Stokes' Theorem on Smooth Manifolds

Home , Directional derivative, Multivariable calculus, Partial derivative, Smoothness

Stokes’ Theorem on Smooth Manifolds Erik Jörgenfelt

Spring 2016 Thesis, 15hp Bachelor of mathematics, 180hp Department of mathematics and mathematical statistics

Abstract A proof of Stokes’ theorem on smooth manifolds is given, complete with prerequisite results in tensor algebra and diﬀerential geometry. The essay assumes familiarity with multi- variable calculus and linear algebra, as well as a basic understanding of point-set topology. Stokes’ theorem is then applied to the conservation of energy-momentum in general relativity under the existence of so called Killing vectors.

Sammanfattning Stokes sats för släta mångfalder bevisas, komplett med nödvändiga resultat från tensoral- gebran och diﬀerentialgeometrin. Uppsatsen förutsätter förtrogenhet med ﬂervariabelanalys och linjär algebra, samt en grundläggande förståelse för allmän topologi. Stokes sats appli- ceras sedan till bevarande av energi-momentum i allmän relativitetsteori under existensen av so kallade Killingvektorer.

Contents

1 Introduction 1

2 Preliminaries 3 2.1 Tensors...... 3 2.2 Smooth Manifolds with Boundary...... 9 2.3 Vector Bundles...... 14

3 Partitions of Unity 17

4 The Tangent Bundle 25

5 Diﬀerential Forms and Stokes’ Theorem 35 5.1 Diﬀerential Forms and Integration...... 35 5.2 Stokes’ Theorem on Smooth Manifolds...... 47

6 Conservation of Energy-Momentum 50 6.1 Regular Submanifolds...... 51 6.2 The Tensor Bundles...... 51 6.3 Lie Derivative...... 53 6.4 Lorentz Manifolds...... 54 6.5 Connection...... 56 6.6 Hodge Duality...... 59 6.7 Conservation of Energy-Momentum and Killing Vectors...... 60

1 Introduction

Stokes’ theorem on smooth manifolds subsumes three important theorems from standard calculus, unifying them and extending the application to any number of dimensions. It does this in the language of differential geometry. According to Katz[6] the unified theorem was first published in 1945 by Élie Cartan in [1], although several previous versions are also listed. It lends its name from the last of the subsumed theorems, which was published by George Stokes as a problem in the Smith’s Prize Exam at Cambridge in 1854[13]. According to Katz he had received a letter with the theorem by Lord Kelvin on July 2. Today we know this theorem as Stokes’ theorem, the Kelvin-Stokes theorem, or the curl theorem. The first published proof is however, again according to Katz, by Hermann Hankel in 1861[4]. The other subsumed theorems are today referred to as Gauss’ theorem or the divergence theorem, and Green’s theorem. Although Gauss did publish three special cases of the former[3], Katz lists Michael Ostrogradsky[10] as the first to state and prove the theorem in a lecture, not published until later, and publish it first in 1831[11]. Green’s theorem was stated first without proof by Augustin Cauchy[2] in 1846, but proved five years later by Bernhard Riemann[12]. Katz also notes that all the aforementioned authors, as well as others publishing related results, were interested in them for applications in physics. Our main theorem, stated below, among other things allows for native application in the formalism of space-time — a 4-dimensional Lorentz manifold — where it is frequently used to work with flux integrals and conservation theorems. In [5] Harrison produces a Stokes’ theorem for non-smooth chains, thus building on the work of Whitney[16], who used TheoremA to define integration over certain non-smooth domains. TheoremA (Stokes’ theorem on smooth manifolds). For any smooth (n − 1)-form ω with compact support on the oriented n-dimensional smooth manifold M with boundary ∂M, we have that Z Z dω = ω, M ∂M where ∂M is given the induced orientation. This essay is structured as follows. In Section2 we cover preliminary results in tensor algebra, and define smooth manifolds with boundary and vector bundles. In Section3 we show the existence of a partition of unity subordinate to any open cover of a smooth manifold. In Section4 we construct the tangent bundle over a smooth manifold. In Section5 we construct smooth differential forms, define oriented manifolds, and define integration over oriented smooth manifolds with boundary. We then prove TheoremA. In Section6 we apply TheoremA in the context of general relativity to prove TheoremB: TheoremB. Let S be a Lorentz manifold with the Levi-Civita connection, with dim(S) = n, which we will refer to as spacetime, endowed with a stress-energy tensor, T . If ξ is a Killing i i j vector then J := Tj ξ is a conserved flux, i.e. I i J dΣi ≡ 0, Σ for any sufficiently nice closed hypersurface Σ. This essay is based on [7], with some inspiration from [14] and [8]. The proof of TheoremB is inspired by [8]. We prove several supporting results that were left without proof in [7], among

1 these Proposition 2.20, Proposition 2.27, and Proposition 5.1; and state and prove Proposition 5.16, which was not included in [7]. Additionally, our definition of smooth manifolds makes the stronger assumption of second countability over paracompactness, which [7] uses. Second countability is necessary in some other applications such as Sard’s theorem. We thus proceed to show, in Proposition 3.3, that our definition implies paracompactness. Furthermore, we make no mention of what Lee refers to as TpMphys, or roughly “the tangent space according to physicists,” defined via transformation laws. Instead we use what Lee refers to as “the kinematic tangent space,” or TpMkin, here referred to as simply TpM, defined via curves, to define the tangent space at the boundary (Definition 4.15), and in Lemma 5.18. This essay also expands on any arguments in [7] that the author found abstruse; either because they presumed a greater familiarity with the subject than can be expected at this level, or because Lee intentionally left parts of the argument to the reader. Finally, the focus on TheoremA allows the material to be restructured to provide a (hopefully) clearer line of reasoning. 2 Preliminaries

2.1 Tensors Here we study alternating covariant tensors. Tensors can be thought of as generalisations of vectors, in as much as it is linear in each argument, and indeed all vectors are also tensors. We will not concern ourselves with this fact here, but rather construct our tensors from vectors and dual vectors, the latter of which are elements of the dual of a vector space. We therefore first define the dual of a vector space, before introducing consolidated tensors. Although we quickly move on to consider only alternating covariant tensors, the dual space will appear again in Section4 in the form of the cotangent space. In Section6 we briefly touch on more general tensors again. Alternating covariant tensors will form the basis which we build our definition of differential forms (Section5) on. As such they are crucial to our development of TheoremA. The exterior product of two alternating covariant tensors is defined and explored to sufficient depth to provide a basis for each space of alternating covariant tensors, and at the end this allows us to give an alternate definition of the determinant of a linear transformation on a vector space and show that it is equivalent to the matrix definition. The last result not only provides a satisfying link between these results and ones known from elementary linear algebra, but will also be used in Section5 when we investigate oriented manifolds. We begin with the definition of the dual of a vector space, and the dual of any basis:

Definition 2.1. The dual space of a vector space V is the set of all linear maps ω : V → R. ∗ We denote it by V , and call its elements dual vectors. If (e1, . . . , en) is a basis of V , and 1 n ∗ i i 1 n ω , . . . , ω are elements of V such that ω (ej) = δj, then ω , . . . , ω is called the dual basis of (e1, . . . , en). That ω1, . . . , ωn is a basis of V ∗ follows from the linearity of its elements, and we will immediately move on to define (consolidated) tensors over vector spaces: Definition 2.2. A (consolidated) tensor on a real valued vector space V is a multilinear map

∗ ∗ t : V1 × ... × Vr × V1 × ... × Vs → R, ∗ ∗ where all Vi = V and all Vj = V . We say that t is contravariant of dimension r and covariant r of dimension s, and write t ∈ Ts (V ). In fact, once we have deﬁned the tensor product and shown its associativity — a property we should expect from a product operator — we will only be dealing with covariant tensors, i.e. 0 tensors that are contravariant of dimension zero, denoted Ts (V ) or just Ts(V ). As an example of a covariant tensor, consider the familiar inner product (or “dot-product”) on Rn. It takes two vectors as arguments, is linear in each argument, and returns a real number — a two-dimensional, covariant tensor. The tensor product combines two tensors to produce a third. In principle it adjoins the two tensors. As such it can easily be seen to have an identity element, but is obviously not commutative.

Deﬁnition 2.3. For tensors t ∈ T r1 (V ) and t ∈ T r2 (V ), we deﬁne the tensor product t ⊗t ∈ 1 s1 2 s2 1 2 T r1+r2 by s1+s2

1 r1+r2 t1 ⊗ t2(θ , . . . , θ ,v1, . . . vs1+s2 ) :=

1 r1 r1+1 r1+r2 t1(θ , . . . , θ , v1, . . . , vs1 )t2(θ , . . . , θ , vs1+1, . . . , vs1+s2 ).

3 Proposition 2.4. Let t ∈ T r1 (V ), t ∈ T r2 (V ), t ∈ T r3 (V ). Then 1 s1 2 s2 3 s3

(t1 ⊗ t2) ⊗ t3 = t1 ⊗ (t2 ⊗ t3).

ri ,...,ri Proof. Applying both sides to (α1, . . . , αr1+r2+r3 , v , . . . , v ) and taking t 1 k to mean 1 s1+s2+s3 sj1 ,...,sj` r r t α i1 , . . . , α ik , v , . . . , v we get sj1 sj`

1,...,r1+r2+r3 1,...,r1+r2 r1+r2+1,...,r1+r2+r3 (t1 ⊗ t2) ⊗ t3 = (t1 ⊗ t2) (t3) 1,...,s1+s2+s3 1,...,s1+s2 s1+s2+1,...,s1+s2+s3 = (t )1,...,r1 (t )r1+1,...,r1+r2 (t )r1+r2+1,...,r1+r2+r3 1 1,...,s1 2 s1+1,...,s1+s2 3 s1+s2+1,...,s1+s2+s3 = (t )1,...,r1 (t ⊗ t )r1+1,...,r1+r2+r3 1 1,...,s1 2 3 s1+1,...,s1+s2+s3 1,...,r1+r2+r3 = t1 ⊗ (t2 ⊗ t3) 1,...,s1+s2+s3

Our previous example of a tensor was in fact a metric tensor, and all metric tensors are symmetric (see Section6 for more information on metric tensors), meaning that on interchanging the arguments the result is unchanged. We will instead be using alternating (or anti-symmetric, or skew symmetric) tensors, see Definition 2.5. Perhaps the most familiar example of the alternating property is the vector product (or “cross product”) of two vectors — the generalization of which is the exterior product (Definition 2.8) we define later — but this is not a tensor, since it produces a vector rather than a real number. Let us make the definition precise, and hopefully things will be made clear:

Deﬁnition 2.5. A covariant tensor t ∈ Ts(V ) is alternating if

t(v1, . . . , vs) = sgn (σ)t(vσ(1), . . . , vσ(s)) for all v1, . . . , vs ∈ V and all permutations σ of (1, . . . , s).

k We will henceforth let Lalt(V ) denote the set of alternating covariant tensors of dimension k 0 on a real vector space V . We take Lalt(V ) to be equivalent to R. Given any covariant tensor, we can construct an alternating covariant tensor in such a way that any already alternating tensor is left unchanged. This is done with the antisymmetrization map defined next (Definition 2.6). We will use it as a means to form a product of alternating covariant tensors, and Lemma 2.7 further expresses how it does not affect already alternating tensors, even in a product. The lemma will be used in Proposition 2.9 before it can be forever forgotten.

k k Deﬁnition 2.6. The antisymmetrization map Alt : Tk(V ) → Lalt(V ) is deﬁned by 1 X Altk(ω)(v , . . . , v ) := sgn(σ)ω(v , . . . , v ), 1 k k! σ1 σk σ∈Sk where Sk is the symmetric group of k letters.

0 0 Lemma 2.7. For α ∈ Tk (V ) and β ∈ T` (V ), Altk+` Altk α ⊗ β = Altk+` (α ⊗ β) Altk+` α ⊗ Alt` β = Altk+` (α ⊗ β) ,

4 and consequently Altk+` Altk (α) ⊗ Alt` (β) = Altk+` (α ⊗ β) .

0 Proof. For σ ∈ Sk and any T ∈ Tk (V ) let (σT )(v1, . . . , vk) := T vσ1 , . . . , vσk , whence

Altk (σT ) = sgn (σ) Altk (T ) .

Now, by the deﬁnition of the antisymmetrization map

   1 X Altk+` Altk (α) ⊗ β = Altk+`  sgn (σ)(σα) ⊗ β k!   σ∈Sk   1 X = Altk+` sgn (σ)(σα ⊗ β) k!  σ∈Sk 1 X = sgn (σ) Altk+` (σα ⊗ β) . k! σ∈Sk

0 By extending σ into σ ∈ Sk+` deﬁned by ( σ(i), if i ≤ k σ0(i) = i, if i > k, we ﬁnd σα ⊗ β = σ0 (α ⊗ β) and sgn σ = sgn σ0. Thus

1 X Altk+` Altk (α) ⊗ β = sgn σ0 Altk+` σ0 (α ⊗ β) k! σ∈Sk 1 X = sgn σ0 sgn σ0 Altk+` (α ⊗ β) k! σ∈Sk 1 X = Altk+` (α ⊗ β) k! σ∈Sk = Altk+` (α ⊗ β) . By symmetric reasoning Altk+` α ⊗ Alt` (β) = Altk+` (α ⊗ β), and chained application of these results give us the ﬁnal part of the lemma.

The exterior product between alternating covariant tensors is an antisymmmetric tensor k product. Investigation of its properties is revealing about the structure of Lalt(V ), and we will spend most of the remainder of this subsection doing so. When we later define a similar product between differential forms, it will be entirely based on the Definition 2.8. We follow Definition 2.8 with Proposition 2.9, which exhibits three fundamental properties of the wedge product.

5 k1 k2 Deﬁnition 2.8. Given ω ∈ Lalt(V ) and η ∈ Lalt(V ) we deﬁne the wedge product or exterior k1+k2 product ω ∧ η ∈ Lalt (V ) by (k + k )! ω ∧ η := 1 2 Altk1+k2 (ω ⊗ η). k1! k2!

k ` m Proposition 2.9. Let α ∈ Lalt(V ), β ∈ Lalt(V ), and γ ∈ Lalt(V ). Then k ` k+` (i) ∧ : Lalt(V ) × Lalt(V ) → Lalt (V ) is R-bilinear; (ii) α ∧ β = (−1)k`β ∧ α; (iii) α ∧ (β ∧ γ) = (α ∧ β) ∧ γ.

0 0 ` Proof. We start by proving (i). For c, c ∈ R and β ∈ Lalt(V ) we have, by the deﬁnition (k + `)! α ∧ (cβ + c0β0) (v , . . . , v ) = Altk+`(α ⊗ (cβ + c0β0))(v , . . . , v ) 1 k+` k! `! 1 k+` 1 X = sgn(σ)α(v , . . . , v )(cβ + c0β0)(v , . . . , v ) k! `! σ1 σk σk+1 σk+` σ∈Sk+` c X = sgn(σ)α(v , . . . , v )β(v , . . . , v ) + k! `! σ1 σk σk+1 σk+` σ∈Sk+` c0 X + sgn(σ)α(v , . . . , v )β0(v , . . . , v ) k! `! σ1 σk σk+1 σk+` σ∈Sk+` = c(α ∧ β) + c0(α ∧ β0).

Clearly the symmetric argument goes through unedited for the ﬁrst term, which gives us (i). To prove (ii) we let τ ∈ Sk+` be such that τ(1, . . . , k + `) = (k + 1, . . . , k + `, 1, . . . , k). Then (k + `)! (α ∧ β)(v , . . . , v ) = Altk+`(α ⊗ β)(v , . . . , v ) 1 k+` k! `! 1 k+` 1 X = sgn(σ)α(v , . . . , v )β(v , . . . , v ) k! `! σ1 σk σk+1 σk+` σ∈Sk+` 1 X = sgn(σ)α(v , . . . , v )β(v , . . . , v ) k! `! στ`+1 στ`+k στ1 στ` σ∈Sk+` 1 X = sgn(τ) sgn(στ)β(v , . . . , v )α(v , . . . , v ). k! `! στ1 στ` στ`+1 στ`+k σ∈Sk+`

To see this recall that sgn(στ) = sgn(σ) sgn(τ) and that sgn(τ) sgn(τ) = 1 for all σ, τ ∈ Sk. We thus have

(α ∧ β)(v1, . . . , vk+`) = sgn(τ)(β ∧ α)(v1, . . . , vk+`), since if σ varies over all of Sk+` then so does στ. Now, since τ obviously contains k` inversions, sgn(τ) = (−1)k`. This gives us (ii). Finally, for (iii) we simply compute

(k + ` + m)! α ∧ (β ∧ γ) = Alt α ⊗ (β ∧ γ) k!(` + m)!

