The Existence of Riemannian Metrics on Real Vector Bundles

The Existence of Riemannian Metrics on Real Vector Bundles

Jan-Ola Collin

Bachelor Thesis, 15hp Bachelor in Mathematics, 180hp

Fall 2018

Department of Mathematics and Mathematical Statistics

Abstract

In this thesis we present a self-contained proof of the existence of Riemannian metrics on real vector bundles.

Sammanfattning

I denna uppsats presenterar vi ett självständigt bevis på existensen av

Riemannskametriker på reella vektorbuntar.

Contents

1 Introduction 1

2 Differentiable Manifolds 5

3 Functions on and Between Manifolds 9

3.1 Smooth Functions on Manifolds . . . . 9 3.2 Smooth Maps Between Manifolds . . . 11 3.3 Diffeomorphisms . . . 13

4 The Tangent Space 15

5 Partitions of Unity 25

5.1 The General Case . . . 28

6 The Existence of Riemannian Metrics 31

6.1 The Tangent Bundle as a Manifold . . . 31 6.2 The Tangent Bundle as a Vector Bundle . . . 35

Acknowledgement 41

A Point-Set Topology 43

Bibliography 57

Chapter 1

Introduction

“All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced.”

– Gauss One important fundamental tool in differential geometry are the symmetric tensors, and the most significant example of a symmetric tensor is a Riemannian metric on a differentiable manifold [8]. A differentiable manifold is just a topological space on which we have the ability to differentiate mathematical objects. This is enough if we are interested only in differential topology, but to be able to measure the lengths of as well as determine the angles between tangent vectors, we need to introduce additional structure. The Riemannian metric provides us with this extra structure [10]. In simple terms; if we for all points p in a differentiable manifold M , define an inner product h·, ·i

on the fiber E

over p to the vector bundle π : E → M over M , then the collection of all these inner products constitute the Riemannian metric. From this we see that a Riemannian metric is not the same mathematical object as a metric in the sense of metric spaces, but the two concepts are closely related.

Our goal in this thesis is to prove the existence of Riemannian metrics on real vector bundles. To do this we first need to verify the existence of a smooth partition of unity on a given differentiable manifold. A partition of unity is a technical tool that can help one “patch” together locally defined objects with some desirable properties to obtain a globally defined object that also has the desired properties. We will use this tool in the proof of the existence of Riemannian metrics. Then we show that every differentiable manifold M has a tangent bundle T M which is a differentiable manifold in its own right [13], and that the tangent bundle T M is in turn a special case of a smooth vector bundle (Theorem 6.10). With this as a background we then reach the main result of this thesis which is the following theorem:

Theorem 6.15. On every real vector bundle there exist a Riemannian metric.

According to Morita in [9], geometry is the science of figures and the figures that are treated in modern geometry are called manifolds. Furthermore, Morita claims that the notion of manifolds was introduced by Bernhard Riemann in his lecture at Göttingen University in 1854, entitled: “Ueber die Hypothesen, welche der Geome- trie zu Grunde liegen” [11], and that in this talk the geometry of manifolds with

1

2

Riemannian metrics was also initiated. In others words, the branch of mathematics called differential geometry was now born.

In [5], Jost argue that even the concept of a differentiable manifold was in some vague manner implicitly contained in Bernhard Riemann’s lecture at Göttingen.

Roughly speaking, a differentiable manifold of dimension n is a manifold of dimen- sion n with a differentiable structure. In [9], Morita explains that in the study of differentiable manifolds, the tangent space at each point plays a fundamental role.

Neither vector fields nor differential forms could be defined without tangent spaces.

To quote Shigeyuki Morita in [9] (page 169):

On the other hand, a look at a manifold immersed in a Euclidean space clearly shows that the tangent spaces do not exist in isolation but move smoothly as points move on the manifold.

What Morita means is that because of this it is in some sense natural to consider the set of all tangent spaces, and this set is called the tangent bundle of a manifold.

Next level of abstraction is to look at the vector bundle. This structure is simply a generalization of the tangent bundle. In simple terms, if M is a differentiable manifold and if T

M is the tangent space to M at a point p, then the union T M of all the T

M as p varies over M is a vector bundle called the tangent bundle of M [10].

In summary, one can say that the three components that form the essence of this thesis are differentiable manifolds, vector bundles and Riemannian metrics, and with the help of partition of unity and Riemann’s indispensable heritage in differential geometry, we will show the existence of Riemannian metrics on real vector bundles.

Now some words about the terminology of the thesis. We treat the terms “smooth”

and “differentiable” as synonymous throughout this thesis. Some authors use the word “infinitely differentiable” to mean what we call “differentiable”. We will use the latter word for simplicity. Some authors use the word “smooth” to mean merely con- tinuously differentiable. We say that a function is smooth (or differentiable) if each of its component functions has continuous partial derivatives of all orders (Defini- tion 2.3). Lastly, usually we do not make a difference between a map and a function, but throughout this thesis we will reserve the term “function” for a map whose range is R or R

for some n > 1. We will use the word “mapping” to describe any kind of map, such as a map between arbitrary manifolds.

This thesis assumes familiarity with linear algebra and real analysis, and contains some concepts from multilinear algebra and abstract algebra (in case of need see e.g. [1–4]). A large part of this thesis rests on point-set topology, and therefore this thesis is provided with a detailed appendix about this field, in order to review some basic definitions and classical theorems see Appendix A.

This thesis is structured as follows. In Chapter 2 we construct a differentiable manifold by defining its two layers of structure, i.e. first we define a topological space with certain properties, then we define a differentiable structure on it. In Chapter 3 we investigate smooth functions on, and between, differentiable manifolds.

