...... 8

3 Manifolds 9 3.1 Manifolds ...... 9 3.2 Coordinates and parameterizations ...... 9 3.3 Chart and atlas ...... 9 3.4 Differentiable structures ...... 9 3.5 Maps of manifolds ...... 9 3.6 Immersion and embedding ...... 9

4 Tangent spaces 10 4.1 Partial derivatives ...... 10 4.2 asdfasdfasdf ...... 11 4.3 asdfasdfasdf ...... 11

5 Vector bundles 12

6 Flows 13

2 1 Categories

Category theory is sometimes ([Lang]) referred to as “abstract nonsense”. However, it is simply an ab- straction of some very concrete things. Namely, we are abstracting the concept of structures (groups, vector spaces, topological spaces, etc.) and mappings (homomorphisms, linear transformations, homeomorphisms, etc.). One oddity is that the definitions below are specifically designed to avoid mentioning the elements of those structures. The primary reason for presenting categories in this paper is as a mnemonic device: for example, category ∗ theory helps us remember the difference between f∗ and f in differential geometry. Definition 1.1. A category C consists of a class of objects and sets of morphisms or arrows between those objects:

• For every ordered pair A, B of objects there is a set HomC(A, B) of morphisms from A to B. This is called a hom set.

• There is a law of composition: if f ∈ HomC(A, B) and g ∈ HomC(B,C) (which is to say f : A → B and g : B → C), then there is an element g ◦ f of HomC(A, C).

• The hom sets are disjoint: If A1 and A2 are different, or if B1 and B2 are different, then HomC(A1,B1) and HomC(A2,B2) have no morphisms in common. • Composition of morphisms is associative: if f : A → B, g : B → C, and h : C → D, then h ◦ (g ◦ f) = (h ◦ g) ◦ f.

• Each object A has an identity morphism, written 1A. Since we are defining morphisms without reference to specific elements of an object, we can’t say something like “. . . such that 1A(x) for all x ∈ A”. Rather, an identity morphism on an object A is characterized by the property that for all other objects B, and for all f : A → B and all g : B → A,

f ◦ 1A = f and 1A ◦ g = g.

xxx put a picture here

Some familiar examples are:

• Objects are groups and morphisms are group homomorphisms. • Objects are rings and morphisms are ring homomorphisms. • Objects are vector spaces and morphisms are linear transformations.

A perhaps less familiar example is a partially ordered set. Let P be a partially ordered set, with operation 4. Then the poset P is itself the entire category. Objects of P are the elements of P . Morphisms of P are quite literally arrows: there is an arrow from a to b if a 4 b. Composition of arrows is nothing more than the transitivity property of posets. The identity morphisms are provided by the reflexivity property of posets, namely, a 4 a for all a ∈ P . xxx divisibility lattice in Z.

Another example is a single group G. There is a single object which is all of G; the hom set HomG(G, G) is all of G. That is, the morphisms are the elements of G. The composition of two morphisms x and y is the usual product xy. The identity morphism is the identity element of G. Lang’s cat1, cat2, cat3.

3 1.1 Functors xxx forgetful functor xxx lang’s fun1, fun2 Definition 1.2.

1.2 Hom functors xxx covt and ctvt Definition 1.3.

1.3 Pushouts and pullbacks

Definition 1.4.

4 2 Linear algebra and tensors

Everything said in this section is applicable to finite-dimensional vector spaces over an arbitrary field. However, for this paper, the base field is always R.

2.1 Basis and dual space

Let V be a finite-dimensional vector space V over R, say of dimension m. Then V has many bases; in particular, it has a standard basis {e1,..., em} with elements of the form

e1 = (1, 0, 0,..., 0), e2 = (0, 1, 0,..., 0),..., em = (0, 0, 0,..., 1).

Definition 2.1. The dual space of V is the set V ∗ of all linear transformations from V to R.

Recall that V ∗ is a vector space over R. It has the same dimension, namely m, as V . Definition 2.2. An element of a dual space is called a linear functional.

Definition 2.3. For any basis {v1,..., vm}, there is a basis for the dual space (i.e. a dual basis)

∗ ∗ {v1,..., vm}

∗ where the vi functions are defined to have value

∗ vi (vj) = δij where the δ is Kronecker’s, i.e. δij = 1 when i = j, 0 otherwise.

Thus, the standard basis has a standard dual basis

∗ ∗ ∗ {e1,..., em} with ei (ej) = δij.