6 (k + ` + m)! (` + m)! = Alt α ⊗ Alt (β ⊗ γ) k!(` + m)! `! m! (k + ` + m)! = Alt α ⊗ Alt (β ⊗ γ) . k! `! m! By lemma 2.7 Alt α ⊗ Alt (β ⊗ γ) = Alt α ⊗ (β ⊗ γ), so we end up with

(k + ` + m)! α ∧ (β ∧ γ) = Alt α ⊗ (β ⊗ γ) . k! `! m! Symmetric computation gives us (k + ` + m)! (α ∧ β) ∧ γ = Alt (α ⊗ β) ⊗ γ , k! `! m! and by the associativity of the tensor product (proposition 2.4), we have (iii).

We note here that combining properties (i) and (iii) of the proposition yields k-linearity in R of a wedge product of k terms. From Proposition 2.9 follow several interesting results, and apart from Corollary 2.11 they will all be used in Section5. It is the following that is the culmination of this subsection.

1 k ∗ 1 Lemma 2.10. Let α , . . . , α be elements of V = Lalt(V ), and let v1, . . . , vk be elements of V . Then

1 k α ∧ · · · ∧ α (v1, . . . , vk) = det A,

h i i i i where A = aj is the k × k matrix whose ij:th entry is aj = α (vj). Proof. From proposition 2.9, we have α1 ∧ · · · ∧ αk = k! Alt α1 ⊗ · · · ⊗ αk , or,

1 k X 1 k α ∧ · · · ∧ α (v1, . . . , vk) = sgn(σ)α (vσ1 ) ··· α (vσk )

σ∈Sk = det A, where A is as above.

k 1 ∗ For Corollary 2.11, which we will use to ﬁnd a basis of Lalt(V ) from one of Lalt(V ) = V (Proposition 2.12), it is convenient to ﬁrst introduce the Levi-Civita symbol:  1 if j , . . . , j is an even permutation of i , . . . , i ,  1 k 1 k i1,...,ik = j1,...,jk −1 if j1, . . . , jk is an odd permutation of i1, . . . , ik,  0 otherwise.

1 n Corollary 2.11. Let (e1, . . . , en) be a basis for a vector space V , and let (e , . . . , e ) be its dual basis for V ∗. Then

ei1 ∧ · · · ∧ eik (e , . . . , e ) = i1...ik . j1 jk j1...jk

7 Proof. Suppose there is some j` ∈/ {i1, . . . , ik}. Then the matrix A from lemma 2.10 will contain a zero column and det A = 0. Suppose instead that j1, . . . , jk is an even permutation of i1, . . . , ik. Then A will be the matrix resulting from an even number of row inversions of the identity matrix. Since each inversion adds a factor of −1 to the determinant and the identity matrix has a determinant of 1, clearly det A = 1. Similarly, if j1, . . . , jk is an odd permutation of i1, . . . , ik, then A is the result of an odd number of row inversions. Thus det A = −1.

Let us now introduce multi-index notation

I = (i1, i2, . . . , ik)

I i1 ik and write eI for (ei1 , . . . , eik ) and e for e ∧ · · · ∧ e , where e1, . . . , en as usual is the basis of some vector space V , and e1, . . . , en is the basis of the dual space V ∗. Thus we could, with some I I abuse of notation, write e (eJ ) = J , where I and J are taken to be multi-indices of the same I i k length, or similarly α (vJ ) = det α (vj) . Additionally we let In be the set of strictly increasing multi-indices of length k between 1 and n:

k In = I = (i1, . . . , ik) : 1 ≤ i1 < ··· < ik ≤ n

1 n ∗ 1 Proposition 2.12. If (e , . . . , e ) is a basis for V = Lalt(V ), then the set

n I k o e : I ∈ In

k k n k is a basis for Lalt(V ). Thus dim Lalt(V ) = k , and in particular dim Lalt(V ) = 0 for k > n. Proof. To prove linear independence suppose that

X I 0 = cI e . k I∈In

k Applying both sides to eJ , J ∈ In, we get, by Corollary 2.11

X I 0 = cI e (eJ ) k I∈In X I = cI J k I∈In

= cJ . where the last equality follows from the fact that the only permutation of a strictly increasing sequence that is also a strictly increasing sequence is the identity permutation (an even permu- I k tation). Since this holds for any J, it holds for all J, and e : I ∈ In is linearly independent. I k k To prove that e : I ∈ In spans Lalt(V ), we ﬁrst note that if two alternating k-linear maps k agree on eI , ∀I ∈ In, then by k-linearity and the alternating property (both from proposition k P I k 2.9), they are equal. Now let f ∈ L (V ) and g = k f(eI )e . Then for J ∈ alt I∈In In

X I g(eJ ) = f(eI )e (eJ ) k I∈In

8 X I = f(eI )J k I∈In

= f(eJ ).

k Thus f and g agree on all eJ , so f = g, and since f was arbitrary, all elements of Lalt(V ) can be written as a linear combination of eI . We end this subsection with a look at the determinant of a linear transformation on some vector space. As with the preceding results, Proposition 2.14 will be used in Section5. In particular it will be used to show a degree of equivalence between a new definition of the Jacobian determinant and the standard one, known from multi-variable calculus. n Definition 2.13. By Proposition 2.12 the dimension of Lalt(V ) is one if V is an n-dimensional vector space. Thus any non-zero element provides a basis. Let λ ∈ L(V ; V ), that is a lin- ∗ n n ∗ ear transformation on V . Then λ : Lalt(V ) → Lalt(V ), defined by λ (ω)(v1, . . . , vn) := n ω λ(v1), . . . , λ(vn) , is a linear transformation on Lalt(V ), so there is a unique number det(λ) ∈ R such that λ∗(ω) = det(λ)ω

n for any ω ∈ Lalt(V ). This number is called the determinant of λ. h i i Proposition 2.14. Let V be some vector space, and let the matrix A = aj represent some λ ∈ L(V ; V ). Then det(λ) = det(A), where the matrix determinant is deﬁned as usual. Pn i Proof. Let (e1, . . . , en) be some basis of V , and write λ(ej) = i=1 ajei. Then, if we denote the dual basis by e1, . . . , en, we have ∗ 1 n 1 n λ (e ∧ · · · ∧ e )(e1, . . . , en) = (e ∧ · · · ∧ e ) λ(e1), . . . , λ(en) h i i = det e λ(ej)

h i i = det aj = det(A).

1 n ∗ 1 n On the other hand (e ∧ · · · ∧ e )(e1, . . . , en) = 1, so λ (e ∧ · · · ∧ e )(e1, . . . , en) = det(λ).

2.2 Smooth Manifolds with Boundary We now turn to the most fundamental object of differential geometry — smooth manifolds. Although smooth manifolds are special topological manifolds we will make no further mention of this fact. Instead we will define smooth manifolds directly via the existence of an atlas of smooth charts. The majority of this subsection will consist of definitions necessary to prove and understand TheoremA. However we will also produce three results we carry with us for later, namely Proposition 2.20 (used in Sections4 and5), Proposition 2.27 (used without explicit mention throughout the text), and Proposition 2.28 (used in Section3). These are included here because they are relevant in their own right to produce a satisfactory understanding of smooth manifolds with boundary. We begin with establishing the link to Rn. This link is fundamental insofar that being locally Euclidean, perhaps better named locally Cartesian due to the integral role coordinates play, is a defining property of smooth manifolds. Definition 2.15 will later be extended to bijections between smooth manifolds.

9 Definition 2.15. A bijection between open sets U ⊂ Rn and V ⊂ Rm is called a diffeomorphism if f and f −1 are both smooth. A diffeomorphism can thus be seen to provide an (infinitely) smooth deformation of U to V , and vice versa. A chart (Definition 2.16) can similarly be seen to provide just that; a way to map some region of a set to the familiar Cartesian coordinates in n dimensions, just like a chart of some region on Earth does so in two dimensions. Except in this case we are interested in the mathematical properties. This is indeed how charts are used in differential geometry: they allow us to transition to Cartesian coordinates and back, thus effectively charting the region mathematically. Definition 2.16. Let M be a set. A chart on M is a bijection of a subset U ⊂ M onto an open subset of some Rn. We will often denote a chart as an ordered pair (U, φ) where U is the domain of the bijection φ, and may also refer to the domain itself as a chart. We will also have occasion to use the notation (U, φ) = (U, u1, . . . , un), which should be taken to mean ui = xi ◦ φ. Here xi : Rn → R denotes the i:th canonical coordinate map on Rn, defined by (x1, . . . , xn) 7→ xi. Thus ui represents the i:th coordinate map on U. Definition 2.17. Two charts, (U, φ) and (V, ψ), are said to be C∞-compatible if either U ∩V = ∅ or if φ ◦ ψ−1 (with suitable restrictions) is a diffeomorphism. Definition 2.17 provides an equivalence relation between charts on a set M. Any set of compatible charts that cover M is called an atlas: n Definition 2.18. Let A = {Uα, φα}α∈A be a collection of R -valued charts on a set M. We call A an Rn-valued smooth atlas if: S (i) α∈A Uα = M.

(ii) (Uα, φα) and (Uβ, φβ) are compatible ∀α, β ∈ A. A set is said to be locally Euclidean if there exists an atlas on it. Since compatibility is an equivalence relation, if a chart is compatible with some chart in an atlas, it is compatible with all charts in the atlas. Including the chart in the atlas thus produces a new atlas. This procedure can be extended to hold between two atlases, giving us a derivative equivalence relation between atlases:

Definition 2.19. Two atlases A1, A2 on M are said to be equivalent if their union, A1 ∪ A2 is also an atlas. An equivalence class of atlases is called a smooth structure on M, and we will often refer to a smooth structure as simply a differentiable structure. In other words, if any chart from the first atlas is compatible with any chart from the second atlas, they belong to the same differentiable structure. Each atlas thus gives rise to a unique differentiable structure. A smooth manifold will consist of a set M together with a smooth structure on M, but since the union of all atlases in such a structure is also an atlas, and is uniquely determined by the original atlas, we may use this maximal atlas interchangeably with the differentiable structure. Firstly however, we will prove a result that provides a link between any atlas and the induced topology: the topology generated by sets that are domains of charts in any atlas in the differentiable structure. The proof that these sets provide a topological basis is trivial, and requires only that we show that each intersection and open restriction (via charts) is also the domain of some chart. We omit the explicit statement, but see e.g. [7]. The reason for its importance is that we will require a manifold to have an (induced) topology that is exactly Hausdorff and second countable.

10 Proposition 2.20. Let M be a set with a smooth structure given by an atlas A. Then

(i) If for every two distinct points p, q ∈ M, we have that either p and q are respectively in disjoint chart domains Uα and Uβ from the atlas, or they are both in a common chart domain, then the induced topology is Hausdorﬀ.

(ii) If A is countable, or there is an atlas contained in A that is, then the topology induced is second countable.

Proof. We start by proving (i). If p, q ∈ M are distinct points such that they are contained in respectively disjoint chart domains Uα and Uβ then these very domains are the necessary neighbourhoods. Suppose instead that they are contained in the domain of a common chart (U, φ). Then φ(p) and φ(q) are distinct points in some Rn, whence there are open sets φ(p) ∈ O, −1 −1 φ(q) ∈ B such that O ∩ B = ∅. Take V = φ (O) and W = φ (B). Clearly (V, φ |V ) and (W, φ |W ) are compatible with (U, φ) and thus all charts in A, whence they are charts in some atlas in the diﬀerentiable structure. Then V and W are the necessary neighbourhoods. To prove (ii) we let A = (Ui, φi) be the countable atlas, and suppose O ⊂ M is open. i∈N Obviously O is covered by {Ui} . If every Ui with O ∩ Ui 6= is contained in O we are done, i∈N ∅ so suppose there is some Uj such that O ∩ Uj 6= ∅ and Uj 6⊂ O. Then O ∩ Uj is open, whence n n φj(O ∩ Uj) is open in R . Since R is second countable it follows that φj(O ∩ Uj) is the union of a countable family of open balls B (i ∈ ). If we denote V = φ−1(B ), then (V , φ | ) i N j(i) j i j(i) j Vj(i) is a compatible chart for all i, and by adjoining these charts to A we get a new countable atlas that provides a countable cover for O where Uj has been replaced by sets Vj(i), which are all contained in O. This process can be repeated until O is the union of chart domains of some countable atlas.

We will use Proposition 2.20 in Proposition 4.18 and Proposition 5.1 to show that we are dealing with smooth manifolds (Deﬁnition 2.21), which we are now ready to deﬁne:

Definition 2.21. A smooth manifold is a set M and a C∞ structure on M such that the topology induced by the structure is Hausdorff and second countable. If the charts are in Rn then we say that the dimension of the manifold, dim(M) = n, or simply that M is a smooth n-manifold. We will not explicitly refer to the differentiable structure, but be satisfied with that there is such a structure chosen. Instead we refer to a chart as admissible if it belongs to some atlas in the differentiable structure.

Smooth manifolds are the most fundamental object in differential geometry. Although this is somewhat obscured by the abstract definition, the differentiable structure provides a mathematical description of the geometry of the underlying set, up to diffeomorphism. If endowed with a connection, we can then investigate the curvature of the set, but even without a connection, the topology can be investigated thoroughly. We shall not set out on this endeavor, but refer the interested reader to e.g. [7]. Note that although the charts allow us to investigate the properties of a manifold locally, in general there is no global chart. In Section3 we develop a method to “stitch” properties that hold locally (in each chart) together. This will allow us to conclude that they are in fact global properties. As a first example of a smooth manifold, consider the unit circle in R2, S1, completely parametrized by (x, y) = (cos θ, sin θ), θ ∈ [−π, π). We may thus create a chart, φ, on S1, by using the parameter θ. Say we take the map to be defined by p 7→ θ. The image must be an open interval in R, so the chart may at most be defined for points corresponding to the parameter θ ∈ (−π, π). Thus the chart may not cover the point (−1, 0). To complete our atlas, we

11 need a second chart, ψ, defined similarly by another parametrization, say (x, y) = (cos θ, sin θ), θ ∈ [0, 2π). This chart may be defined for points corresponding to the parameter θ ∈ (0, 2π), omitting the point (1, 0). These charts are compatible, for we find in their intersection ( θ for θ ∈ (0, π), φ ◦ ψ−1(θ) = θ − 2π for θ ∈ (π, 2π).

By Proposition 2.20, the induced topology is clearly second countable, but we cannot yet conclude that it is Hausdorff: the points (−1, 0) and (1, 0) are not in a common chart domain, nor are they in respectively disjoint chart domains. The remedy is however simple, requiring only the introduction of a third chart, which we may find by a third parametrization interval. We do not state it explicitly. As another example, consider an n-dimensional, real vector space V . If (e1, . . . , en) is a basis Pn i of V , any vector v ∈ V can be written uniquely as v = i=1 v ei. We find a chart on V to Rn by taking the map v 7→ (v1, . . . , vn). This chart covers all of V , whence it is an atlas, and from Proposition 2.20 we immediately get that the induced topology is Hausdorff and second countable. From now on, any vector space will be assumed to be real, and implicitly understood to be smooth manifolds. Having now defined smooth manifolds we can use the differentiable structure to define the notion of smooth maps between manifolds: Definition 2.22. Let M and N be smooth manifolds. We say that a map f : M → N is smooth at p ∈ M if there exist admissible charts (V, ψ) for N and (U, φ) for M such that: (i) p ∈ U (ii) f(U) ⊂ V (iii) ψ ◦ f ◦ φ−1 is smooth. We say that f is smooth if it is smooth at every point p ∈ M. We will often use the notation C∞(M) for the set of smooth functions from a smooth manifold M to R, but we leave this consideration for now and note that Definition 2.22 naturally extends our notion of diffeomorphism (Definition 2.15) to smooth manifolds in general, and we renew the definition: Definition 2.23. A bijection between smooth manifolds is a diffeomorphism if f and f −1 are both smooth. A smooth manifold with boundary is to a smooth manifold what a closed or half-closed interval is to an open interval. We need this notion for TheoremA. The reader will recognize the following definitions as the equivalents of Definition 2.16, Definition 2.18, and Definition 2.21. Definition 2.24. A half-space chart φ for a set M is a bijection of some subset U of M onto n an open subset of some half-space Rx1≤0. Definition 2.25. A half-space atlas A = (Uα, φα) α∈A is a collection of half-space charts on a set M such that S (i) α∈A Uα = M

(ii) (Uα, φα) and (Uβ, φβ) are compatible ∀α, β ∈ A.