In Chapter 4 we look at the tangent spaces to a differentiable manifold, and define a tangent vector as a point-derivation of the algebra of germs of smooth functions at a point on a differentiable manifold. In Chapter 5 we prove the the existence of a smooth partition of unity subordinate to an open cover on a differentiable manifold.

This is an important tool that we will use in Theorem 6.15. In Chapter 6 we define

CHAPTER 1. INTRODUCTION 3

the tangent bundle to a differentiable manifold, and show that it is a smooth vector bundle. Lastly, we state and prove Theorem 6.15.

This thesis is based on [7, 8, 13]. Chapter 2, 3 and 5 is mainly inspired by [13],

and Chapter 4 and 6 is based on [7, 8, 13].

Chapter 2

Differentiable Manifolds

In this chapter we will construct a differentiable manifold by defining its two layers of structure, i.e. first we define a topological space with certain properties, then we define a so-called differentiable structure on it. We start by introducing topological spaces that are Hausdorff, second countable and locally Euclidean, i.e. topological manifolds. Then we describe the idea of a maximal smooth atlas, the differentiable structure, wich in turn will take us from topological manifolds to differentiable man- ifolds.

In simple terms, a n-dimensional topological manifold is a topological space that in and around each point resembles a common, n-dimensional Euclidean space. Ev- ery point on a topological manifold has a neighborhood that is homeomorphic to an open subset of R

. A topological manifold with a differentiable structure, i.e. a differentiable manifold, will allow us to carry out computations as we are used to in a Euclidean space, and therefore we can transfer many of the familiar concepts, such as calculus, from R

to a differentiable manifold. This chapter is based on [13].

We start by defining the locally Euclidean property of a topological space.

Definition 2.1. A topological space M is locally Euclidean of dimension n if every point p in M has a neighborhood U such that there is a homeomorphism φ : U → R

from U onto an open subset of R

. We call the pair (U, φ) a chart, U a coordinate neighborhood or a coordinate open set, and φ a coordinate map or a coordinate system on U .

We move on to define a topological manifold.

Definition 2.2. A topological manifold M is a Hausdorff, second countable and a locally Euclidean space. It is said to be of dimension n if it is locally Euclidean of dimension n.

From the definition of a topological manifold, we know that each point p ∈ M is contained in the domain of some chart (U, φ) on M . If φ(p) = 0, we say that the chart (U, φ) is centered at p ∈ U . If (U, φ) is any chart whose domain contains p and φ(p) 6= 0, it is easy to obtain a new chart centered at p by subtracting the constant vector φ(p).

The following definition is a reminder and will be our primary tool for study- ing higher-dimensional manifolds. We denote the standard coordinates on R

by r

, ..., r

and we let p = (p

, ..., p

) be a point in R

. We follow the conventions of

5

6

differential geometry by letting the indices on the coordinates to be superscripts and not subscripts.

Definition 2.3. Let k be a nonnegative integer and U an open subset of R

. A real-valued function f : U → R is said to be C

at p ∈ U if its partial derivatives

∂

f

∂x

· · · ∂x

of all orders j ≤ k exist and are continuous at p. The function f : U → R is smooth (or differentiable) at p if it is C

for all k ≥ 0; in other words, its partial derivatives

∂

f /∂x

· · · ∂x

of all orders exist and are continuous at p. A vector-valued function f : U → R

is said to be smooth at p if all of its component functions f

, ..., f

are smooth at p. We say that f : U → R

is smooth on U if it is smooth at every point in U .

Now we will define smooth-compatible charts, one of the most important ideas in the construction of differentiable manifolds. This kind of charts will in turn give rise to a smooth atlas on a given manifold, which in turn, by choosing a maximal smooth atlas, will give us a differentiable structure on that specific manifold. Just to clarify, when we speak of “compatible charts” we will always mean smooth-compatible charts.

Let us establish the following before we look at the definition of compatible charts.

Let (U, φ : U → R

) and (V, ψ : V → R

) be two charts of a topological manifold.

Because of the fact that U ∩V is open in U and φ : U → R

is a homeomorphism onto an open subset of R

, the image φ(U ∩ V ) will be an open subset of R

. Likewise, ψ(U ∩ V ) will also be an open subset of R

.

Definition 2.4. Two charts (U, φ : U → R

), (V, ψ : V → R

) of a topological manifold are smooth-compatible if the two maps

φ ◦ ψ

: ψ(U ∩ V ) → φ(U ∩ V ), ψ ◦ φ

: φ(U ∩ V ) → ψ(U ∩ V )

are smooth. If U ∩ V is nonempty, then these two maps are called the transition functions between the charts. If U ∩V is empty, then the two charts are automatically smooth-compatible.

We will now define a smooth atlas. In simple terms, if we have a collection of pairwise compatible charts that cover a given topological manifold, we have a smooth atlas on that manifold. Let us proceed to the definition.

Definition 2.5. A smooth atlas on a locally Euclidean space M is a collection U = {(U

, φ

)} of pairwise smooth-compatible charts that cover M , i.e. such that M = S

α

U

.

Note, we say that a chart (V, ψ) is compatible with an atlas {(U

, φ

)} if it is compatible with all the charts (U

, φ

) in the atlas.

Just to clarify, when we speak of an “atlas” we will always mean a smooth atlas.

Let us proceed to the next lemma claiming that if two charts are both compatible

with an atlas, they are compatible with each other.

CHAPTER 2. DIFFERENTIABLE MANIFOLDS 7

Lemma 2.6. Let {(U

, φ

)} be an atlas on a locally Euclidean space. If two charts (V, ψ) and (W, σ) are both compatible with the atlas {(U

, φ

)}, then they are com- patible with each other.