Let λ : V → R and v ∈ V . Given the above basis, we may write

m m X ∗ X λ = aiei and v = bjej. i=1 j=1

∗ ∗ Since the ei ’s are linear functions, and since ei (ej) = δij, we have

m  m  m m m X ∗ X X X ∗ X λ(v) = aiei  bjej = aibjei (ej) = aibi. i=1 j=1 i=1 j=1 i=1

Now let f : V → W be a linear transformation, and let λ : W → R be a linear functional, i.e. λ ∈ W ∗. We can map from V to R by going through W first. That is, for v ∈ V , f(v) is in W , and so we can apply λ to it:

5 V H λ ◦ f HHj f R ¨¨* ?¨ λ W

Formally, λ ◦ f : V → R. Given λ ∈ W ∗, we have obtained λ ◦ f ∈ V ∗. Post-composing λ by f is said to be a pullback of λ from W ∗ to V ∗. Another way to look at this is that, given f : V → W , we have a map from W ∗ to V ∗:

f : V → W f ∗ : W ∗ → V ∗ f ∗(λ) = λ ◦ f (f ∗λ)(v) = λ(fv).

In this way, f ∗ takes functionals to functionals:

V V ∗ 6 f f ∗ ? W W ∗

xxx have a map from V to V ∗. xxx have a map from W to W ∗. xxx note arrow-reversing; refer to the contravariant hom functor here.

∗ Is it true that f (φ) = φ ◦ f and f∗(φ) = f ◦ φ? Check Lang.

2.2 Algebras

xxx write axioms in various ways: ring and subring; vector space w/ multiplication; third way? Examples: extension fields; poly rings; matrix rings

2.3 Tensors

(1) Tensor products from abstract algebra. xxx cite chapter and verse from DF, hungerford, grove, etc. (2) Old-fashioned tensors from physics (“transform according to . . . ”). (3) Tensors as extent-r arrays

(4) Tensors as r-linear functions T : V r → R.

xxx Discuss the last first. Show all are equivalent.

Definition 2.4. Let V be a vector space over R.A tensor is a k-linear map from V k to R. The number k is called the order or degree of the tensor. Often, we call a tensor of order k simply a k-tensor.

6 The term “k-linear” means that a tensor is linear in each slot. For example, for a 2-tensor λ, for all u, v, w ∈ V ,

λ(u1 + u2, v) = λ(u1, v) + λ(u2, v)

λ(u, v1 + v2) = λ(u, v1) + λ(u, v2) λ(au, v) = aλ(u, v) λ(u, bv) = bλ(u, v)

As a consequence:

λ(au, bv) = abλ(u, v)

λ(u1 + u2, v1 + v2) = λ(u1, v1) + λ(u1, v2) + λ(u2, v1) + λ(u2, v2)

and so on. xxx linear functional example xxx dot product example xxx write matrix for dotpr xxx 2x2 det example; matrix xxx 3x3 det example; write 3x3x3 array.

2.4 Pushouts and pullbacks

xxx ([Spivak1], p. 77)

Let f : V → W be a linear transformation, and let λ : W k → R be a k-tensor, i.e. λ ∈ T k(W ). We can map k from V to R by going through W first. That is, for (v1,..., vk) ∈ V ,(f(v1), . . . , f(vk)) is in W , and so we can apply λ to it:

V k H λ ◦ f HHj f R ¨¨* ?¨ λ W k

Formally, λ ◦ f : V → R. Given λ ∈ W ∗, we have obtained λ ◦ f ∈ V ∗. Post-composing λ by f is said to be a pullback of λ from W ∗ to V ∗. Another way to look at this is that, given f : V → W , we have a map from W ∗ to V ∗:

f : V → W f ∗ : W ∗ → V ∗ f ∗(λ) = λ ◦ f (f ∗λ)(v) = λ(fv).

In this way, f ∗ takes functionals to functionals:

7 V T k(V ) 6 f f ∗ ? W T k(W )

2.5 Tensor algebras

8 3 Manifolds

3.1 Manifolds

Definition 3.1. A manifold is a metric space with the property that for all q ∈ M, there is some neigh- borhood U of q and some non-negative integer m(q) such that U is homeomorphic to Rm. If m(q) is the same for all q ∈ M, then we say that M has dimension m.

For this course we consider only connected manifolds. In particular, all our manifolds have a dimension. (For a counterexample, consider the plane z = 1 in R3 along with the line y = z = 0. This has two disconnected components: the plane with dimension 2 and the line with dimension 1.)