12 Deﬁnition 2.26. An n-dimensional smooth manifold with boundary is a pair (M, A) consisting of a set M together with a maximal half-space atlas, A, of Rn valued half-space charts. The boundary of M is denoted ∂M and is the set of points whose image under any admissible chart is contained in the boundary of the associated half-space. The interior of M is the set M \ ∂M, and is denoted by int M.

Definition 2.22 and Definition 2.23 extend obviously to smooth manifolds with boundary, and we shall not state them explicitly. Proposition 2.27 is highly satisfying in its own regard, as it shows that any one atlas in a differentiable structure is enough to completely determine the boundary of the manifold, and that any one chart is enough to completely determine whether or not points in its domain are boundary points. We shall use it without explicit mention moving forward.

Proposition 2.27. If p ∈ ∂M for some smooth n-manifold with boundary M. Then for every n admissible chart (U, φ) on M such that p ∈ U we have φ(p) ∈ ∂Rx1≤0.

n −1 Proof. By the deﬁnition there is a chart (V, ψ) such that ψ(p) ∈ ∂Rx1≤0. Then f = ψ ◦ φ : n φ(U ∩V ) → ψ(U ∩V ) is a diﬀeomorphism between open sets in Rx1≤0, and f φ(p) = ψ(p). Then ∂f 1 = 0 for all j ∈ {1, . . . , n} because φ(p) is a local (global) maximum for f 1 = x1◦f. Thus ∂xj φ(p) n Df φ(p) ∈ Rx1=0, or in other words Df φ(p) is singular, whence f is not locally invertible at n n n φ(p) in R . But f must be invertible in Rx1≤0 so we must have φ(p) ∈ ∂Rx1≤0.

Proposition 2.28 allows us to work on the boundary as its own manifold. We shall use this in Proposition 3.10, not to mention it is fundamental to formulating TheoremA. Obviously there is a corresponding result for the interior, and we will use that without explicit statement, referring instead to Proposition 2.28.

Proposition 2.28. If M is an n-manifold with boundary, then ∂M is an (n − 1)-manifold.

n Proof. Let p ∈ ∂M. By deﬁnition, there is a chart (U, φ) with φ(p) ∈ ∂Rx1≤0. The image of the n n n−1 restriction φ |U∩∂M is contained in ∂Rx1≤0. If we let f : ∂Rx1≤0 → R be a linear isomorphism, and deﬁne

φ∂M := f ◦ φ,

n−1 then (U ∩ ∂M, φ∂M ) is an R -valued chart for ∂M. The family of charts obtained in this way must obviously cover ∂M, and for any two overlapping charts (U ∩∂M, φ∂M ) and (V ∩∂M, ψ∂M ), we have

−1 −1 φ∂M ◦ ψ∂M = (f ◦ φ) ◦ (f ◦ ψ) = f ◦ φ ◦ ψ−1 ◦ f −1.

As the composition of smooth maps is smooth, the family of charts obtained in this way is an Rn−1-valued atlas for ∂M.

Although this is not stated explicitly, the reader should note that the smooth structure on ∂M in the proof of Proposition 2.28 is completely determined by the smooth structure on M. When we talk about the boundary as its own manifold, it will be implied that its structure is inherited thus from the main manifold.

13 2.3 Vector Bundles We shall never be needing vector bundles explicitly, but since the author found the subject much more intuitively accessible with their formalism, they are included here. We use them implicitly in Proposition 4.18 and Proposition 5.1. They can be given a far more extensive analysis, but for our purposes the briefest of introductions shall suffice. As such this subsection will contain only a few necessary definitions — five to be precise. Definition 2.32 is the main result. We will define fiber bundles, sections, local trivializations, vector bundles, and local frame fields. We start with defining fiber bundles, of which vector bundles are a special case. Up to diffeomorphism, a fiber bundle links to each point in a manifold another manifold. The new manifold may express properties that each point in the original manifold could have; indeed, we shall immediately after introduce sections, which then can be viewed as assigning one such property to each point. Furthemore, combining the original manifold and the new produces a third manifold, which then describes all of space and its properties. The definition is more precise, and more abstract: Definition 2.29. Let F,M, and E be smooth manifolds and let π : E → M be a smooth map. The quadruple (E, π, M, F ) is called a smooth fiber bundle if for each p ∈ M there is an open set U ⊂ M containing p and a diffeomorphism f : π−1(U) → U × F such that the following diagram commutes:

f π−1(U) U × F

π pr1 U

Here pr1 :(p, q) 7→ p is the projection map from U × F to U. E is called the total space, π the bundle projection, M the base space, and F is called the typical fiber. For each p ∈ M, the set −1 Ep := π (p) is called the fiber over p. We have already described how a section may be viewed as assigning properties, or more precisely assigning a point from each fiber to each point in the base space. Two familiar examples would be scalar fields (F = R) and vector fields (E = TM, see Proposition 4.18), but tensor fields, and in particular differential forms (Definition 5.2) are also sections of fiber bundles. In fact, these are all sections of vector bundles. For trivial examples of fiber bundles that are not also vector bundles, take a typical fiber F that is not a vector space, and a base space M. The bundle (M ×F, pr1,M,F ) is called a product bundle, or a trivial bundle. The observant reader will have noticed that M × F is not properly defined as a manifold. This can be remedied, but we shall not do so here, as it is not necessary to understand the principle. Examples include the cylinder and the two-torus. In both cases M = S1 is the circle, and in the cylinder F is a line segment while in the two-torus F is also S1. In these cases it is perhaps most natural to identify the “properties” we mentioned earlier with position along the fiber. As a final note, observe that the Möbius strip and the Klein bottle are both non-trivial examples of fiber bundles with the same base space and corresponding typical fiber. Definition 2.30. A smooth section of a smooth fiber bundle ξ = (E, M, π, F ) over an open set U ⊂ M is a smooth map σ : U → E such that π ◦ σ = idU . The set of smooth sections of ξ is denoted by Γ(ξ). Definition 2.31 provides a way to exhibit the structure of the total space, given that the structure of the base space and the typical fiber is known, by finding the diffeomorphisms in

14 Definition 2.29. It also makes some of the discussion preceding Definition 2.29 more clear, and sets us up for the definition of a vector bundle (Definition 2.32). Definition 2.31. The maps f occuring in Definition 2.29 are called local trivializations of the bundle. Under suitable restriction f = (π, fb), where fb : π−1(U) → F is a smooth map such that f | : E → F is a diffeomorphism. We call f the principal part of the local trivialization, and b Ep p b say that (U, f) is a bundle chart. A family of bundle charts (Uα, fα) α∈A, such that {Uα}α∈A cover M is said to be a bundle atlas. Now we are ready to define vector bundles. We have already touched on how we will use them. In fact we will construct a vector bundle known as the tangent bundle in Section4, though we shall not exhibit it very closely, and in Section5 we exhibit a family of vector bundles that are indeed central to TheoremA. Note that we use the trivial fact that all vector spaces (over R) are also smooth manifolds, which we showed in Subsection 2.2. Definition 2.32. Let V be a finite-dimensional real vector space. A smooth vector bundle with typical fiber V is a fiber bundle (E, π, M, V ) such that

−1 (i) For all p ∈ M the set Ep := π (p) has the structure of a real vector space, isomorphic to V . (ii) Every p ∈ M is in the domain of some bundle chart (U, f) such that for each q ∈ U the map f | : E → V is a vector space isomorphism where f = (π, f). b Eq q b The bundle charts (U, f) are called vector bundle charts, and is assumed to be linear on fibers. Similarly, a collection of vector bundle charts that cover M is called a vector bundle atlas. Obviously condition (ii) above guarantees the existence of a vector bundle atlas on any vector bundle. As promised, vector bundles are special cases of fiber bundles, and they appear naturally whenever we are dealing with vector spaces in relation to a manifold. Our final task before moving on to Section3 is to define local frame fields. As with the rest of this subsection, Definition 2.33 is never explicitly needed, but it does facilitate discussion of certain subjects, notably near the end of Section4.

Deﬁnition 2.33. An ordered set of sections (σ1, . . . , σn) of a vector bundle ξ = (E, π, M, V ) over an open set U ⊂ M is called a local frame ﬁeld over U if for all σ ∈ Γ(ξ) we have

n X i σ |U = s σi, i=1 for some smooth functions si ∈ C∞(U). Thus local frame ﬁelds ﬁll much the same function for vector bundles as a basis does for a vector space.

3 Partitions of Unity

In this section we show that every smooth manifold is paracompact, regular, and normal. We then produce the main results of this section: Proposition 3.9 and Corollary 3.10, stating that for any open cover of a manifold, with our without boundary, there is a partition of unity subordinate to it. We shall make these notions precise shortly. Partitions of unity can be used to extend local properties globally. Indeed, we shall do so throughout the text, and in particular in our proof of TheoremA. Before we define a partition of unity we will also show the existence of what may be called a cut-off function (Lemma 3.7). Firstly, however, we will consider some topological properties of smooth manifolds. We begin by showing paracompactness (Proposition 3.3), but for that we will need two preliminary results: we first show that for any smooth manifold there is a basis with compact closure of its elements, and then we use that to show that there is a locally finite open cover with compact closure of its elements. Proposition 3.3 then follows without much trouble.

Lemma 3.1. Every smooth manifold M has a countable basis all of whose elements have compact closure.

Proof. Because M is second countable, it has a countable basis B. Now consider the subcollection S of elements in B that have compact closure Given any open subset U ⊂ M and point p ∈ U, choose an open neighbourhood V of p such that V ⊂ U and the closure of V is compact. Since M is locally Euclidean, this is always possible. Furthermore, since B is a basis, there is an open set B ∈ B such that

p ∈ B ⊂ V ⊂ U.

Then B ⊂ V . Because V is compact, so is B, whence B ∈ S. Since U and p were arbitrary, we have shown that S is a basis.

Lemma 3.1 should come as no surprise, given the intimate connection between smooth n- manifolds and Rn, which is indeed used in the proof, along with second countability of course. The topological property we use, which follows trivially from M being locally Euclidean, is that M is locally compact. Note therefore that Lemma 3.1, Lemma 3.2, and Lemma 3.5, as well as Proposition 3.3 and Proposition3.4, would all go through for any second countable Hausdorff space that is locally compact, and in fact the Hausdorff property is not needed until Proposition 3.4. Although Lemma 3.2 is a bit more involved it follows directly from Lemma 3.1. Indeed, given a countable basis with compact closure of all of its elements we can always construct a countable, locally finite cover such that all its elements have compact closure. The proof shows how this may be carried out.

Lemma 3.2. Every smooth manifold M admits a countable, locally ﬁnite cover U = {Ui} i∈N such that Ui is compact ∀i ∈ N.

Proof. By Lemma 3.1 M has a countable basis B = {Bi} with each Bi compact. Set V0 = B0. i∈N Then, by the definition of compactness V 0 is covered by finitely many of the Bi’s. Define i0 to be the smallest integer ≥ 1 such that

V 0 ⊂ B0 ∪ B1 ∪ · · · ∪ Bi0 .

17 Now suppose that V1,...,Vm have been deﬁned, each with compact closure. As before, by compactness, V m is covered by ﬁnitely many of the Bi’s. If im is the smallest integer ≥ m + 1 such that

V m ⊂ B0 ∪ · · · ∪ Bim 6⊂ B0 ∪ · · · ∪ Bim−1 , then we set

Vm+1 = B0 ∪ · · · ∪ Bim .

If there is no such im we have that B0 ∪ · · · ∪ Bim−1 = Vm = M, and we immediately have a ﬁnite cover V = Vj : j ∈ N, j ≤ m such that for each Vj ∈ V, V j is compact. Otherwise, since the ﬁnite union of compact sets is compact, and hence

V m+1 ⊂ B0 ∪ · · · ∪ Bim is a closed subset of a compact set, V m+1 is compact. Since im ≥ m + 1, Bm+1 ⊂ Vm+1, we have [ [ M = Bi ⊂ Vi ⊂ M. S Thus M = Vi. Now define V−1 = ∅ and let Uj = Vj+1 \ V j−1, j ∈ N. For an arbitrary p ∈ M, we have p ∈ Uk where k is the smallest non-negative integer such that p ∈ Vk+1. Clearly Uk ∩ U` = ∅ unless ` = k − 1, ` = k or ` = k + 1. Thus Uj is a locally finite cover of M, and since U j j∈N is a closed subset of the compact set V j+1, it is compact. See figure1 for an example of the sets used in this proof. Proceeding with Proposition 3.3 we note that it follows directly from Lemma 3.2. Once again we are in a position to say that it should come as no surprise that we can use the cover from Lemma 3.2, U, to construct a locally finite refinement of any open cover. This is because U is locally finite in itself and all its elements have compact closure. The countable part, of course, has been passed along from Lemma 3.1 to Lemma 3.2 to Proposition 3.3, originating with the second countability of all smooth manifolds. Proposition 3.3. Every open cover V of a smooth manifold M has a (countable and) locally finite refinement. In effect, M is paracompact. Proof. Let U = Uj be a countable, locally finite cover, by Lemma 3.2, such that U j is j∈N compact for all i ∈ N. For each p ∈ M there are only finitely many of the sets Uj that contain p, by local finiteness. Then let Vp ∈ V be such that p ∈ Vp and take \ Wp = Uj ∩ Vp.

{j:p∈Uj } n o For each k, the collection Wp : p ∈ U k is an open cover of U k. By compactness, U k is covered

0 1 mk by finitely many of these sets. Denote these sets by Wk ,Wk ,...,Wk . The collection of all such i sets W = Wk as k and i vary is clearly a countable open cover of M that refines V. i Furthermore, Wk ⊂ Uj for some j ∈ N by construction, and each such Uj can be taken to i contain only a finite amount of sets Wk ∈ W. For suppose this is not the case, then some U` i 0 0 contains an infinite amount of sets Wk, say W . W , possibly together with a finite amount of 0 elements W ∈ W such that W/∈ W , form an open cover of U `. By compactness, there is a

18 M

B3 B0 = V0 B4

Figure 1: Example sets from the proof of Lemma 3.2. The sets V0 = B0, V1 = B0 ∪ B1 ∪ B2, and V2 = V1 ∪ B3 ∪ B4 form an ever growing family of sets with compact closures. The set U0 = V1 and the set U1 = V2 \ V0 is marked in grey. The set Ui in general is formed by “cutting a hole” in Vi+1 that is smaller than Vi but larger than Vi−1.

finite subcover. This subcover contains a finite amount of elements from W0, say Wf0. Replace the subset W0 ⊂ W by Wf0 to form the collection Wf, which is still a countable open cover of M i that refines U. We can do this for any set Uj that contains an infinite amount of sets Wk ∈ W. By the above argument W must be locally finite because U is.

Having now recovered paracompactness, which e.g. [7] took as the deﬁnition, we proceed to prove Lemma 3.5, often called the shrinking lemma, which shows that it is possible to “shrink” any locally ﬁnite open cover of any space such that any two closed sets are contained in mutually disjoint open sets, and still retain an open cover. As alluded to, we shall need to show that the induced topology of a smooth manifold allows for the separation of any two closed sets by mutually disjoint open sets (Proposition 3.4) to prove Lemma 3.5, but will otherwise have no more need of it. Note that it is property (ii) that we are after, but that property (i) follows naturally from the proof and is stated separately for clarity.