Proof. Let p ∈ V ∩ W . We need to show that σ ◦ ψ

is smooth at ψ(p). Since {(U

, φ

)} is an atlas for M , p ∈ U

for some α. Then p is in the triple intersection V ∩ W ∩ U

. Note that, σ ◦ ψ

= (σ ◦ φ

) ◦ (φ

◦ ψ

) is smooth on ψ(V ∩ W ∩ U

), hence at ψ(p). Since p was an arbitrary point of V ∩ W , this proves that σ ◦ ψ

is smooth on ψ(V ∩ W ). Similarly, ψ ◦ σ

is smooth on σ(V ∩ W ).

To construct a differentiable manifold we have to define a differentiable structure on our topological manifold. In general, there will be many possible choices of atlases that give that “same” differentiable structure. The solution is the maximal atlas.

Definition 2.7. An smooth atlas M on a locally Euclidean space is said to be maximal if it is not contained in a larger atlas; in other words, if U is any other atlas containing M, then U = M. A maximal smooth atlas is also called a differentiable structure.

Now we have all the tools needed to construct a differentiable manifold, let us proceed.

Definition 2.8. A differentiable manifold is a topological manifold together with a maximal smooth atlas.

Before we proceed, let us say something about the topological properties compact- ness and connectedness in the context of differentiable manifolds. A differentiable manifold does not have to be connected. We say that a differentiable manifold have dimension n if all of its connected components have dimension n. The dimension of a differentiable manifold is well-defined, i.e. if n 6= m then we know that an open subset of R

can not be homeomorphic to an open subset of R

. For a proof see e.g.

p. 89 in [13]. Lastly, a differentiable manifold M is said to be compact if every com- ponent is compact, M is said to be noncompact if at least one of the components are noncompact. For more information on these topological properties, see Appendix A.

For the curious reader who wants to immerse himself in this area, see e.g. [15].

We do not have to find a maximal atlas to determine if a topological manifold is a differentiable manifold, it is enough to find any atlas. The following theorem assures us that this is the case.

Theorem 2.9. Any atlas U = {(U

, φ

)} on a locally Euclidean space is contained in a unique maximal atlas.

Proof. Adjoin to the atlas U all charts (V

, W

) that are compatible with U. By Lemma 2.6 the charts (V

, W

) are compatible with one another. So the enlarged collection of charts is an atlas. Any chart compatible with the new atlas must be compatible with the original atlas U and so by construction belongs to the new atlas.

This proves that the new atlas is maximal.

Let M be the maximal atlas containing U that we have just constructed. If M

is

another maximal atlas containing U, then all the charts in M

are compatible with

U and so by construction must belong to M. This proves that M

⊂ M. Since both

are maximal, M

= M. Therefore, the maximal atlas containing U is unique.

8

The next theorem is the last one for this chapter and claims that it is possible to construct a differentiable manifold M × N by taking the Cartesian product of two given differentiable manifolds M and N . This construction is called a product manifold.

Theorem 2.10. If {(U

, φ

)} and {(V

, ψ

)} are smooth atlases for the manifolds M and N of dimensions m and n, respectively, then the collection

{(U

× V

, φ

× ψ

: U

× V

→ R

× R

)}

of charts is a smooth atlas on M × N . Therefore, M × N is a smooth manifold of dimension m + n.

Proof. See e.g. p. 20 in [7].

From here and onward, when we write “manifold” we will always mean a differentiable manifold. And remember that, throughout this thesis “smooth” and

“differentiable” will always mean the same thing.

We conclude this chapter with a brief summary. To find out if a topological space

is a differentiable manifold we have to check that the space in question is Hausdorff,

second countable and has a maximal smooth atlas.

Chapter 3

Functions on and Between Manifolds

3.1 Smooth Functions on Manifolds

We have constructed differentiable manifolds. In this chapter we will investigate smooth functions (and maps) on, and between, manifolds. With the help of our compatible charts and differentiable structure, we can transfer the ideas of smooth functions in R

to our manifolds. This chapter is based on [13].

We begin by defining the concept of coordinates on a chart to a given differentiable manifold.

Definition 3.1. If (U, φ : U → R

) is a chart of a differentiable manifold, we let x

= r

◦ φ be the ith component of φ and write φ = (x

, ..., x

) and (U, φ) = (U, x

, ..., x

). Thus, for p ∈ U, (x

(p), ..., x

(p)) is a point in R

. The functions x

, ..., x

are called coordinates or local coordinates on U .

Note, sometimes we omit the p and it becomes possible for the notation (x

, ..., x

) to represent coordinates on the open set U or to be a point in R

. This will be clear from the context and should not cause any confusions. We now move on to define a smooth function on a manifold.

Definition 3.2. Let M be a differentiable manifold of dimension n. A function f : M → R is said to be smooth (or differentiable) at a point p in M if there is a chart (U, φ) about p in M such that f ◦ φ

, a function defined on the open subset φ(U ) of R

, is smooth at φ(p). The function f is said to be smooth on M if it is smooth at every point of M .

In the definition above we did not assume f : M → R to be continuous. Though, if f is smooth at p ∈ M , then f ◦ φ

: φ(U ) → R, being a smooth function at the point φ(p) in an open subset of R

, is continuous at φ(p). As a composite of continuous functions, f = (f ◦ φ

) ◦ φ is continuous at p (see e.g. Theorem A.22).

Due to the fact that we are only interested in functions that are smooth on an open set, there is no loss of generality in assuming at the beginning that f is continuous.

Note, we denote the set of all smooth functions f : M → R on a manifold M by C

(M ), and because sums and constant multiples of smooth functions are smooth,

9

10 3.1. SMOOTH FUNCTIONS ON MANIFOLDS

C

(M ) is a vector space. And of course, the set of all smooth functions f : R

→ R on R

is denoted by C

(R

).

The following definition provides us with a useful notation that we frequently call upon.