3.2 Coordinates and parameterizations

Definition 3.2. Let M be an m-dimensional manifold. A coordinate function is a map from M to Rm ... Definition 3.3. Let M be an m-dimensional manifold. A parameterization is a map from Rm to M ... graph coords xitn functions yx−1

3.3 Chart and atlas

3.4 Differentiable structures

3.5 Maps of manifolds

Definition 3.4. Let f : M → N be a map of manifolds. A coordinate expression for f is y ◦ f ◦ x−1, where x and y are coordinate functions on M and N, respectively. Definition 3.5. A map of manifolds f : M → N is differentiable if for all coordinates (x, U) of M and (y, V ) of N, y ◦ fx−1 is a differentiable function from Rm to Rn.

3.6 Immersion and embedding

9 4 Tangent spaces

` TM = q∈M TM|q

TM|q = {v} Df

f∗

xxx note f : M → N; f∗ : TM → TN. Arrows go the same direction so this is a contravariant functor?

M TM

f f ∗ ? ? N TN

m n f∗ : T R → T R : vq 7→ [Df(p)(v)]f(q) put commutative diagram (sp2 p. 65) here df df is a 1-tensor; f∗ is not? sp1 p. 89: df(q)(vq) = Df(q)(v)

i m M ,→ R → N; f∗ : TM|q → TN|q.

f box: TM →∗ TN

πM πN

f M → N

Triv’n of TM: TM =∼ M × Rm as isom of vector bundles?

4.1 Partial derivatives

−1 ∂/∂xi|q ↔ x (ei|x(q)) def vector operating on function

x1, . . . , xm is a coordinate for M.

Basis for TM|q: ∂Dx1|q, . . . , ∂/∂xm|q.

∗ Basis for TM|q : dx1|q, . . . , dxm|q. With dxi(∂/∂xj) = δij.

10 4.2 asdfasdfasdf

Formalize with i : M → Rm: M = ker F , TM = ker DF .

4.3 asdfasdfasdf

Symmetric 2-tensor: g : V → V → R. g(vq → wq) = vq · wq. P g = g(∂/∂xi, ∂/∂xj)dxi ⊗ dxj

11 5 Vector bundles

Ω0(E) = {sections of E}. Is a vector space.

s1, s2 : B → E;(s1 + s2)(b) = s1(b) + s2(b).

Section times function: f ∈ C∞(B, R) and s ∈ Ω0(E). fs = (fs)(b) = f(b)s(b) where f(b) is the scalar and s(b) is the vector. So, Ω0(E) is a C∞(B) module. Ω0(TM) = {vector fields}. Definition 5.1. Ω0(T ∗M) = cotangent bundle.

= tensor fields of order 1. E.g. differential df of f : M → R. df is a linear functional on each tangent space.

∗ ∗ f : TM → T R: f (vq) = df(vq)|f(q).

Directional derivative of f in the vq direction. Ω0(T 2(T ∗ M)) = {order-2 tensor fields}.

2 g ∈ T (T ∗ M): g(vq, wq) = v · w.

Triangle f : M → N, φ : N → R; pullback to M: f ∗ : C∞(M) → C∞(N) with f ∗φ = φ ◦ f. ∗ ∗ ∗ f (φ1 + φ2) = f (φ1) + f (φ2).

∗ ∗ ∗ f (φ1 ◦ φ2) = f (φ1) ◦ f (φ2). f ∗ :Ω0(T (T ∗N)) → Ω0(T r(T ∗M)):

f ∗ box T r(T ∗(M)) ← T r(T ∗N))

πM , πN

f M → N.

∗ f (ω|q(fv1q, . . . , vrq) = ω|f(q)(f∗(v1q), . . . , f∗(vrq))

12 6 Flows

Definition 6.1. complete

13 References

[DF] D.S. Dummit and R.M. Foote. Abstract Algebra (2nd ed.). John Wiley and Sons, 1999. [Grove] L.C. Grove. Algebra. Dover, 2004. [Lang] S. Lang. Algebra (3rd ed.). Springer, 2002. [Spivak1] M. Spivak. Calculus on Manifolds. Perseus, 1965. [Spivak2] M. Spivak. A Comprehensive Introduction to Differential Geometry. Publish or Perish, 2005.

14 Index A arrows ...... 3

C category ...... 3 complete ...... 13 composition ...... 3 coordinate ...... 9 coordinate expression ...... 9 cotangent bundle ...... 12

D degree ...... 6 differentiable ...... 9 dimension ...... 9 dual basis ...... 5 dual space ...... 5

H hom set ...... 3

I identity morphism ...... 3

K k-tensor ...... 6 Kronecker delta ...... 5

L law of composition ...... 3 linear functional ...... 5

M manifold ...... 9 metric space ...... 9 morphisms ...... 3

O objects ...... 3 order ...... 6

P parameterization ...... 9 pullback ...... 6, 7

S standard basis ...... 5 standard dual basis ...... 5

T tensor ...... 6

15