Proposition 3.4. For any smooth manifold M:

(i) If p ∈ M and A ⊂ M is a closed set such that p∈ / A there exist open sets U, V ⊂ M such that p ∈ U, A ⊂ V , and U ∩ V = ∅.

(ii) If A, B ⊂ M are disjoint closed sets, there exist open sets X,Y ⊂ M such that A ⊂ X, B ⊂ Y , and X ∩ Y = ∅.

In eﬀect, M is regular and normal.

19 M

W p A

Figure 2: The separation of the point p from the closed set A by mutually disjoint open sets. Here U is the open set containing p marked with darker gray, and V is the open set containing A marked with lighter gray. For each set Vp0 that makes up the open cover of A and intersects W there is a corresponding set Up0 containing p such that Up0 ∩ Vp0 = ∅. The union of the sets Vp0 form V , and the intersection of the sets Up0 form U. Note that here we have already taken a locally ﬁnite reﬁnements (see the text).

Proof. To prove (i) we note that M is Hausdorff, and we define, for each p0 ∈ A, open sets 0 0 0 0 0 0 0 0 Up ,Vp ⊂ M such that p ∈ Up , p ∈ Vp , and Up ∩ Vp = ∅. Clearly Vp = Vp p0∈A covers A. Furthermore Vp ∪ M \ A forms an open cover of M. Now, by Proposition 3.3 there is locally finite refinement of our open cover of M, which we call O. Let P = {U ∈ O : U ∩ A 6= ∅}. Then P is a locally finite refinement of Vp. Thus there is some open set W containing p such that only a finite amount of members in P intersect W . Each of these members is contained in some Vp0 . For each member, choose one such Vp0 and let 0 T = p ∈ A : Vp0 is one of our chosen sets . Then we define (see Figure2)   \ U := W ∩  Up0  , p0∈T [ V := P, P ∈P whence U, V fulfills (i). We now turn our attention to (ii) and note that, by (i), for each a ∈ A there exist open sets Ua,Va such that a ∈ Ua, B ⊂ Va, and Ua ∩ Va = ∅. Clearly U = {Ua}a∈A covers A, and furthermore U ∪ M \ A forms an open cover of M. Again we find a locally finite refinement of our cover of M, say O and we form the set Q = {U ∈ O : U ∩ A 6= ∅}. Let X be the union of all elements in Q. Then for any b ∈ B there exist an open set Wb around b that intersects only a finite amount of members of Q. We let S T be similar to T in the proof of (i) and take Db = Wb ∩ a∈S Va. Finally we take Y to be the union of all the sets Db. Then X,Y fulfills (ii). We will refer to property to in Proposition 3.4 by saying that M is normal. Although our statement of Lemma 3.5 does demand a countable and locally finite open cover, any open cover

20 of a smooth manifold can obviously be reﬁned thus, by second countability and Proposition 3.3. This is exactly what we will later do, in the proof of Proposition 3.9. Lemma 3.5 (Shrinking Lemma). Let U = Uj j∈ be a locally ﬁnite open cover of a smooth N manifold M. Then there exists another open cover V = Vj of M such that V j ⊂ Uj. j∈N Proof. Consider the set

Aj = {Vk : k < j} ∪ {Uk : k ≥ j} .

Then A0 = U and so obviously covers M. Now assume that Aj covers M, and consider Aj = A \ U . Let B = M \ S A. Then B ⊂ U is closed. By Proposition 3.4 M is normal, j j j A∈Aj j j so there are open sets Vj and Wj such that Bj ⊂ Vj and M \ Uj ⊂ Wj and Vj ∩ Wj = ∅. Thus Vj ⊂ Uj and in particular V j ⊂ Uj, since M \ Wj is a closed set containing Vj, whence V j ⊂ M \ Wj. Written concisely

Bj ⊂ Vj ⊂ Vj ⊂ Uj, whence Aj ∪ Vj = Aj+1 is also an open cover of M. Now, let

O = lim Aj, j→∞ and suppose that O does not cover M. Then there is some p not in any of the elements of O. Since U is a locally ﬁnite cover, there are a non-zero, ﬁnite amount of sets Uj such that p ∈ Uj.

Denote the largest integer j such that p ∈ Uj by jp. Since Ajp+1 covers M by induction, and

Uj ∈/ Ajp+1 for all j ≤ jp, we must have p ∈ Vj for some j ≤ jp. Thus O must cover M.

That concludes our topological considerations, and we move on to the following simple result, which we will need to show the existence of so called cut-oﬀ functions (Lemma 3.7). Essentially, Lemma 3.6 deﬁnes a function on R that is smooth, zero for all non-positive arguments, and positive for all positive arguments (more can be said, but these are the properties that are used in Lemma 3.7).

Lemma 3.6. The function g : R → R deﬁned by

( −1 e−t if t > 0, g(t) = 0 otherwise, is C∞.

Proof. We begin by noting that

lim g(t) = lim g(t) = 0, t→0+ t→0− and that if t 6= 0 g(t) is obviously smooth. It remains to show that g(t) is smooth at t = 0. We have

−1 g(h) − g(0) e−h lim = lim , h→0+ h h→0+ h

21 and, in general, for n ∈ N

−1 e−h lim = 0. h→0+ hn Thus g(h) − g(0) g(h) − g(0) lim = 0 = lim . h→0+ h h→0− h So g(t) is at least C1. Now assuming that g(t) is Ck we have

( −1 t−2ke−t if t > 0, g(k)(t) = 0 otherwise.

So by the same argument as for C1, g(t) is at least Ck+1, whence by induction g(t) is C∞. In the proof of Lemma 3.7 we deﬁne a function g(t), which is a product of the function from Lemma 3.6 with its inversion through the origin (both translated). This is obviously a smooth function, and by the fundamental theorem of calculus, integrating yields another smooth function, depending on the limits of integration. This, in principle simple idea, is the key to proving Lemma 3.7. The rest is just compensating for complications. Lemma 3.7. Let M be a smooth manifold and let K ⊂ M be compact, such that K ⊂ O for some open set O. Then there exists a smooth function f : M → R such that f(p) ≡ 1 if p ∈ K and f(p) ∈ [0, 1] if p ∈ O, with compact support in O. Proof. We perform the proof in three steps:

Special case 1: Assume that M = Rn and that O = B(0,R) and K = B(0, r) for 0 < r < R. Then take R R g(t)dt φ(p) = |x| , R R r g(t)dt where

( −1 −1 e−(t−r) e(t−R) if r < t < R, g(t) = 0 otherwise.

Then g(t) is smooth by lemma 3.6, and thus φ(p) is smooth. Translation immediately gives the result for a ball centered at some arbitrary point.

Special case 2: Assume again that M = Rn, and let K ⊂ O be as in the hypothesis. For each point p ∈ K let Up ⊂ O be an open ball centered at p, and let Kp ⊂ Up be a closed ball centered at p. Then the interiors of the Kp balls form an open cover for K, so by compactness we can reduce to a finite subcover. Thus we have a finite family {Ki} of S closed balls of various radii such that K ⊂ Ki, and with corresponding concentric open balls Ui ⊂ O. For each Ui let αi be the corresponding function from special case 1. Then Y f(p) = 1 − 1 − αi(p) i fulfills the lemma.

22 General case: If K is contained in the domain of some chart (U, φ), then f ◦ φ fulﬁlls the lemma. If K is not contained in such a chart, then we may take a ﬁnite number of charts (U1, φ1),..., (Uk, φk) and compact sets K1,...,Kk with Sk (i) K ⊂ i=1 Ki, (ii) Ki ⊂ Ui, Sk (iii) i=1 Ui ⊂ O. Now, by special case 2 and the considerations outlined at the beginning of the general case, let βi be any smooth function that identically 1 on Ki and identically 0 on M \ Ui. Then

k Y f(p) = 1 − 1 − βi(p) i=1 fulﬁlls the requirements of the lemma.

We now finally define a partition of unity, and what it means for it to be subordinate to an open cover, before we show that we may always choose a partition of unity subordinate to any open cover of a smooth manifold. That the same holds for any smooth manifold with boundary then follows from Proposition 2.28. Definition 3.8. A partition of unity on a smooth manifold M is a collection of smooth functions

{ϕα}α∈A on M such that

(i) 0 ≤ ϕα ≤ 1 for all α ∈ A. (ii) The collection of supports supp (ϕα) α∈A is locally ﬁnite. P (iii) α∈A ϕα(p) = 1 for all p ∈ M.

If U = {Uα}α∈A is an open cover of M and supp (ϕα) ⊂ Uα for all α ∈ A, then we say that {ϕα}α∈A is a partition of unity subordinate to U. Thus a partition of unity subordinate to an open cover simply partitions unity in such a manner that each part is contained in the corresponding chart. This is what (by Proposition 3.9 and Corollary 3.10) allows us to conclude that many properties that hold locally also hold globally, for by an appropriate partition of unity we may demote the global statement to a family of local ones. This procedure requires that we have an object that we can partition with our partition of unity, and that the property holds naturally in the zero case. It is applied in the proofs of Lemma 5.19 and TheoremA.

Proposition 3.9. Let U = {Uα}α∈A be an open cover of a smooth manifold M. Then there is a partition of unity {ρα}α∈A subordinate to U. Proof. By Lemma 3.1 we may first take a refinement of U such that all elements of our refinement have compact closure. Then, by Proposition 3.3, we make an additional refinement that is (countable and) locally finite. We denote this cover by V = Vj j∈ . N Now, by Lemma 3.5 let W = Wj be another cover of M such that W j ⊂ Vj. Since j∈N W j is a closed subset of the compact set V j it is also compact. By Lemma 3.7 we may take non-negative functions 0 ≤ ψ ≤ 1 such that supp ψ ⊂ V and ψ | ≡ 1. Let j j j j W j X ψ := ψj. j∈N

23 P For each p ∈ M the sum ψj(p) is finite, by V being locally finite, and ψ(p) > 0. Let j∈N ψ φ := j . j ψ Then φj is a partition of unity subordinate to V. j∈N Now, remembering that V is a refinement of U, we let f : N → A be such that Vj ⊂ Uf(j) for all j ∈ N, and take X ρα := φj. j∈f −1(α)

Then {ρα}α∈A is a partition of unity subordinate to U. Since a smooth manifold with boundary can be split into its interior and its boundary (Propo- sition 2.28), Corollary 3.10 easily follows.

Corollary 3.10. Let U = {Uα}α∈A be an open cover of a smooth manifold with boundary M. Then there exists a partition of unity {φα}α∈A subordinate to U.

Proof. Let V = {Uα ∩ int M : α ∈ A} and let W = {Uα ∩ ∂M : α ∈ A}. Then V is an open cover of int M and W is an open cover of ∂M. Since int M and ∂M are smooth manifolds (without boundary) there exist partitions of unity {ψα}α∈A subordinate to V and {ρα}α∈A subordinate to W. Deﬁne

φα := ψα + ρα.

Because int M and ∂M are disjoint, {φα}α∈A is then a partition of unity subordinate to U.

24 4 The Tangent Bundle

This section first defines the tangent spaces at a point in a smooth manifold via curves (Definition 4.2). We show that a tangent space is a vector space (Proposition 4.5) and that it is isomorphic to the derivation space at a point (Corollary 4.13), also defined (Definition 4.8). We are thus free to express tangent vectors as derivations, which we frequently do. We also take the opportunity to show that the tangent spaces together form a vector bundle over the manifold — the tangent bundle (Proposition 4.18). This allows a definition of smooth vector fields as smooth sections of the tangent bundle, but we shall not explore this fact here. Instead we need it later to define the induced orientation (Definition 5.20), as well as to make certain notions used elsewhere in Section5 precise. We also take with us the definition of the tangent map (Definition 4.20), the cotangent space (Definition 4.22), and the differential of smooth functions at a point (Definition 4.23), as well as a few straight-forward results concerning these. Our definition of tangent vectors at a point (Definition 4.1) identifies them with the velocities of curves passing through the point. This is a very intuitive definition, and we use it to define tangent vectors at boundary points (Definition 4.15) and to prove Lemma 5.18.

Deﬁnition 4.1. Let p be a point in a smooth n-manifold M. Suppose we have two curves c1, c2 mapping into M, each with open interval domains containing 0 ∈ R, and with c1(0) = c2(0) = p. We say that c1 is tangent to c2 at p if for all smooth functions f : U → R such that U is an open 0 0 neighbourhood of p, we have (f ◦ c1) (0) = (f ◦ c2) (0). This is obviously an equivalence relation on the set of all such curves. We deﬁne a tangent vector at p to be an equivalence class under this relation.

If M = Rn the tangent vectors at a point are simply the vectors in Rn originating at the point, and all tangent vectors can be translated without problem to any other point. In general, this is not the case, which can be made obvious by considering the circle, S1. However, it is intuitively clear that each tangent space (Deﬁnition 4.2) is linearly isomorphic to every other tangent space. In fact, this is always the case, and we prove it implicitly when proving Proposition 4.5.

Deﬁnition 4.2. If p is a point in a smooth n-manifold M, the tangent space at p, TpM, is the set of all tangent vectors at p. The observant reader will have noticed that we do not yet know whether or not the tangent space is a vector space, so we really should not be talking about vector isomorphisms. This is true, but as we have already promised, the necessary result appears in Proposition 4.5. The preceding discussion was placed her to facilitate a qualitative understanding of the tangent spaces. Before moving on to Proposition 4.5 we will need Lemma 4.3 and Proposition 4.4. Lemma 4.3 is intuitive, and in particular it shows that (φ ◦ c)0(0) ∈ Rn for some chart φ is independent of the choice of chart.

0 0 Lemma 4.3. c1 is tangent to c2 as deﬁned above if and only if (f ◦ c1) (0) = (f ◦ c2) (0) for all smooth Rk-valued functions f deﬁned on a neighbourhood of p. 1 k 0 0 i 0 i 0 Proof. If f = (f , . . . , f ), then (f ◦ c1) (0) = (f ◦ c2) (0) if and only if (f ◦ c1) (0) = (f ◦ c2) (0) 0 0 for i = 1, . . . , k. Thus (f ◦ c1) (0) = (f ◦ c2) (0) if c1 is tangent to c2 at p.

0 We note that by Lemmas 3.7 and 3.5 c1 and c2 are equivalent if and only if (f ◦ c1) (0) = 0 k (f ◦ c2) (0) for all globally deﬁned smooth functions f : M → R . Moving on, we note that Proposition 4.4 is a general result, but we place it here for continuity, as we shall only be needing it in Proposition 4.5. This is because the vector space structure of

25 TpM is non-trivial to exhibit. Fortunately, the derivation space provides a simple way to solve this.

Proposition 4.4. Suppose that S is a set and that {Vα}α∈A is a family of n-dimensional vector spaces. Further suppose that for each α ∈ A there is a bijection bα : Vα → S. If for every −1 α, β ∈ A the map bβ ◦ bα : Vα → Vβ is a linear isomorphism, then there is a unique vector space structure on S such that each bα is a linear isomorphism. −1 −1 Proof. Deﬁne addition on S by s1 + s2 := bα bα (s1) + bα (s2) . This deﬁnition is independent of the choice of α. Indeed

−1 −1 −1 −1 −1 −1 bα bα (s1) + bα (s2) = bα bα ◦ bβ ◦ bβ (s1) + bα ◦ bβ ◦ bβ (s2)

−1 −1 −1 = bα ◦ bα ◦ bβ bβ (s1) + bβ (s2)

−1 −1 = bβ bβ (s1) + bβ (s2) .

−1 Then we deﬁne scalar multiplication by a · s := bα abα (s) , and show it to be independent on the choice of α in the same way as we did for addition. Since Vα is a vector space, S must inherit the vector space structure thus.

We now use Proposition 4.4 to show that TpM is a vector space. From the proof we shall also pick out a basis for TpM (Corollary 4.6). Proposition 4.5. Let p be a point in a smooth n-dimensional manifold M. The tangent space at p, TpM, is a vector space.