Definition 3.3. The Kronecker delta δ is defined by

δ

=

 1 if i = j, 0 if i 6= j.

Let us move on to the next definition. Simply put, if we have a function f of a variable y, where y itself is a function of another variable x, then we can express f as a function of x. We will call this composite function the pullback of f by the function y.

Definition 3.4. Let N and M be sets. Let F : N → M be map and h a function on M . The pullback of h by F is the composite function h ◦ F : N → R. We denoted the pullback by F

h, i.e.

F

h : N → R.

Note, in this context a function f on a manifold M is smooth on a chart (V, ψ) if, and only if, its pullback (ψ

)

f by ψ

is smooth on the subset ψ(V ) of Euclidean space.

We will now define partial derivatives. Let f be a smooth function and (U, φ) a chart on a manifold M of dimension n.

Definition 3.5. For p ∈ U , we define the partial derivative ∂f /∂x

of f with respect to x

at p to be

∂

∂x

f := ∂f

∂x

(p) := ∂(f ◦ φ

)

∂r

(φ(p)) := ∂

∂r

(f ◦ φ

).

Note, since p = φ

(φ(p)), we can express the equation above in the form

∂f

∂x

(φ

(φ(p))) = ∂(f ◦ φ

)

∂r

(φ(p)).

Thus, as functions on φ(U ),

∂f

∂x

◦ φ

= ∂(f ◦ φ

)

∂r

.

Lastly, the partial derivative ∂f /∂x

is smooth on U because its pullback (∂f /∂x

) ◦ φ

is smooth on φ(U ).

In the next theorem we see that partial derivatives on a manifold satisfy the same property ∂r

/∂r

= δ

as the coordinate functions r

on R

.

Theorem 3.6. Suppose (U, x

, ..., x

) is a chart on a manifold. Then ∂x

/∂x

= δ

. Proof. At a point p ∈ U , by the definition of ∂/∂x

|

,

∂x

∂x

(p) = ∂(x

◦ φ

)

∂r

(φ(p)) = ∂(r

◦ φ ◦ φ

)

∂r

(φ(p)) = ∂r

∂r

(φ(p)) = δ

.

CHAPTER 3. FUNCTIONS ON AND BETWEEN MANIFOLDS 11

In Theorem 3.8 we will see three equivalent statements that relate smooth func- tions, charts and atlases on a given manifold. But first of all we state a remark that we will use in the theorem.

Remark 3.7. The definition of the smoothness of a function f at a point is indepen- dent of the chart (U, φ), for if f ◦ φ

is smooth at φ(p) and (V, ψ) is any other chart about p in M , then on ψ(U ∩ V ),

f ◦ ψ

= (f ◦ φ

) ◦ (φ ◦ ψ

), which is smooth at ψ(p).

Theorem 3.8. Let M be a differentiable manifold of dimension n, and f : M → R a real-valued function on M . The following are equivalent:

(i) the function f : M → R is smooth;

(ii) the manifold M has an atlas such that for every chart (U, φ) in the atlas, f ◦ φ

: R

⊃ φ(U ) → R is smooth;

(iii) for every chart (V, ψ) on M , the function f ◦ ψ

: R

⊃ ψ(V ) → R is smooth.

Proof. We will prove the theorem as a cyclic chain of implications.

(ii) ⇒ (i): This follows directly from the definition of a smooth function, since by (ii) every point p ∈ M has a coordinate neighborhood (U, φ) such that f ◦ φ

is smooth at φ(p).

(i) ⇒ (iii): Let (V, ψ) be an arbitrary chart on M and let p ∈ V . By Remark 3.7, f ◦ ψ

is smooth at ψ(p). Since p was an arbitrary point of V , f ◦ ψ

is smooth on ψ(V ).

(iii) ⇒ (ii): By Definition 3.2.

Note, if we want to prove the smoothness of a real-valued function, it is sufficient that one of the conditions in Theorem 3.8 applies to the charts of any atlas to conclude that the same condition apply to each chart on the given manifold.

3.2 Smooth Maps Between Manifolds

We move on to define a smooth map between manifolds.

Definition 3.9. Let N and M be differentiable manifolds of dimension n and m, respectively. A continuous map F : N → M is smooth (or differentiable) at a point p in N if there are charts (U, φ) about p in N and (V, ψ) about F (p) in M such that the composition ψ ◦ F ◦ φ

, a map from the open subset φ(F

(V ) ∩ U ) of R

to R

, is smooth at φ(p). The continuous map F : N → M is said to be smooth if it is smooth at every point of N .

Note, in the definition above we assumed F : N → M to be continuous to guarantee that F

(V ) is an open set in N . Hence, smooth maps between manifolds are by definition continuous.

Now we will use the smooth-compatibility property to show that the smoothness

of a map F : N → M at a point is independent of the choice of charts and is therefore

well-defined.

12 3.2. SMOOTH MAPS BETWEEN MANIFOLDS

Theorem 3.10. Suppose F : N → M is smooth at p ∈ N . If (U, φ) is any chart about p in N and (V, ψ) is any chart about F (p) in M , then ψ ◦ F ◦ φ

is smooth at φ(p).

Proof. From Definition 3.9 we have the following, since F is smooth at p ∈ N , there are charts (U

, φ

) about p in N and (V

, W

) about F (p) in M such that ψ

◦ F ◦ φ

is smooth at φ

(p). By the smooth-compatibility of charts in a differentiable structure, both φ

◦ φ

and ψ ◦ ψ

are smooth on open subsets of Euclidean spaces. Hence, the composite

ψ ◦ F ◦ φ

= (ψ ◦ ψ

) ◦ (ψ

◦ F ◦ φ

) ◦ (φ

◦ φ

) is smooth at φ(p).