Proof. For each chart (Uα, φα) deﬁne

n bα : R → TpM v 7→ [γv]

−1 where γv : t 7→ φα φα(p) + tv) for t in a suﬃciently small interval containing 0. Then we have

0 d −1 (φα ◦ γv) (0) = φα ◦ φα φα(p) + tv dt t=0

d = φα(p) + tv dt t=0 = v.

n Suppose that [γv] = [γw] for v, w ∈ R . Then by Lemma 4.3, we have 0 0 v = (φα ◦ γv) (0) = (φα ◦ γw) (0) = w, whence bα is injective. Next we show that bα is surjective. Let [c] ∈ TpM be represented by c :(−ε, ε) → M, and let 0 n v := (φα ◦c) (0) ∈ R . Then we have bα(v) = [γv], but [γv] = [c] since for any smooth real-valued function f deﬁned near p we have

0 d −1 (f ◦ γv) (0) = f ◦ φα φα(p) + tv dt t=0 −1 = D f ◦ φα φα(p) · v

26 −1 0 = D f ◦ φα φα(p) · (φα ◦ c) (0) −1 = D f ◦ φα ◦ (φα ◦ c) (0)

= (f ◦ c)0(0).

Thus bα is surjective, whence it is bijective. From Lemma 4.3 we know that the map [c] 7→ 0 −1 (φα ◦ c) (0) is well-deﬁned and from above we see that the map is exactly bα . Thus

−1 d −1 bβ ◦ bα(v) = φβ ◦ φα φα(p) + tv dt t=0 −1 = D φβ ◦ φα φα(p) · v.

−1 So bβ ◦ bα is a linear isomorphism, whence by Proposition 4.4 TpM is a vector space.

We have already proven Corollary 4.6 when we proved Proposition 4.5, but we state it explicitly because it is an important result that may otherwise be missed.

Corollary 4.6. Let p be a point in a smooth n-manifold M, and let (U, φ) be a chart containing p. Then the tangent vectors [γi] deﬁned on a suﬃciently small interval (−ε, ε) by

−1 γi : t 7→ φ φ(p) + tei ,

n where ei is the i:th vector in the standard basis of R , form a basis of TpM. Proof. An isomorphism maps basis vectors to basis vectors, and by the proof of Proposition 4.5 there is an isomorphism that maps ei to [γi].

With Proposition 4.5 and Corollary 4.6 in hand, the reader should have formed a clear picture of what the tangent space at a point is. For example, the tangent spaces of the two-sphere, S2, are identical to the tangent planes as we are used to consider them. We now turn our attention to what we call the derivation space at a point. As already stated, we will exhibit a linear isomorphism between the derivation space at a point and its tangent space. We will do this by showing that a generalization of the standard partial derivative provides a basis for the derivation space, and ﬁnd an injective, linear map between this basis and the basis from Corollary 4.6.

Definition 4.7. Let M be a smooth manifold, and let p ∈ M be fixed. A derivation at p is a ∞ ∞ linear map vp : C (M) → R that fulfills the Leibniz law ∀f, g ∈ C (M):

vp(fg) = g(p)vp(f) + f(p)vp(g).

Lemma 4.10 shows that two properties that we recognize from the standard (directional) derivatives in Rn follow from linearity and the Leibniz law, and then Proposition 4.12 shows that all derivations are in fact generalizations of directional derivatives. Proceeding in this manner may appear confusing at ﬁrst, but once completed, the picture is much clearer than that which would have been produced if we started with directional derivatives.

Deﬁnition 4.8. If p is a point in a smooth n-manifold M, the derivation space at p, DpM, is the set of all derivations at p.

27 That the derivation space fulfills the axioms for a vector space follows immediately from the definition, when the operations are defined in the obvious way, namely (up+vp)(f) = up(f)+vp(f)

∂ and (avp)(f) = avp(f). That i ∈ DpM, defined below (Definition 4.9), follows from the ∂u p usual product rule. Definition 4.9. Let M be a smooth n-manifold, and let p ∈ M be contained in a chart (U, φ) = (U, u1, u2, . . . , un). Define the operator

∂ ∞ i : C (M) → R ∂u p

−1 f 7→ Di f ◦ φ , φ(p)

n where Dig denotes the i:th partial derivative of a function g : R → R. Lemma 4.10 is important because it shows that the derivations at a point consider only the local behaviour of a function. Not surprisingly this leads to Proposition 4.11, showing that we must only consider any neighbourhood of a point p to completely determine DpM.

Lemma 4.10. Let vp ∈ DpM. Then

∞ (i) if f, g ∈ C (M) are equal on some neighbourhood of p, then vp(f) = vp(g);

∞ (ii) if h ∈ C (M) is constant on some neighbourhood of p, then vp(h) = 0.

Proof. Since vp is a linear map (i) is equivalent to the statement that if f = 0 on a neighbourhood U of p, then vp(f) = 0. Obviously vp(0) = 0, so we let β be a cut-oﬀ function such that sup β ⊂ U and β(p) = 1. Then βf ≡ 0 so

0 = vp(βf) = f(p)vp(β) + β(p)vp(f)

= vp(f).

With (i) proven, we can assume that h ≡ c globally on M, where c is some constant. Then vp(c) = c · vp(1) by linearity, and

vp(1) = 1 · vp(1) + 1 · vp(1)

= 2vp(1), whence vp(1) = 0 =⇒ vp(c) = 0. Proposition 4.11. Let M be a smooth n-manifold and let p ∈ U ⊂ M, where U is open. Then DpU is isomorphic to DpM. Proof. Deﬁne

Φ: DpU → DpM vp 7→ vep, ∞ where vep(f) := vp(f |U ). Now suppose that vep = 0, and let h ∈ C (U). Then there exists a cut-oﬀ function β with support in U such that βh extends by zero to f ∈ C∞(M) and f ≡ h on some neighbourhood of p. Then by Lemma 4.10, vp(h) = vp(f |U ) = vep(f) = 0. Since h was arbitrary, we must have vp = 0, so Φ is injective.

28 Now let wp ∈ DpM, and let h, β, and f be as above. Take vp(h) := wp(f). Then let βb be another cut-off function as above, and let fb be the similar extension of βhb . Then f ≡ fb on some neighbourhood of p, and so by Lemma 4.10 wp(f) = wp(fb). This shows that vp is independent of the choice of cut-off function. ∞ If we now let f ∈ C (M) be arbitrary, and define F as the extension by zero of βf |U to M, where β is some cut-off function as above, then

vep(f) := vp(f |U ) := wp(F )

= wp(f). Where the last equality follows from Lemma 4.10 since f and F agree on some neighbourhood of p. Thus Φ is also surjective, whence it is bijective. Because Φ obviously preserves vector addition and scalar multiplication, Φ is an isomorphism. We are now ready to show that the partial derivatives deﬁned in Deﬁnition 4.9 provides a basis for DpM. As a bonus we will also immediately get the components of any derivation at p in this basis. Proposition 4.12. Let M be an n-manifold and (U, φ) = (U, u1, . . . , un) a chart with p ∈ U. ∂ ∂ Then ,..., is a basis for DpM. Additionally for each vp ∈ DpM we have ∂u1 p ∂un p n X i ∂ vp = vp(u ) ∂ui i=1 p Proof. By Proposition 4.11 we can assume that φ(U) is convex in Rn. Now, for x ∈ φ(U) let y := x−φ(p), and for some f ∈ C∞(M) let g = f ◦φ−1 ◦y−1. Then by the fundamental theorem of calculus Z 1 d g(x) = g(0) + g(tx)dt 0 dt  n  Z 1 X ∂ ∂ = g(0) + g(tx) xi(tx) dt  ∂xi ∂t  0 i=1 n X Z 1 ∂ = g(0) + xi g(tx)dt ∂xi i=1 0 n X i := g(0) + x gi(x), i=1 whence f(q) = g ◦ y ◦ φ(q) n X i = f(p) + x φ(q) − φ(p) fi(q), i=1 ∂ for some smooth functions fi. Now, applying to both sides yields ∂uj p n ∂f X ∂ i i ∂fi = 0 + x φ(q) − φ(p) fi(p) + x φ(p) − φ(p) ∂uj ∂uj ∂uj p i=1 p

29 n X ∂ui = f (p) ∂uj i i=1 n X i = δjfi(p) i=1

= fj(p).

Finally, applying vp ∈ DpM to f yields

n X i vp(f) = 0 + vp(u )fi(p) + 0vp(fi) i=1 n X i ∂f = vp(u ) . ∂ui i=1 p

P i ∂ P i ∂ We have thus shown that vp = vp(u ) , so the the set spans DpM. Now let a = ∂ui p ∂ui p 0 be the zero derivation, and apply it to uj. Then

n n X ∂uj X ai = aiδj ∂ui i i=1 i=1 = aj = 0.

Since this holds for an arbitrary j, the set is also linearly independent. The second assertion has already been shown above.

By now, the reader should have a pretty clear picture of the derivation space at a point, and the promised isomorphism between the derivation space and the tangent space at a point should come as no surprise. Corollary 4.13 exhibits the isomorphism between TpM and DpM: Corollary 4.13. The derivation space at p is isomorphic to the tangent space at p, and by extension to Rn. Proof. Deﬁne the linear map

Φ: TpM → DpM

[c] 7→ Dc, where Dc is the directional derivative given by f(c(h)) − f(p) Dcf = lim h→0 h = (f ◦ c)0(0), for f ∈ C∞(M). By deﬁnition, and by Lemmas 3.7 and 3.5, Φ is well-deﬁned and injective. ∂

Additionally = Dγi , where γi is the i:th component of the basis for TpM in Corollary ∂ui p 4.6. Thus Φ maps basis to basis, so by linearity and injectivity, Φ is a linear isomorphism. Since n isomorphy is transitive, and TpM is isomorphic to R , so is DpM.

30 Corollary 4.13 allows us to express tangent vectors as derivations, and we shall do this henceforth without referring to DpM. We note that we could instead have taken DpM as the definition of TpM but this would have left us without a geometric understanding of the tangent space. Fur- thermore, as stated earlier, we use Definition 4.2 in Definition 4.15 and in Lemma 5.18. However, we purposefully did not include other equivalent definitions, since although useful in some situations, we shall not be needing them. In Lemma 4.14 we return, in spirit, to Definition 4.2. This formalism allows us to show that we may easily define tangent vectors (and thus derivations) at boundary points. The discussion following Lemma 4.14 and Definition 4.15 makes this point clearer. Lemma 4.14. Let M be a smooth n-manifold and let c :(a, b] → M be a smooth curve into dc M, let f ∈ C∞(M), and let c :(a, b + ε) → Rn be a smooth extension of f ◦ c. Then e is e dt b independent of the choice of extension. ∞ 1 Proof. Since ec is C , in particular it is C . Then c(t) − c(b) c(t) − c(b) lim e e = lim e e t→b+ t t→b− t

dc = e . dt b

Now, the right hand side depends only on the behaviour of ec in (a, b], and the lemma follows. dc This allows us to define (f ◦ c)0(b) := e, and we may make a similar definition for curves dt c :[a, b) → M. We use this to define the tangent space at points in the boundary of a smooth manifold: Definition 4.15. Let M be a smooth n-manifold with boundary, and let p ∈ ∂M. We define TpM to be the tangent space formed by curves c into M defined on an interval (a, 0] or [0, a) such that c(0) = p, according to Definition 4.2 and Lemma 4.14. The tangent vectors at p are elements of TpM.

It then follows automatically that TpM behaves the same, as determined by charts, whether or not p is a boundary point. We also note that by Proposition 2.28 ∂M is a (n − 1)-manifold, so Tp∂M is a (n − 1)-dimensional vector space, and since any vp ∈ Tp∂M is also in TpM it is a subspace of TpM. We now define the tangent map at a point, which is used to define the tangent map between tangent bundles (Definition 4.20). Definition 4.16. Let f : M → N be a smooth function between manifolds, and consider a point p ∈ M. Define the tangent map of f at p, Tpf : TpM → Tf(p)N, such that for vp ∈ TpM Tpf(vp) (g) = vp(g ◦ f), for all g ∈ C∞(N). Proposition 4.17 is a chain rule for the tangent map at a point, and it obviously carries over to Definition 4.20. Proposition 4.17. Let f : M → N and g : N → P be smooth maps between manifolds. For each p ∈ M Tp(g ◦ f) = Tf(p)g ◦ Tpf.

31 ∞ Proof. Let h ∈ C (P ) and vp ∈ TpM be arbitrary. Then Tp(g ◦ f)(vp) (h) = vp(h ◦ (g ◦ f))

= vp((h ◦ g) ◦ f) = Tpf(vp) (h ◦ g) = Tf(p)g Tpf(vp) (h).

The tangent bundle (Definition 4.19), which we construct in Proposition 4.18, is a vector bundle that finally unifies the disparate family of tangent spaces. In terms of the qualitative discussion regarding bundles in subsection 2.3, the tangent bundle obviously contains the generalization of vector properties from Rn. The main reason for the additional complexity is that not all manifolds are vector spaces. Proposition 4.18. Let M be a smooth n-manifold, and define [ TM := TpM. p∈M

2n Then T M, πTM ,M, R is a vector bundle, where πTM is the map deﬁned by mapping any element of TpM to p.

1 n −1 Proof. First deﬁne for each chart (U, φ) = (U, u , . . . , u ) the set TU := πTM (U) ⊂ TM. Then deﬁne, for vp ∈ TpM, the map

1 n 1 n φe : vp 7→ u (p), . . . , u (p), v , . . . , v ,

i i i where v := vp(u ) = du (vp). Thus

1 n 1 n φe = u ◦ πTM , . . . u ◦ πTM , du , . . . , du .

We also have the tangent map T φ : TU → TV where V ⊂ Rn, obtained by combining the maps n Tqφ for all points q ∈ U. Thus we may make the identiﬁcation Tφ(p)V = φ(p) × R . Now let ∞ n [γ] = vp, and observe, for some g ∈ C (R ) Tpφ(vp) (g) := vp(g ◦ φ) = (g ◦ φ ◦ γ)0(0), whence, under our identiﬁcation

d Tpφ(vp) = φ(p), (φ ◦ γ) . dt t=0

n Thus, if we denote by ei the i:th component of the standard basis on R , then

 n  X i Tpφ(vp) = φ(p), v ei i=1 = (u1(p), . . . , un(p), v1, . . . , vn),

32 but this last expression is just φe(vp). So we have T φ = φe, and for each chart (U, φ) on M we can thus deﬁne a chart (T U, T φ) on TM. That T φ(TU) is open in R2n is obvious by considering the deﬁnitions. The set of such charts on TM obviously cover TM, and if (T U, T φ), (T V, T ψ) are two such charts with TU ∩ TV 6= ∅. Then U ∩ V 6= ∅, and on the overlap we get the coordinate transitions T ψ ◦ (T φ)−1 :(x, v) 7→ (y, w) where

y = ψ ◦ φ−1(x),

d w = (ψ ◦ γ) dt t=0

d −1 = (ψ ◦ φ ◦ φ ◦ γ) dt t=0 d = D ψ ◦ φ−1 · (φ ◦ γ) x dt = D ψ ◦ φ−1 · v. x

Thus TU and TV are compatible for any two charts U, V . Again that T φ(TU ∩ TV ) and T ψ(TU ∩ TV ) are open in R2n is obvious by the same considerations as for T φ(TU). Thus each atlas on M gives rise to an atlas on TM, and the thus induced topology is Hausdorﬀ and second countable by Proposition 2.20, because the topology of M is. Finally observe that the principal parts of the local trivializations have been given in the proof of Proposition 4.5. Thus we can easily deﬁne the local trivializations for each chart on M, whence the vector bundle structure has been exhibited.

Deﬁnition 4.19. The vector bundle in 4.18 is called the tangent bundle of the manifold M.

The tangent bundle allows us to combine the tangent maps at a point (Deﬁnition 4.16 to form the tangent map (Deﬁnition 4.20).

∞ Deﬁnition 4.20. Given f ∈ C (M,N) for smooth manifolds M,N, the tangent maps Tpf combine to give a map T f : TM → TN, which is linear on each ﬁber. This is the tangent map of f.

Corollarly 4.21 depends on Proposition 4.17, which is notationally simpler in terms of Defi- nition 4.20. We need it for Definition 5.3, so that the pull-back of a smooth differential form is again smooth.

Corollary 4.21. For any smooth map f : M → N between smooth manifolds M and N, of dimensions m respectively n, the tangent map T f is smooth.

Proof. Note that T f : TM → TN and consider any charts (U, φ) on M and (V, ψ) on N. Then for any g ∈ C∞(Rn) we have, by identifying T Rm = Rm × Rm

T ψ ◦ T f ◦ (T φ)−1(x, v) · g = T (ψ ◦ f ◦ φ−1)(x, v) · g = v g ◦ ψ ◦ f ◦ φ−1 . x Now, the ﬁnal expression is simply a directional derivative of a smooth function in standard calculus, whence it depends smoothly on x and on v. Since g was arbitrary this must hold in general.

33 The cotangent space (Definition 4.22) is the dual space (Definition 2.1) of the tangent space, and allows for a vector bundle structure, similarly to the tangent space. However, as it also appears as a special case of Proposition 5.1, we shall not make any explicit reference to the thus constructed cotangent bundle. Definition 4.22. Let M be a smooth manifold, and let p ∈ M. We denote the dual space of ∗ TpM by Tp M and call it the cotangent space. In Rn we are not used to distinguishing tangent vectors from cotangent vectors (or simply co-vectors), since both are intrinsically identified with elements of the vector space Rn. However, while tangent vectors appear naturally as velocity vectors, cotangent vectors appear naturally as normal vectors to hypersurfaces. To see this consider a hypersurface Σ ⊂ M defined locally by f(u1, . . . , un) = 0, where u1, . . . , un are coordinate maps in some chart containing a point p ∈ Σ. Then dfp (Definition 4.23) is a co-vector such that any (tangent) vector vp tangent to Σ at p (if v = [c] then c ⊂ Σ and c(0) = p) fulfills dfp(vp) = 0. Just as Definition 4.16 combines to give Definition 4.20, our definition of differentials at a point (Definition 4.23) combine to give differentials in Definition 5.4. However, we must postpone such considerations until after Proposition 5.1. Lemma 4.24 then obviously carries over to Definition 5.4, and we shall use this to prove Proposition 5.9. Similarly, Proposition 4.25 actually gives us a local frame field (Definition 2.33) for the cotangent bundle, and thus by Proposition 2.12 a local frame field for the alternating bundle of Proposition 5.1. Definition 4.23. Let M be a smooth manifold and let p ∈ M. For f ∈ C∞(M) we define the differential of f at p to be the linear map dfp : TpM → R given by

dfp(vp) = vp(f),

∗ for all vp ∈ TpM. Thus dfp ∈ Tp M. Lemma 4.24. Let M be a smooth manifold and let f, g ∈ C∞(M). For each p ∈ M

d(fg)p = g(p)dfp + f(p)dgp.

Proof. For each vp ∈ TpM we have

d(fg)p(vp) = vp(fg)

= g(p)vp(f) + f(p)vp(g)

= g(p)df(vp) + f(p)dg(vp).

∂ ∂ Proposition 4.25. For some manifold M, let ,..., be a basis of the tangent ∂u1 p ∂un p 1 n ∗ space TpM at a point p. Then dup, . . . , dup is the dual basis of the cotangent space Tp M. Proof. By deﬁnition i ∂ ∂ i dup = u ∂uj p ∂uj p i = δj.

34 5 Diﬀerential Forms and Stokes’ Theorem

5.1 Differential Forms and Integration Differential forms allows us to generalize integrals to smooth manifolds. They can be given a more direct geometrical interpretation via the Grassman algebra, but to retain compactness we shall not delve into this subject here, and instead refer the interested reader to e.g. [7]. Differential forms are defined as sections of a vector bundle, exhibited in Proposition 5.1. The author would also like to note that it is entirely possible to define bundle-valued differential forms, but that the generalization, although straight-forward in principle, requires some extra work, and is altogether unimportant for our purposes here. It is however frequently used in physics, and we shall perform the generalization in Section6. We then define the exterior product of differential forms (Definition 5.6) via the exterior product of alternating tensors (Definition 2.8); and the exterior derivative (Definition 5.10) via the differentials at a point (Definion 4.23). From there we move on to define orientations of a manifold, and the induced orientation of its boundary. This allows us to define integration of differential forms over manifolds, which is the final piece of the puzzle needed to formulate TheoremA. Along the road we will produce one result, namely Lemma 5.11, which is used directly in Subsection 5.2. Other results, although many times notable in their own regard, are used only in this section. We begin by constructing the necessary vector bundle:

Proposition 5.1. Let M be a smooth n-manifold, and deﬁne

k [ k Lalt(TM) := Lalt(TpM). p∈M

k d n Then (Lalt(TM), π, M, R ) is a vector bundle, where d = k and π is the map deﬁned by mapping k any element of Lalt(TpM) to p.

1 n S k −1 Proof. Let (U, φ) = (U, u , . . . , u ) be a chart on M, and let Ue := p∈U Lalt(TpM) = π (U). n d Then, if d = k , we have a map f : Ue → U × R , given by αp 7→ (p, a), where a is some d-tuple. Now by Proposition 2.12, if we let ∂1, . . . , ∂n denote the basis of TpM with respect to our chart (see Proposition 4.12), we have

X I αp = αp(∂I )dup. k I∈In

k We can order the set In by letting I < J if and only if ` is the lowest index such that i` 6= j` k and i` < j`. If i` = j` for all ` ∈ {1, . . . , n} then clearly I = J. Thus In = {I1,I2,...,Id} where j < ` if and only if Ij < I`. We may hence denote αp(∂I` ) by αp(`) and let a = αp(1), . . . , αp(d) .

d Now let fb : Ue → R be given by αp 7→ a, and deﬁne fe := (φ ◦ π, fb). Then (U,e fe) is a chart k d on Lalt(TM). If (V, ψ) is another chart on M and g : Ve → V × R is the corresponding map, then U ∩ V = ∅ if and only if Ue ∩ Ve = ∅, and if Ue ∩ Ve 6= ∅ we have the transition map

35 −1 −1 −1 −1 fe◦ ge = (φ ◦ ψ , fb◦ gb ). For φ ◦ ψ the smoothness is guaranteed since (U, φ) and (V, ψ) are compatible charts, and

 d  −1 X Ij fb◦ gb (r1, . . . , rd) = fb rjdvp  j=1

 d d  X Ij X Ij =  rjdvp (∂I1 ),..., rjdvp (∂Id ) j=1 j=1

k Thus the transition maps are (smooth) linear maps and our charts on Lalt(TM) are all compati- k ble, and since they obviously provide a cover, Lalt(TM) is a smooth (n+d)-manifold. That (U, f) k d and (V, g) are the corresponding vector bundle charts is now obvious, and so (Lalt(TM), M, π, R ) is a vector bundle.

Just like vector fields can be defined as smooth sections of the tangent bundle, TM, differential k k-forms are defined as sections of Lalt(TM). In particular, then, co-vector fields are 1-forms, and vice versa, and smooth functions are 0-forms. Differential forms are, however, more difficult to understand intuitively than vector fields are, but if we recall the results from Subsection 2.1, we can convince ourselves that differential k-forms may be thought of as some sort of measurement of an object determined by an ordered set of k vectors. Such an object corresponds to an oriented k-plane element in the same way that a vector corresponds to a line element via the velocity vector. Further understanding is best gained from the Grassman algebra, which we do not exhibit here for brevity. The interested reader is instead referred to e.g. [7]. However, to connect with the way we explained that 1-forms naturally appear (as normal vectors to hypersurfaces) we will present an incomplete picture. We will also use the exterior product (Definition 5.6), but present it here because of its use in understanding differential forms. Suppose that a k-form can be written as the exterior product of k 1-forms (such a k-form is said to be decomposable). Each 1- form determines a hypersurface locally at each point, but this says nothing of its magnitude. Let each 1-form determine a family of (identical) hypersurfaces, and let the “distance” between two hypersurfaces in the same family be inversely proportional to the magnitude of the 1-form, such that the 1-form “measures” how many hypersurfaces a vector, or alternatively a line element, pierces (where a vector tangential to the hypersurface is thus intuitively taken to pierce none). A decomposable 2-form similarly determines a family of “tubes”, and measures how many tubes an oriented 2-surface element cover. A decomposable k-form can be taken to determine a family of “cells” (limited in k dimensions), and measure how many such cells an oriented k-plane element fill. The orientation allows for positive or negative values, depending on how it matches the order of multiplication of the 1-forms, or correspondingly the orientation of the cells. Each k-form can be written as a sum of decomposable k-forms.

k k Deﬁnition 5.2. We denote the sections Γ(M; Lalt(TM)) by Ω (M) and sections over U ⊂ M by Ωk(U). We will let Ω(M) denote the direct sum of all Ωk(M). Elements of Ωk(M) are called diﬀerential k-forms over M.

Although not necessary for our purposes note that Ω(M) can be trivially extended to a graded algebra, see e.g. [7] for the details. We mention it here to clarify the concept of a graded derivation in Deﬁnition 5.7.

36 Pull-backs and the related concept of push-forwards are important in differential geometry, as they provide means to relate geometrical objects in one manifold to those in another via smooth functions. However, we shall only be needing the pull-back of differential forms: Definition 5.3. Let f : M → N be a smooth map between smooth manifolds M and N. The r pull-back of a differential form ω ∈ Ω (N) by f is defined for all v1, . . . , vr ∈ TpM by

∗ f ω(v1, . . . , vr)(p) := ω(T f · v1, . . . , T f · vr).

We now combine the point differentials of Definition 4.23 to form the differential. We shall see later that the differential is really just a special case of the exterior derivative (Definition 5.10). The notion of exact 1-forms also exist for forms in general, with the exterior derivative taking the role of the differential. This notion is essential for De Rham cohomology, but outside the scope of this text, so we only mention it here for completeness, and refer the interested reader to e.g. [7].

∞ Definition 5.4. Given f ∈ C (M) on a smooth manifold M, the differentials dfp combine to give a map df : TM → R. This is a 1-form called the differential of f. Any 1-form that can be written as the differential of a smooth function is said to be an exact 1-form. Like Definition 5.4 is a special case of Definition 5.10, so too is Lemma 5.5 a special case of Lemma 5.11, which we use directly in Subsection 5.2. Both show that the operator in question commutes with pull-back. Remembering that the pull-back is used to relate geometric objects in one manifold to those in another, it is clear that this implies that it does so in a manner that preserves the differential, and later the exterior derivative. Alternatively, one can view it as being the differential, and later the exterior derivative, that is natural in such a way that it commutes with the pull-back. Lemma 5.5. The differential is natural with respect to pull-back. In other words, if f : M → N is a smooth map between smooth manifolds M and N, and g ∈ C∞(N), then d(f ∗g) = f ∗dg.

Proof. Let v ∈ TpM and write q = f(p), then

∗ (f dg) v = dg (Tpf · v) p q = (Tpf · v)g (q) = v(g ◦ f)(p)

= d(f ∗g) v. p Since p and v were arbitrary, the statement is true without restrictions. The last definition to carry over previous definitions to the language of differential forms is Definition 5.6, defining the exterior product. It extends Definition 2.8.

Deﬁnition 5.6. Let ω ∈ Ωk1 (M) and η ∈ Ωk2 (M), for a smooth manifold M. Then we deﬁne the exterior product ω ∧ η ∈ Ωk1+k2 (M) by

(ω ∧ η)(p) := ω(p) ∧ η(p).

The definition of a (natural) graded derivation (Definition 5.7) is used to show that it is determined entirely by its action on functions and exact 1-forms (Lemma 5.8). This leads to Proposition 5.9, which in turn allows us to define the exterior derivative (Definition 5.10).

37 Deﬁnition 5.7. A (natural) graded derivation of degree r on Ω(M), for some smooth manifold M, is a family of maps, one for each open set U ⊂ M, denoted by DU : Ω(U) → Ω(U) such that

k k+r DU :Ω (U) → Ω (U), and such that

(i) DU is R-linear; kr k (ii) DU (α ∧ β) = DU (α) ∧ β + (−1) α ∧ DU (β) for all α ∈ Ω (U) and β ∈ Ω(M); (iii) for V ⊂ U the following diagram commutes, where the downward arrow represent restrictions:

Ωk(U) DU Ωk+r(U)

Ωk(V ) DV Ωk+r(V )

We will denote all the maps DU by D.

Lemma 5.8. Suppose D1 and D2 are graded derivations of degree r on Ω(M) for some smooth n-manifold M. If they agree when applied to functions and exact 1-forms, then D1 = D2.

Proof. By the deﬁnition, if D1 and D2 agree on chart domains, then they agree globally. Let (U, φ) = (U, u1, . . . , un) be a chart on M. Then every element of Ωk(U) is a sum of elements of the form fdui1 ∧ · · · ∧ duik , where f ∈ Ω0(M), but by the deﬁnition

i1 ik i1 ik i1 ik D1(fdu ∧ · · · ∧ du ) = D1(f) ∧ du ∧ · · · ∧ du ± fD1(du ∧ · · · ∧ du )

i1 ik i1 ik = D2(f) ∧ du ∧ · · · ∧ du ± fD1(du ∧ · · · ∧ du ).

i The second term can be similarly expanded using the deﬁnition, and then the elements D1du i i1 i can be replaced by D2du . The result is obviously equal to D2(fdu ∧ · · · ∧ du k ). By R-linearity they then agree for all diﬀerential forms.

The demands made in Proposition 5.9 may seem arbitrary, or more specifically the first one may. Hopefully it is clear why extending Definition 5.4 to a natural graded derivation would be advantageous. If nothing else, its use in TheoremA could be viewed as an argument as to why the exterior derivative (Definition 5.10) that comes from this is sensible, but this fails to express the role the exterior derivative can play in understanding the structure of a smooth manifold (with or without metric). The first demand is not as arbitrary as it may seem, though. Start with the pictorial view of differential forms presented near the start of this section, where a k-form on an n-manifold represents a grid of k-cells extending in (n − k) dimensions. Say we want to know if any of these k-cells end somewhere in any of the (n − k) dimensions, and furthermore would like to place a new hypersurface there, turning them into (k + 1)-cells. Maybe we can agree that this may be represented by a degree one derivation. Now, there is a topological fact that states that the (oriented) boundary of an (oriented) boundary is zero; it is not difficult to convince yourself of this fact in lower dimensions by considering simple objects (see Figure3), and for manifolds we have already shown that the boundary is without boundary (Proposition 2.28). So we would expect that applying the degree one derivation again would result in zero.