The following theorem provides a procedure to control smoothness of a map without specifying a particular point on the manifold.

Theorem 3.11. Let N and M be differentiable manifolds, and F : N → M a continuous map. The following are equivalent:

(i) the map F : N → M is smooth;

(ii) there are atlases U for N and B for M such that for every chart (U, φ) in U and (V, ψ) in B, the map

ψ ◦ F ◦ φ

: φ(U ∩ F

(V )) → R

is smooth;

(iii) for every chart (U, φ) on N and (V, ψ) on M , the map ψ ◦ F ◦ φ

: φ(U ∩ F

(V )) → R

is smooth.

Proof. (ii) ⇒ (i): Let p ∈ N . Suppose (U, φ) is a chart about p in U and (V, ψ) is a chart about F (p) in B. By (ii), ψ ◦ F ◦ φ

is smooth at φ(p). By Definition 3.9, F : N → M is smooth at p. Since p was an arbitrary point of N , the map F : N → M is smooth.

(i) ⇒ (iii): Suppose (U, φ) and (V, ψ) are charts on N and M respectively such that U ∩ F

(V ) 6= ∅. Let p ∈ U ∩ F

(V ). Then (U, φ) is a chart about p and (V, ψ) is a chart about F (p). By Theorem 3.10, ψ ◦ F ◦ φ

is smooth at φ(p). Since φ(p) was an arbitrary point of φ(U ∩ F

(V )), the map ψ ◦ F ◦ φ

: φ(U ∩ F

(V )) → R

is smooth.

(iii) ⇒ (ii): By Definition 3.9.

Let us proceed to the next theorem which claims that the composition of smooth maps is smooth.

Theorem 3.12. If F : N → M and G : M → P are smooth maps of manifolds, then the composite G ◦ F : N → P is smooth.

Proof. Let (U, φ), (V, ψ) and (W, σ) be charts on N, M and P respectively. Then σ ◦ (G ◦ F ) ◦ φ

= (σ ◦ G ◦ ψ

) ◦ (ψ ◦ F ◦ φ

).

Since F and G are smooth, by Theorem 3.11 (i) ⇒ (iii), σ ◦ G ◦ ψ

and ψ ◦ F ◦ φ

are smooth. As a composite of smooth maps of open subsets of Euclidean spaces,

σ ◦ (G ◦ F ) ◦ φ

is smooth. By Theorem 3.11 (iii) ⇒ (i), G ◦ F is smooth.

CHAPTER 3. FUNCTIONS ON AND BETWEEN MANIFOLDS 13

3.3 Diffeomorphisms

Remember, a continuous bijective map whose inverse is also continuous is called a homeomorphism (Definition A.34). It is time to introduce another fundamental map that will follow us throughout this thesis.

Definition 3.13. A diffeomorphism between manifolds N and M is a smooth bijective map F : N → M whose inverse F

is also smooth. We say that N and M are diffeomorphic if there exists a diffeomorphism between them.

We know from the locally Euclidean property to a differentiable manifold M , that every point p ∈ M has a neighborhood U such that there is a homeomorphism φ from U onto an open subset of R

. The theorem below claims that φ in fact are a diffeomorphism. But before we look at Theorem 3.15, we have to define the identity map.

Definition 3.14. If A is a set, the identity map 1

: A → A is defined to be the map with domain and codomain A which satisfies

1

(p) = p for all points p in A.

Theorem 3.15. If (U, φ) is a chart on a manifold M of dimension n, then the coordinate map φ : U → φ(U ) ⊂ R

is a diffeomorphism.

Proof. By definition, φ is a homeomorphism, so it suffices to check that both φ and φ

are smooth. To test the smoothness of φ : U → φ(U ), we use the atlas {(U, φ)} with a single chart on U and the atlas {(φ(U ), 1

)} with a single chart on φ(U ). Since 1

◦ φ ◦ φ

: φ(U ) → φ(U ) is the identity map, it is smooth. By Theorem 3.11 (ii) ⇒ (i), φ is smooth.

To test the smoothness of φ

: φ(U ) → U , we use the same atlases as above.

Since φ ◦ φ

◦ 1

= 1

: φ(U ) → φ(U ), the map φ

is also smooth.

The following theorem shows us that if we have an open subset U of a manifold M and a diffeomorphism F : U → F (U ) ⊂ R

, we can conclude that (U, F ) forms a chart in the maximal atlas on M .

Theorem 3.16. Let U be an open subset of a manifold M of dimension n. If F : U → F (U ) ⊂ R

is a diffeomorphism onto an open subset of R

, then (U, F ) is a chart in the maximal atlas on M .

Proof. For any chart (U

, φ

) in the maximal atlas of M , both φ

and φ

are smooth by Theorem 3.15. As composites of smooth maps, both F ◦ φ

and φ

◦ F

are smooth (Theorem 3.12). Hence, (U, F ) is compatible with the maximal atlas. By the maximality of the atlas, the chart (U, F ) is in the atlas.

We conclude this chapter with a brief summary. We have seen that smooth

functions (and maps) on and between manifolds are independent of choice of charts

and is therefore well-defined. Theorem 3.8 gave us the relationship between smooth

functions, charts and atlases. We saw in Theorem 3.11 that we do not have to

specifying a particular point on a manifold to verify the smoothness of a map, instead

we can do this verification with help of our charts. Finally, we have seen that the

composition of smooth maps is smooth (Theorem 3.12), and that our coordinate

maps in fact are diffeomorphisms (Theorem 3.15).

Chapter 4

The Tangent Space

We want to make sense of linear approximations on manifolds. To do this we need to introduce the concept of the tangent space at a point in a manifold. The tangent space can be thought of as a sort of “linear model” for the manifold near the point.