38 -form on 0 and boundaries, a . For a d M of , then we should expect ) will not pass through the M n M coincides with the diﬀerential we have ) df , . . . , u U ( 1 k Ω

. U, u I ∈ ) α must pass straight through the manifold, , du I ( ) that end inside ) = ( ) the 1-form M ω ∧ ) du I ( The boundary of the cube is made up by six U, φ M I = 0 (

( dα α d 39 ∞ k n k n I I ◦ -cells that pass through the boundary, i.e. the right C ∈ ∈ d X X I I 1) -manifold. There is a unique degree one derivation ∈ n − = =

f n ( α α φ d and for each chart -cells (representing M φ 1) d ⊂ − be a smooth U n ( M to be the usual differential. For φ Let , and we define d such that ) can be viewed as a statement of a clear connection between ) M ( M ∞ = 0 squares, and theHowever, boundary each edge of is the counted boundary twice, in is opposite made directions, up so summing of gives the zero. edges of each square. The boundary of a boundary is zero. Ω( C d ◦ ∈ → d ) I We define an operator α M This view also matches the differential, if we allow the “density of boundaries”Thus of a scalar we simply define Figure 3: : Ω( where U d from Definition 5.4. Proof. thus being counted twice withaccounts opposite for signs the and fact summedboundary, to that zero. a since cell Via each entirely orientation, endTheoremA, this contained is view in is also the slightly counted, interior more and of complicated, again but withProposition we opposite have 5.9. just signs. found the The essence mathematics of behind it. and such that for each open field (0-form), orsurfaces, equivalently i.e. a smooth itsthe rate function, exterior of to derivative. change. be Having equated formed with this the picture, density we of can glimpse its the levels importance of fact that perhaps putsA Theorem asin counting a the new number of light.that Indeed, this if number equals we intuitively thehand view number side. the of left Because hand any side cell that does not end inside That this is an R-linear map follows immediately from the definition of the differential and Lemma 4.10 in conjunction with the Leibniz rule. To show that it satisfies property (ii) of P I k P J ` Definition 5.7, consider α = k αI du ∈ Ω (U) and β = k βJ du ∈ Ω (U). Then I∈In J∈In    X I X J  dφ(α ∧ β) = dφ  αI du ∧ βJ du  k k I∈In J∈In    X I J  = dφ  αI βJ du ∧ du  k I,J∈In X I J = (dαI )βJ + αI (dβJ ) ∧ du ∧ du k I,J∈In      X I  X J X I  k X J  =  dαI ∧ du  ∧ βJ du + αI du ∧ (−1) dβJ ∧ du  k k k k I∈In J∈In I∈In J∈In = d(α) ∧ β + (−1)kα ∧ dβ, by Lemma 4.24 and Proposition 2.9. ∞ Pn ∂2f i j For any f ∈ C (M) we have dφdφf = dφdf = i,j=1 ∂ui ∂uj du ∧ du = 0 because ∂2f i j ∂ui ∂uj is symmetric in i, j and du ∧ du is antisymmetric in i, j. From above it then follows ∞ that dφ(df1 ∧ · · · ∧ dfk) = 0 for any functions fi ∈ C (M), whence it follows that dφ ◦ dφα = P I k dφ k dαI ∧ du = 0 for any α ∈ Ω (U). I∈In Clearly dφ has the desired properties on the chart (U, φ), and we can define a similar operator on each chart. Now consider two different charts (U, φ) and (V, ψ) such that U ∩ V 6= ∅. That property (iii) is satisfied is obvious from our definition, and then since, for any f ∈ C∞(M), dφf = dψf = df and dφdf = dψdf = 0, dφ |U∩V = dψ |U∩V by Lemma 5.8. Thus we can combine the individual operators for charts to give a well-defined operator with the desired properties. Definition 5.10. The degree one graded derivative introduced in Proposition 5.9 is called the exterior derivative. Lemma 5.11 extends Lemma 5.5 to the exterior derivative, as promised. The proof uses Lemma 5.5 and the fact that by definition the pull-back is distributive over addition and wedge products. Lemma 5.11. Given any smooth map f : M → N between smooth manifolds M and N, we have that d is natural with respect to the pull-back:

f ∗(dη) = d(f ∗η), for all η ∈ Ω(N). Proof. By Lemma 5.5 we know that the result is true if η is a 0-form. Because of property (iii) in Definition 5.7, we need only prove the statement for a differential form defined in the domain of some chart (U, φ) = (U, u1, . . . , un). By linearity we may assume that η = gdui1 ∧ · · · ∧ duik , for some g ∈ C∞(U) since an arbitrary form on U is a sum of forms of this type: f ∗(dη) = f ∗ d(gdui1 ∧ · · · ∧ duik )

40 = f ∗(dg ∧ dui1 ∧ · · · ∧ duik ) = d(f ∗g) ∧ d(f ∗ui1 ) ∧ · · · ∧ d(f ∗uik ) = d(g ◦ f) ∧ d(ui1 ◦ f) ∧ · · · ∧ d(uik ◦ f) = d (g ◦ f)d(ui1 ◦ f) ∧ · · · ∧ d(uik ◦ f) = d(f ∗η).

We now turn our attention to orientable manifolds, and the orientations thereof. In fact, these notions depend on similar notions for vector bundles, and the tangent bundle in particular. However, we shall not expand on this for brevity. Definition 5.12. A smooth n-manifold M is said to be orientable if there is a nowhere vanishing form $ ∈ Ωn(M). We call such a form a volume form. Two volume forms $ and ϑ are said to be equivalent if ϑ = f$ for some positive function f ∈ C∞(M), and we denote the equivalence class of a volume form $ by [$]. An orientation of M is an equivalence class of volume forms. If an orientation is chosen, the manifold is said to be oriented. If we restrict ourselves to connected manifolds, it is clear that there are exactly two orientations, and each orientation determines, up to the sign of permutations, an order on the local frame fields of the tangent bundle. In other words, an orientation of a manifold determines a unique orientation of each tangent space, as we know it from linear algebra. This is of course true even in non-connected manifolds, since we may treat each connected component separately in this case, but the similarity is more clear in the connected case. Positively oriented atlases (Definition 5.13) are convenient to work with, and we later show that for any manifold (with boundary) of dimension greater than one, we can always find a positively oriented atlas. This simplifies our proofs in Subsection 5.2 notationally. Definition 5.13. Let M be a smooth n-manifold. An atlas A = (Uα, φα) α∈A for M is said to be an oriented atlas if for all (U, φ) = (U, u1, . . . , un) ∈ A

∂ ∂ $(p) ,..., > 0 ∂u1 p ∂un p for all p ∈ U and some orientation [$]. A is a positively oriented atlas for (M, [$]). An orientation preserving diffeomorphism between two oriented manifolds is simply a diffeomorphism that equates the two orientations, via pull-back. This gives rise to a new definition of the Jacobian determinant of multi-variable calculus. In fact, we will later show (Proposition 5.16) that our new definition reduces to the old one for the standard volume forms on Rn. Definition 5.14. Let f : M → N be a diffeomorphism between smooth oriented n-manifolds (M, [$]) and (N, [ϑ]). We say that f is orientation preserving if f ∗ϑ ∈ [$]. Specifically, because n n ∗ each fiber of the bundles Lalt(TM) and Lalt(TN) is 1-dimensional, f ϑ = J$,ϑ(f)$ for some ∞ J$,ϑ(f) ∈ C (M). This function is called the Jacobian determinant of f with respect to $ and ϑ, and the diffeomorphism is orientation preserving if it is everywhere positive. Before moving on, we require the result of Proposition 5.15 for Definition 5.14 to make sense. It shows that the previous claim, that an orientation preserving diffeomorphism equates the two orientations via pull-back, is in fact true, and not just so for specific representatives of the orientations.

41 Proposition 5.15. Let f :(M, [$]) → (N, [ϑ]) be a diﬀeomorphism between smooth oriented n-manifolds. The sign of J$,ϑ(f) is independent of the choice of volume forms $ ∈ [$] and ϑ ∈ [ϑ]. Proof. First suppose ϑ = aϑ0 where a is an everywhere positive function. Then

∗ J$,ϑ(f)ω = f ϑ = f ∗(aϑ0) = (a ◦ f)(f ∗ϑ0)

= (a ◦ f)J$,ϑ0 (f)$, and since (a ◦ f) is everywhere positive by virtue of a being so, we must have sgn J$,ϑ(f) = sgn J$,ϑ0 (f) . Now suppose that $ = b$0 where b is an everywhere positive function. Then

∗ J$,ϑ(f)$ = f ϑ 0 = J$0,ϑ(f)$ 0 = bJ$,ϑ(f)$ , so again sgn J$,ϑ(f) = sgn J$0,ϑ(f) by virtue of b being everywhere positive. We next use Proposition 2.14 to show that, to some extent, our new definition of the Jacobian determinant (Definition 5.14) corresponds to the one known from multi-variable calculus. This result should come as no surprise to the reader, and we use it in Lemma 5.24 when we perform a change of variables. We see that they are the same whenever our volume forms are the coordinate basis of Ωn(Rn). As it stands, the new Jacobian also takes care of the change of variables for the integrand, the function, as we would view it in multivariable calculus. Proposition 5.16. Let f :(M, [$]) → (N, [ϑ]) be a diffeomorphism between smooth oriented n-manifolds. For any charts (U, φ) = (U, u1, . . . , un) of N, and (V, ψ) = (V, v1, . . . , vn) of M −1 1 n 1 n such that f (U) ⊂ V , we may write ϑ |U = gdu ∧ · · · ∧ du and $ |V = hdv ∧ · · · ∧ dv , where g and h are nowhere vanishing smooth functions. Then g ◦ f J (f) | = det(Df), $,ϑ f −1(U) h where det(Df) is defined via the charts. Proof. The transformation f ∗ :Ω1(N) → Ω1(M) is given locally by

f ∗dui = d(ui ◦ f) n i X ∂ u ◦ f = dvj. ∂vj j=1 Thus by Proposition 2.14 we have det(f ∗) = det(Df), and we ﬁnd, under suitable restrictions

∗ J$,ϑ(f)$ = f ϑ = g ◦ f det(f ∗)dv1 ∧ · · · ∧ dvn g ◦ f = det(Df)$ h

42 The notion of outward pointing vectors (Definition 5.17) on the boundary of a smooth manifold is needed to define the induced orientation. Lemma 5.18 then provides a proof of the usual property: that it suffices to check any chart containing a point p to determine whether or not a vector vp ∈ TpM is outward pointing. We use it in Lemma 5.19.

Definition 5.17. Let M be a smooth n-manifold with boundary, and let p ∈ ∂M. A vector vp ∈ 1 n TpM ⊂ TM |∂M is called outward pointing if in some half-space chart (U, φ) = (U, u , . . . , u ) 1 containing p we have du (vp) > 0. A smooth section χ of TM |∂M is called outward pointing if χ(p) is outward pointing for each p. Inward pointing is defined analogously. The proof of Lemma 5.18 uses Definition 4.15, and in the process makes it very clear why outward pointing vectors are called outward pointing, and vice versa. We therefore do not attempt to clarify this qualitatively, but move on directly to Lemma 5.18: Lemma 5.18. Let M be a smooth n-manifold with boundary, and let p ∈ ∂M. A vector 1 vp ∈ TpM is outward pointing if and only if du (vp) > 0 for all half-space charts (U, φ) = 1 n 1 (U, u , . . . , u ) containing p, and inward pointing if and only if du (vp) < 0 for all half-space charts containing p.

1 1 1 1 Proof. We first recall that du (vp) = vp(u ) and note that u : M → (−∞, 0] with u (p) = 0. 1 Thus if vp = [c] for some curve c :(a, 0] → M or c : [0, a) → M with c(0) = p, then vp(u ) > 0 if and only if c(t) ∈/ ∂M on some interval (−ε, 0), where ε > 0. Thus whether or not vp is outward pointing is a property of the vector, and independent of charts. Similarly, vp is inward pointing if and only if c(t) ∈/ ∂M on some interval (0, ε). Lemma 5.19 is important because it tells us that an orientation of a smooth manifold with boundary always defines an induced orientation of the boundary (Definition 5.20). From our understanding of what it means for a vector to be outward pointing, it is clear that such a vector always exists on every boundary point of a smooth manifold with boundary. Lemma 5.19 simply shows that such vectors always can be combined to form a smooth vector field on the boundary. Intuitively, this seems obvious, and the proof is not much more complicated. Lemma 5.19. Let M be a smooth n-manifold with boundary. Then outward pointing sections of TM |∂M always exist.

Proof. Let {ρα}α∈A be a smooth partition of unity subordinate to some half-space atlas of M, (Uα, φα) α∈A, and deﬁne X ∂ χ := ρ . α ∂u1 α∈A α Then for p ∈ ∂M we have

X ∂ χ(p) = ρα(p) . ∂u1 α∈A α p

Now let (V, ψ) = (V, v1, . . . , vn) be some half-space chart in the atlas. Then

! 1 X 1 ∂ dv χ(p) = ρα(p)dv ∂u1 α∈A α p > 0,

43 because ρα(p) ≥ 0 for all α, with ρα(p) > 0 for at least one α, and

! 1 ∂ dv 1 > 0 ∂uα p

∂ for all such α, since for all U containing p we have du1 = 1 > 0 so the result follows α α 1 ∂uα p by Lemma 5.18.

The induced orientation of the boundary of a smooth manifold is deﬁned by the orientation of the manifold and an outward pointing section. In terms of the orientations of the tangent spaces, the induced orientation can be represented by an ordered basis that matches the original orientation whenever it is completed with an outward pointing vector in the ﬁrst slot. Figure3 serves to exemplify this in a familiar manner: we can deduce from the orientations of the surfaces (boundary) that the cube has the standard righthanded orientation, and we observe that this is true for all surfaces.

Deﬁnition 5.20. Let (M, [$]) be an oriented smooth n-manifold with boundary, and suppose n−1 that χ is an outward pointing section of TM |∂M . Let $ ∈ [$]; then deﬁne ıχ$ ∈ Ω (∂M) by

ıχ$(p)(v2, . . . , vn) := $(p)(χ(p), v2, . . . , vn). Clearly ıχ$ is nowhere vanishing, and clearly ϑ ∈ [$] ⇐⇒ ıχϑ ∈ ıχ$ . The orientation thus deﬁned on ∂M is called the induced orientation.

Although our definition of integration of a differential form over a manifold includes a sign- dependence (see further Definition 5.25), it is convenient to work with atlases of positively oriented charts. Lemma 5.21 shows that such an atlas always exist for manifolds of dimensions greater than one. We use this implicitly in Subsection 5.2.

Lemma 5.21. If M is a smooth n-manifold with non-empty boundary and n ≥ 2, then there is an atlas of positively oriented charts.

Proof. If a chart (U, φ) = (U, u1, . . . , un) is not positively oriented then the chart (U, ψ) = (U, u1, −u2, u3, . . . , un) is. However, if n = 1 we cannot take (U, ψ) = (U, −u1) because −u1 : M → [0, ∞) and is thus not a half-space chart.

We now turn our attention to integration, for which we will be using only diﬀerential forms with compact support:

Definition 5.22. The support of a differential form ω ∈ Ωk(M), where M is a smooth manifold, is defined as the closure of the set p ∈ M : ω(p) 6= 0 and is denoted by sup (ω). The set of all k k-forms that have compact support is denoted by Ωc (M), and the set of all k-forms with compact k support contained in a subset U ⊂ M is denoted by Ωc (U).

We first define integration of differential forms over Rn. Note that we only define integration for top forms, and that the integration corresponds with our previous understanding of differential forms as a sort of “measure” of (in this case) oriented volume elements, by considering that we measure the manifold in a way characteristic to the form, rather than measuring the function that determines the form.

44 n n 1 n Definition 5.23. Let ω ∈ Ωc (U), where U ⊂ R is open. Let (u , . . . , u ) be the standard coordinates on U. Then ω = fdu1 ∧ · · · ∧ dun for some smooth function f with compact support in U. We define Z Z ω := f(u)du1 ··· dun, U U where the latter integral is the Riemann (or Lebesgue) integral of f. For this we extend f by zero to all of Rn and integrate over a sufficiently large closed n-cube containing the support of f in n R its interior. If instead U ⊂ Rx1≤0 then we define U ω by the same formula. Lemma 5.24 shows that the integration is independent of the choice of coordinates. When we define integration over manifolds in Definition 5.25, this translates to independence of what chart is chosen.

Lemma 5.24. Let f : V → U be an orientation preserving diﬀeomorphism between open subsets n n U, V of R , and let ω ∈ Ωc (U ∩ V ) Then Z Z ω = f ∗ω. U V If f is not orientation preserving, we have instead Z Z ω = − f ∗ω. U V Proof. Suppose ﬁrst that f is orientation preserving. We have without restriction that ω = gdu1 ∧ · · · ∧ dun for some smooth function g. Let u1, . . . , un denote the standard coordinates on U, and let v1, . . . , vn denote the standard coordinates on V . Then by the classical change of variable formula: Z Z ω = g(u)du1 ··· dun U U Z 1 n = g ◦ f(v) det(Df) dv ··· dv , V

0 1 n 1 n but by Proposition 5.16 det(Df) = Jη0,η(f) > 0 where η = dv ∧· · ·∧dv and η = du ∧· · ·∧du . Thus we may drop the absolute signs to get Z Z ω = g ◦ f(v) det (Df) dv1 ··· dvn U V Z = g ◦ f(v) det (Df) dv1 ∧ · · · ∧ dvn V Z = f ∗ω V Finally note that if f is not orientation preserving, then, again by Proposition 5.16, we introduce a minus-sign when we drop the absolute signs.