The goal of this chapter is to find a way to formulate tangent vectors in R

that will generalize to manifolds. One way to do this is to define a tangent vector on a differentiable manifold M as a point-derivation of C

(M ), i.e. the algebra of germs of smooth functions at p. This chapter is based on [7, 8, 13].

We begin by defining tangent vectors in Euclidean space. Later in this chapter, when we have the fundamentals, we define tangent vectors on a manifold.

Definition 4.1. We define the tangent space at the point p ∈ R

to be the set T

R

= {(p, v) : v ∈ R

},

and we call an element of this space for a tangent vector at p in R

.

We write a point in R

as p = (p

, ..., p

) and a vector in the tangent space T

R

as

v

= (p, v) or





 v

.. . v





 or [v

, ..., v

]

,

where T is the ordinary transpose of a matrix. We think of (p, v) as the vector v

with its initial point at p. If it is clear from the context which point the vector v is tangent to, we sometimes denote a tangent vector only by v. The set T

R

is a real vector space under the operations

(p, v) + (p, w) := (p, v + w), c(p, v) := (p, cv),

where w ∈ R

and c ∈ R. We denote the standard basis for T

R

(or R

) by e

, ..., e

, and therefore we can write v

= P

i=1

v

e

for some v

∈ R.

Before we proceed we will make an “identification” between T

R

and R

, and to be able to do this we need the following definition.

15

16

Definition 4.2. Two vector spaces V and W over R are isomorphic, denoted by V ∼ = W , if there is a bijection T : V → W which preserves addition and scalar multiplication. That is, for all vectors u, v ∈ V , and all scalars r, s ∈ R,

T (ru + sv) = rT (u) + sT (v).

Because R

is an n-dimensional vector space, the tangent space of R

at a point p ∈ R

is again an n-dimensional vector space. Hence, T

R

is isomorphic to R

.

We move on to define the directional derivative. The Euclidean tangent

vector that we have defined, enables a way of taking “directional derivatives” of functions. This makes it possible for us to characterize tangent vectors as certain operators on functions. In simple terms, any tangent vector v ∈ T

R

yields a map D

: C

(R

) → R, which takes the directional derivative in the direction v at p.

Before we look at the definition, remeber that if (U, φ : U → R

) is a chart on a manifold M , then for p ∈ U we know that (x

(p), ..., x

(p)) is a point in R

(Definition 3.1).

Definition 4.3. The line through a point p = (p

, ..., p

) with direction v = [v

, ..., v

]

in R

has parametrization

c(t) = (p

+ tv

, ..., p

+ tv

),

where t ∈ R. Its ith component c

(t) is p

+ tv

. If f is smooth in a neighborhood of p in R

and v is a tangent vector at p, the directional derivative of f in the direction v at p is defined to be

D

f = lim

t→0

f (c(t)) − f (p)

t = d

dt    

f (c(t)).

By the chain rule,

D

f =

n

X

i=1

dc

dt (0) ∂f

∂x

(p) =

n

X

i=1

v

∂f

∂x

(p). (4.1)

Let us take a closer look at the notation D

f . Note that since v is a vector at p, it is understood that the partial derivatives are to be evaluated at p. Hence D

f is a number and not a function. If we want (4.1) to be written in terms of a map we write

D

=

n

X

i=1

v

∂

∂x

and interpret this notation as the map that sends a function f to the number D

f .

If two functions coinciding on some neighborhood of a point p, they will have

the same directional derivatives at p. This indicates that we should introduce an

equivalence relation on the set of smooth functions defined in some neighborhood

of p.

CHAPTER 4. THE TANGENT SPACE 17

Definition 4.4. A relation on a set S is a subset R of S × S. Given x, y in S, we write x ∼ y if, and only if, (x, y) ∈ R. The relation R is an equivalence relation if it satisfies the following three properties for all x, y, z ∈ S:

i) x ∼ x (reflexivity);

ii) if x ∼ y, then y ∼ x (symmetry);

iii) if x ∼ y and y ∼ z, then x ∼ z (transitivity).

Let us introduce an equivalence relation on the set of smooth functions. This will in turn generate germs of smooth functions which we now will define. Consider the set of all orderd pairs (f, U ), where U is a neighborhood of p and f : U → R is a smooth function. Let (g, V ) be an arbitrary such set constructed in the same way.

Definition 4.5. We say that (f, U ) is equivalent to (g, V ) if there is an open set W ⊂ U ∩V containing p such that f = g when restricted to W . This is an equivalence relation, and the equivalence class of (f, U ), denoted by [(f, U )], is called the germ of f at p. We write C

(R

) for the set of all germs of smooth functions on R

at p.

We move on to define an algebra over R.

Definition 4.6. An algebra over R is a vector space A over R with a multiplication map

µ : A × A → A,

usually written µ(a, b) = a · b, such that for all a, b, c ∈ A and r ∈ R:

i) (a · b) · c = a · (b · c) (associativity);

ii) a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c (distributivity);

iii) r(a · b) = (ra) · b = a · (rb) (homogeneity).

Let us use the same notation as in Definition 4.5. Note, addition and multipli- cation of functions and scalar multiplication with functions, induce corresponding operations on C

(R

). So we can make C

(R

) into an algebra over R by defining the operations:

[(f, U )] + [(g, V )] := [(f + g, U ∩ V )], c[(f, U )] := [(cf, U )],

[(f, U )][(g, V )] := [(f g, U ∩ V )],

where c ∈ R. These operations are well-defined since on some neighborhood of p both sides are equal. Hence they belong to the same equivalence class. We can take the value of a germ at p if we define an evaluation map by

[(f, U )](p) := f (p).

This map is well-defined since every other function in [(f, U )] equals f on a neigh-

borhood of p, and therefore also at p. The set C

(R

) is also a real vector space

under the operations that we just defined above. This leads us to the next definition.

18

Definition 4.7. A map L : V → W between vector spaces over R is called a linear map (or a linear operator ) if for any u, v ∈ V and r, s ∈ R,

L(ru + sv) = rL(u) + sL(v).

We say that the map is R-linear if we want to emphasize which field the scalars belong to.

An algebra over R, such as C

(R

), is an algebraic structure, and an algebra homomorphism is an structure preserving map between algebras that we now will define.

Definition 4.8. If A and A

are algebras over R, then an algebra homomorphism is a linear map L : A → A

that preserves the algebra multiplication: L(ab) = L(a)L(b) for all a, b ∈ A.

We know that we can associate a tangent vector v with the directional derivative D

by v 7→ D

, and thereby we have found a way to characterize tangent vectors as certain operators on functions. Hence, for each tangent vector v at a point p in R

, the directional derivative at p gives a map of real vector spaces

D

: C

(R

) → R.

From (4.1) we get that D

is R-linear and fulfills the Leibniz rule

D

(f g) = (D

f )g(p) + f (p)(D

g), (4.2) just because the partial derivatives ∂/∂x

|

have these kind of properties.

We move on to define a point-derivation of the set of all germs of smooth functions on R

at a point p.

Definition 4.9. A linear map D : C

(R

) → R satisfying the Leibniz rule (4.2) is called a point-derivation of C

(R

) (or a derivation at p). We denote the set of all derivations at p by D

R

.

We should notice two things before continuing. From Definition 4.9 we get that D

is a point-derivation of C

(R

). And, the set D

R

is a real vector space. This can be confirmed by noting that the sum of two derivations at p, and a scalar multiple of a derivation at p, are again derivations at p.

Because the directional derivatives at p are all derivations at p, there exists a map

φ : T

R

→ D

R

, (4.3)

v 7→ D

=

n

X

i=1

v

∂

∂x

.

We know that D

is linear in v, and therefore we get that the map φ is a linear map

of vector spaces. We want show that the linear map φ is an isomorphism of vector

spaces. In that case, we could in a precise way identify the tangent vectors at p with

the derivations at p. It turns out that this is possible and Theorem 4.13 provides an

answer for us. But before we get to this theorem we need some essential tools.

CHAPTER 4. THE TANGENT SPACE 19

Lemma 4.10. If D is a point-derivation of C

(R

), then D(c) = 0 for any constant function c.

Proof. Since we do not know whether every derivation at p is a directional derivative, we need to prove this lemma using only the defining properties of a derivation at p.

By R-linearity, D(c) = cD(1). So it suffices to prove that D(1) = 0. By the Leibniz rule (4.2),

D(1) = D(1 · 1) = D(1) · 1 + 1 · D(1) = 2D(1).

Subtracting D(1) from both sides gives 0 = D(1).

We move on to define a star-shaped set.

Definition 4.11. We say that a subset S of R

is star-shaped with respect to a point p in S if for every x in S, the line segment from p to x lies in S.

Note, any open ball

B(p, ) = {x ∈ R

: ||x − p||< }, where  > 0, is star shaped with respect to the point p.

The following theorem (Taylor’s theorem with remainder) will be used in Theorem 4.13. We know that a smooth function does not have to be real-analytic, and therefore does not have to be equal to its Taylor series (for more information about this see e.g. p. 5 in [13]). Taylor’s theorem with remainder for smooth functions solves this problem for us. We prove the first case in which the Taylor series consists of only the constant term f (p).

Theorem 4.12. Let f be a smooth function on an open subset U of R

star- shaped with respect to a point p = (p

, ..., p

) in U . Then there are functions g

(x), ..., g

(x) ∈ C

(U ) such that

f (x) = f (p) +

n

X

i=1

(x

− p

)g

(x), g

(p) = ∂f

∂x

(p).

Proof. Since U is star-shaped with respect to p, for any x in U the line segment p + t(x − p), 0 ≤ t ≤ 1, lies in U . So f (p + t(x − p)) is defined for 0 ≤ t ≤ 1. By the chain rule we have

d

dt f (p + t(x − p)) =

n

X

i=1

(x

− p

) ∂f

∂x

(p + t(x − p)).

If we integrate both sides with respect to t from 0 to 1, then from the fundamental theorem of calculus we get

f (p + t(x − p))  

1 0

=

n

X

i=1

(x

− p

) ˆ

0

∂f

∂x

(p + t(x − p))dt. (4.4) Let

g

(x) = ˆ

0

∂f

∂x

(p + t(x − p))dt.

20

Because the integrand is smooth in all variables, we can differentiate under the integral infinitely many times, thus g

(x) is smooth and (4.4) becomes

f (x) − f (p) =

n

X

i=1

(x

− p

)g

(x).

Moreover,

g

(p) = ˆ

0

∂f

∂x

(p)dt = ∂f

∂x

(p).

The following theorem states that the linear map φ : T

R

→ D

R

is an isomorphism of vector spaces, and therefore we can in a precise way identify the tangent vectors at p with the derivations at p.

Theorem 4.13. The linear map φ : T

R

→ D

R

defined in (4.3) is an isomorphism of vector spaces.

Proof. To prove injectivity, suppose D

= 0 for v ∈ T

R

. If we apply D

to the coordinate function x

, then from Theorem 3.6 we get

0 = D

(x

) =

n

X

i=1

v

∂

∂x

x

=

n

X

i=1

v

δ

= v

.

Hence, v = 0 and φ is injective.

To prove surjectivity, let D be a derivation at p and let [(f, V )] be a representative of a germ in C

(R

). Making V smaller if necessary, we may assume that V is an open ball, hence star-shaped. By Taylor’s theorem with remainder (Theorem 4.12) there are smooth functions g

(x) in a neighborhood of p such that

f (x) = f (p) +

n

X

i=1

(x

− p

)g

(x), g

(p) = ∂f

∂x

(p).

Applying D to both sides and noting that D(f (p)) = 0 and D(p

) = 0 by Lemma 4.10, we get by the Leibniz rule (4.2)

Df (x) =

n

X

i=1

(Dx

)g

(p) +

n

X

i=1

(p

− p

)Dg

(x) =

n

X

i=1

(Dx

) ∂f

∂x

(p).

This proves that D = D

for v = [Dx

, ..., Dx

]

.

From this theorem we now see that it is possible to identify the tangent vectors at p with the derivations at p. Before we proceed to define tangent vectors on a manifold, let us make an important identification. Under the vector space

isomorphism T

R

∼ = D

R

, we see that the standard basis e

, ..., e

for T

R

corresponds to the set ∂/∂x

|

, ..., ∂/∂x

|

of partial derivatives. From now on, we will make this identification and write a tangent vector v = [v

, ..., v

]

= P

i=1

v

e

as

v =

n

X

i=1

v

∂

∂x

. (4.5)

CHAPTER 4. THE TANGENT SPACE 21

Now we are in a position to define a tangent space at a point p in a manifold M , and we will do this with help of the germ of a smooth function. Just as for R

, we define a germ of a smooth function at p in M to be an equivalence class of smooth functions defined in a open neighborhood of p in M , two such functions being equivalent if they agree on some open neighborhood of p.

Let M be a differentiable manifold, and let p ∈ M be a point. Let N

be the collection of all open neighborhoods of p, and define a set by

S(p, M ) := [

U ∈Np

C

(U ).

Finally, let us introduce an equivalence relation on the set S(p, M ). This will in turn generate germs of smooth functions which we now will define.

Definition 4.14. We say that f, g ∈ S(p, M ) are equivalent if, and only if, f = g on some open neighborhood of p. We denote the set of equivalence classes on S(p, M ) by C

(M ). The elements of C

(M ) are called germs, and the equivalence class of f is denoted by [f ].

We can make C

(M ) into an algebra over R by defining the operations:

[f ] + [g] := [f + g], c[f ] := [cf ], [f ][g] := [f g],

where c ∈ R. These operations is well-defined since on some neighborhood of p both sides are equal. Hence they belong to the same equivalence class. We can take the value of a germ at p if we define an evaluation map by

[f ](p) := f (p).

This map is well-defined since every other function in [f ] equals f on a neighborhood of p, and therefore also at p.

We move on to define a point-derivation of the set of all germs of smooth functions on a manifold M at a point p.

Definition 4.15. Let M be a differentiable manifold, and let p ∈ M be a point.

We define a point-derivation of C

(M ) (or a derivation at p), to be a linear map D

: C

(M ) → R such that the Leibniz rule is satisfied

D

([f ][g]) = D

[f ]g(p) + f (p)D

[g].

Note, by definition we know that a point-derivation of C

(M ) is linear. So, if we define

(D

+ D

)[f ] := D

[f ] + D

[f ],

where D

and D

are point-derivations of C

(M ), it follows that D

+ D

is also a point-derivation. Hence, the set of all point-derivations of C

(M ) is a vector space.

Just to clarify, a point-derivation of C

(R

) is denoted by D and a point- derivation of C

(M ), where M is a manifold, is denoted by D

.

Finally, we move on to define tangent vectors on a manifold M and the space

they live in, the tangent space.

22

Definition 4.16. We define the tangent space at a point p ∈ M , denoted by T

M , to be the vector space of all point-derivations of C

(M ), and we call an element of the tangent space a tangent vector at p in M .

Note, we denote a tangent vector in T

M by v

= (p, v). We see that v

is of the form of an ordered pair (p, v), where p ∈ M and v ∈ T

M . If it is clear from the context which point the vector v is tangent to, we sometimes denote a tangent vector only by v.

The action of a derivation is a differentiation, so it seems that a derivation at p acts on a function that is defined only within some neighborhood U of p. But from definition, the domain of a derivation is C

(M ) and not C

(U ). There is nothing in the definition that immediately allows an element of T

M to act on C

(U ) unless U = M . This leads us to the following remark.

Remark 4.17. If U is an open set containing p in M , then the algebra C

(U ) of germs of smooth functions in U at p is the same as C

(M ). Hence, T

U = T

M . For a proof see e.g. p. 62-63 in [7].

Let f be a smooth function at a point p in a manifold M of dimension n, and (U, φ) a chart about p. Since φ is a function into R

it has n components x

, ..., x

. This means that if r

, ..., r

are the standard coordinates on R

, then x

= r

◦ φ : U → R.

From Definition 3.5 we know that

∂f

∂x

(p) = ∂(f ◦ φ

)

∂r

(φ(p)).

Let us define an operator

|

: C

(M ) → R by

∂

∂x

[f ] := ∂f

∂x

(p).

We want to show that

|

∈ T

M , and therefore we have to show that the

operator

|

is well-defined. If we let h ∈ [f ], then we know that h = f on some neighborhood N of p. Hence, f ◦ φ

= h ◦ φ

on φ(N ). Therefore it holds that

∂(f ◦ φ

)

∂r

(φ(p)) = ∂(h ◦ φ

)

∂r

(φ(p)).

Thus,

(p) =

(p) =

|

[f ], and the operator is well-defined. Lastly, if we apply the product rule we get

∂

∂x

([f ][h]) = f (p) ∂

∂x

[h] + h(p) ∂

∂x

[f ].

Hence,

|

∈ T

M .

The following theorem states that the set {

|

, ...,

|

} is a basis for T

M .