We are now ready to define integration of differential forms over smooth manifolds with boundary. However, Definition 5.25 does require Lemma 5.26 to be well defined.

45 Deﬁnition 5.25. Let M be an oriented smooth n-manifold with boundary, let (U, φ) be a chart n on M, and let ω ∈ Ωc (U). Then Z Z ∗ ω := sgn(φ) φ−1 ω, U φ(U) where ( 1 if (U, φ) is positively oriented sgn φ = −1 otherwise.

n By Lemma 5.24, this deﬁnition is independent of which chart is chosen. If ω ∈ Ωc (M) but sup (ω) is not contained in some chart we choose a partition {ρα}α∈A of unity subordinate to our atlas (Uα, φα) α∈A and deﬁne

Z X Z ω : = ραω M α∈A Uα Z ∗ X −1 = sgn(φα) φα (ραω). α∈A φα(Uα)

Lemma 5.26. The sum in Deﬁnition 5.25 contains a ﬁnite amount of nonzero terms, and the sum is independent of our choice of atlas and smooth partition of unity.

Proof. First, by the definition, for any p ∈ M there is an open set O containing p such that a finite amount of ρα are nonzero on O. Then, as ω has compact support, a finite number of such open sets cover the support, whence only a finite number of ρα are nonzero on sup (ω). Now let (Vβ, ψβ) β∈B be another atlas of M, and let σβ β∈B be a partition of unity subordinate to our new atlas. Then we have   X Z X Z X ραω = ρα σβω α∈A Uα α∈A Uα β∈B X X Z = ρασβω α∈A β∈B Uα∩Vβ   X Z X = σβ ραω β∈B Vβ α∈A X Z = σβω, β∈B Vβ where the sums could be moved precisely because they contain a finite amount of nonzero terms.

46 5.2 Stokes’ Theorem on Smooth Manifolds In proving Stokes’ theorem on smooth manifolds, we ﬁrst prove it for the special cases of Rn and n Rx1≤0. It then follows that the theorem holds locally for any smooth manifold with boundary, and we ﬁnally show that the property extends globally via a partition of unity. Note that the 1-dimensional case is simply the Fundamental Theorem of Calculus. Thus we will concern ourselves only with manifolds of dimension n > 1. By Lemma 5.21 we can thus assume an atlas of positively oriented charts.

Lemma 5.27. For any smooth (n − 1)-form ω with compact support in Rn, Z dω = 0. n R

1 j n Proof. Let ωj = fdx ∧ · · · ∧ dxd ∧ · · · ∧ dx , where the caret denotes omission, f has compact support in Rn, and j = 1, . . . , n. Then for all j Z Z 1 j n dωj = d fdx ∧ · · · ∧ dxd ∧ · · · ∧ dx n n R R Z = df ∧ dx1 ∧ · · · ∧ dxdj ∧ · · · ∧ dxn n R   Z n X ∂f k 1 j n =  k dx ∧ dx ∧ · · · ∧ dxd ∧ · · · ∧ dx  n ∂x R k=1 Z j−1 ∂f 1 n = (−1) j dx ∧ · · · ∧ dx n ∂x R Z j−1 ∂f 1 n = (−1) j dx ··· dx n ∂x R Z Z ∞ ! j−1 ∂f j 1 j n = (−1) j dx dx ··· dxd ··· dx n−1 ∂x R −∞ = 0, by the fundamental theorem of calculus, and since f has compact support in Rn. Now, by proposition 2.12 dx1 ∧ · · · ∧ dxdj ∧ · · · ∧ dxn is a basis of the space of alternating (n − 1)-tensors, n−1 n so all ω ∈ Ωc (R ) can be expressed as an R-linear combination of such ωj, so Z dω = 0. n R

Note that since ∂Rn is empty, Lemma 5.27 is equivalent to the statement Z Z dω = ω, n n R ∂R and we turn our attention to the half-space. n Lemma 5.28. For any smooth (n − 1)-form ω with compact support in Rx1≤0, Z Z dω = ω. n ∂ n Rx1≤0 Rx1≤0

47 1 j Proof. Similar to the proof in Lemma 5.27 we consider the (n − 1)-forms ωj = fdx ∧ · · · ∧ dxd ∧ n n n n n−1 · · ·∧dx , where f has compact support in Rx1≤0. First note that ∂Rx1≤0 = Rx1=0 = {0}×R . Now if j 6= 1, then Z Z 1 j n dωj = d fdx ∧ · · · ∧ dxd ∧ · · · ∧ dx n n Rx1≤0 Rx1≤0 Z Z ∞ ! j−1 ∂f j 1 j n = (−1) j dx dx ··· dxd ··· dx n−1 −∞ ∂u Rx1≤0 = 0, similar to as what we saw in Proposition 5.27. Additionally Z Z Z 0 ! 1 2 j n ωj = fdx dx ··· dxd ··· dx ∂ n n−1 0 Rx1≤0 R = 0. However, if j = 1, then Z Z Z 0 ! 1−1 ∂f 1 2 n dω1 = (−1) 1 dx dx ··· dx n n−1 −∞ ∂x Rx1≤0 R Z = f(0, x2, . . . , xn)dx2 ··· dxn n−1 ZR = fdx2 ∧ · · · ∧ dxn n Rx1=0 Z = ω1, ∂ n Rx1≤0

2 n n where the last equality follows because dx ∧ · · · ∧ dx is positive on ∂Rx1≤0 with the induced orientation. Thus in both cases, Z Z dωj = ωj. n ∂ n Rx1≤0 Rx1≤0 n Note that if the support of f does not meet ∂Rx1≤0 then obviously both sides of the equation equal 2 n n−1 n zero (since then f(0, x , . . . , x ) = 0). As in Proposition 5.27, all (n−1)-forms ω ∈ Ωc (Rx1≤0) can be expressed as an R-linear combination of such ωj, so Z Z dω = ω. n ∂ n Rx1≤0 Rx1≤0

Corollary 5.29 shows that TheoremA holds locally, and follows directly from Lemmas 5.27 and 5.28. Corollary 5.29. Let M be a smooth n-dimensional manifold with boundary, and let (U, φ) be some chart. Then for any smooth (n − 1)-form ω with compact support in U, Z Z dω = dω, U ∂U

48 where ∂U is given the induced orientation. Proof. By the deﬁnition of the integral of a diﬀerential form Z Z ∗ dω = φ−1 dω U φ(U) Z ∗ = d φ−1 ω φ(U) Z ∗ = φ−1 ω ∂φ(U) Z = ω, ∂U where the crucial equality follows from Lemmas 5.27 and 5.28. Armed with corollary 5.29, the proof of TheoremA becomes compact and straightforward. Theorem A (Stokes’ theorem on smooth manifolds). For any smooth (n − 1)-form ω with compact support on the oriented n-dimensional smooth manifold M with boundary ∂M, Z Z dω = ω, M ∂M where ∂M is given the induced orientation.

Proof. Let { (Uα, φα) }α∈A be an atlas of M. Then, by Corollary 5.29, the theorem holds for forms with compact support in each Uα. We use a partition of unity to show that the local property extends globally. In particular, ∞ let {ρα} be a C partition of unity subordinate to {Uα}. Then ραω has compact support in Uα. We also note that (∂M) ∩ Uα = ∂Uα. Therefore Z Z X ω = ραω ∂M ∂M α X Z = ραω (ﬁnite sum) α ∂M X Z = ραω (supp (ραω) ⊂ Uα) α ∂Uα X Z = d(ραω) (by Corollary 5.29) α Uα X Z = d(ραω) (supp d(ραω) ⊂ Uα) α M ! Z X = d ραω (ﬁnite sum) M α Z = dω. M

49 6 Conservation of Energy-Momentum

In the theory of general relativity spacetime is described as an orientable 4-dimensional Lorentz manifold, i.e. a smooth manifold endowed with a non-degenerate metric tensor field (a section of one of the many tensor bundles) of signature (1, −1, −1, −1) or (−1, 1, 1, 1) dependent on the convention. The former is often called the timelike convention while the latter is then referred to as the spacelike convention. Moving on we shall assume the timelike convention. The intrinsic curvature is what gives rise to gravitation, because in an intrinsically curved manifold the geodesics (generalizations of straight lines) do not appear straight. An image popularized by [8] is that of ants on an apple: watching the ants crawl across the surface, the observer marvels at how the ants seem to always find the shortest path, and further observes that as they approach the stem of the apple, the ants seem to be attracted to it, as if it somehow exerted a force on the ants. The observer, of course, understands that there is no such force, but rather it is an effect of the intrinsic curvature of the surface of the apple. The physics of gravitation is described by Einstein’s field equations (multiple because the tensor equation gives rise to ten scalar equations, one from each independent component): 1 R − gR = 8πT. 2 Here we have adopted units that make the speed of light and the gravitational constant both equal to one. R is the Ricci tensor (field), g is the metric tensor, R is the curvature scalar (field), and T is the stress-energy tensor. As physicists frequently do, we dropped the “field” term, and referred to tensor fields simply as tensors. This equation tells us that it is energy (and thus mass) that locally gives rise to the curvature. Note that the left hand side is entirely determined by the metric. Energy-momentum is given by a (tangent) vector P , and among the most fundamental laws of physics are those that state that each (cartesian) component of P is individually conserved in any physical process. We refer to this as conservation of energy-momentum. This has been a long standing experimental fact, and in 1918 Emmy Noether showed that every invariance of a physical action under some smooth transformation group corresponds to a conserved quantity[9], via a vanishing divergence. In particular conservation of energy-momentum corresponds to invariance under Cartesian translation (in each dimension). Of course, in general relativistic formulation, such translational invariance may not always hold, because the metric may depend on any or all coordinate functions in any given frame. However, in a frame where this is not the case, such as a coordinate frame where metric takes on the familiar Minkowski shape (corresponding to a Cartesian behaviour), we would expect the same result. Such frames are referred to as inertial frames (an observer in an inertial frames experiences no acceleration), and similar results are expressed in the equivalence principle, which is the foundation upon which the theory of general relativity is built: “In any and every inertial frame, anywhere and anytime in the universe, all the (non- gravitational) laws of physics must take on their familiar special-relativistic forms” – the equivalence principle.[8] In this section we shall show that invariance of the metric along a vector ξ gives rise to conservation of the ξ component of energy-momentum, in an explicitly general relativistic formulation. For this some additional mathematical concepts need to be introduced. In an effort to reduce unnecessary complexity we shall be brief, and take shortcuts whenever appropriate. See e.g. [7] for additional material concerning Subsections 6.1–6.6, and e.g. [8] for more information regarding the physics.

50 6.1 Regular Submanifolds There are several types of submanifolds, but we will concern ourselves only with the relatively restricted regular submanifolds, or embedded submanifolds. These submanifolds are character- ized by the fact that they respect the smooth structure of the main manifold. This property means that they appear naturally when we consider suﬃciently nice bounded (compact) regions of spacetime. Deﬁnition 6.1. A subset M of a smooth n-manifold M is called a regular submanifold of dimension k if every point p ∈ M is in the domain of some chart (U, φ) such that

k φ(U ∩ M) = φ(U) ∩ (R × {0}), for 0 ∈ Rn−k. Clearly a regular submanifold of dimension k is itself a smooth manifold of dimension k. Because for every chart (U, φ) on M we may form a chart (U ∩ M, pr1 ◦ φ) of M. Such charts obviously cover M, and are compatible because for two intersecting charts φ and ψ we have

−1 −1 −1 (pr1 ◦ φ) ◦ (pr1 ◦ ψ) = pr1 ◦ φ ◦ ψ ◦ pr1 ,

−1 k n where pr1 should be taken to be the inclusion map R ,→ R . As an example, the boundary of any smooth manifold with boundary is by deﬁnition a regular submanifold. Consider a bounded region, M, of spacetime. If no coordinate function is constant in M, then there is a chart around any point p ∈ M such that it may be restricted to a chart entirely contained in M, but still containing p, because spacetime is locally euclidean. If there is a collection of charts covering M such that all charts give rise to (n − k) coordinate functions that are constant on M, then we may similarly shrink any chart containing a point to get a compatible chart that fulﬁlls the condition. Of course, there are bounded regions where the above reasoning does not hold, but these rarely appear in physical situations, and some of them can be treated similarly with additional mathematics (see e.g. [7] for the case of singular k-chains). The generalization to smooth manifolds with boundary is obvious.

6.2 The Tensor Bundles As has already been alluded to general relativity deals frequently with tensor fields, which are sections of tensor bundles. Tensor bundles are vector bundles that can be defined analogously to Proposition 5.1, but we will instead define them via tensor products of the tangent bundle and 1 cotangent bundle (Lalt(TM)), just as the alternating bundles can be defined via exterior products of the cotangent bundle (as in the Grassman algebra). The main reason for the different approach in this section is that we need the extended tensor product in Subsection 6.5. Furthermore, it is particularly straightforward, and immediately gives the local frame fields of any tensor bundle. We begin with defining the tensor product between two vector spaces. Definition 6.2. Let V and W be two vector spaces. Then the tensor product V ⊗ W is defined as the set of all linear combinations of symbols of the form v ⊗ w, where v ∈ V and w ∈ W , subject to the relations

(v1 + v2) ⊗ w = v1 ⊗ w + v2 ⊗ w

v ⊗ (w1 + w2) = v ⊗ w1 + v ⊗ w2 r(v ⊗ w) = rv ⊗ w = v ⊗ rw, for r ∈ R. We can then obviously deﬁne ⊗ : V × W → V ⊗ W by (v, w) 7→ v ⊗ w.

51 It is thus clear that we can informally view the tensor product between two vector spaces as a vector space spanned by the direct product. From Definition 6.2 we can define a tensor product of linear maps between vector spaces (Definition 6.3), which operates between tensor products of vector spaces: Definition 6.3. Let S : V → X and T : W → Y be two linear maps between vector spaces. Then we define the tensor product S ⊗ T : V ⊗ W → X ⊗ Y by

(S ⊗ T )(v ⊗ w) := S(v) ⊗ T (w).

Definitions 6.2 and 6.3 allows us to work on the fibers of a vector bundle and the principle parts of the vector bundle charts. By combining these concepts over all fibers we can define a tensor product between vector bundles:

Lemma 6.4. Given two vector bundles ξ1 = (E1, M, π1,V1) and ξ2 = (E2, M, π2,V2) we let [ E1 ⊗ E2 := E1p ⊗ E2p, p∈M and let π : E1 ⊗ E2 → M be defined as the map that maps any element in a fiber E1p ⊗ E2p to the point p. Then ξ = (E1 ⊗ E2, M, π, V1 ⊗ V2) is a vector bundle known as a tensor product bundle of ξ1 and ξ2. Proof. Let (Uα, φα) α∈A and (Uα, ψα) α∈A be the vector bundle atlases for E1 and E2 respectively. Define φ ⊗ ψ :(E ⊗ E ) | → V ⊗ V by (φ ⊗ ψ ) | := φ | ⊗ ψ | , bα bα 1 2 Uα 1 2 bα bα E1p⊗E2p bα E1p bα E2p for p ∈ Uα, where as usual φbα denotes the principal part of φα. Then we let

φα ⊗ ψα :(E1 ⊗ E2) → Uα × (V1 ⊗ V2) Uα be deﬁned by φα ⊗ ψα := (π, φbα ⊗ ψbα). Now, the transitions maps are given by −1 −1 (φ ⊗ ψ ) ◦ (φ ⊗ ψ ) = φ ⊗ ψ ◦ φ ⊗ ψ |−1 bα bα bβ bβ bα bα bβ bβ E2p E1p⊗E2p E1p⊗E2p E1p E2p E1p −1 −1 = (φbα ◦ φbβ ) ⊗ (ψbα ◦ ψbβ ) E1p E1p E2p E2p

Thus (Uα, φα ⊗ ψα) α∈A is a vector bundle atlas on ξ precisely because (Uα, φα) α∈A and (Uα, ψα) α∈A are vector bundle atlases on ξ1 and ξ2 respectively. The bundle structures on ξ1 and ξ2 gives a bundle structure on ξ thus. To define general tensor bundles on a smooth manifold, we need the associativity of the tensor product between tensor bundles, but this follows trivially from Definition 6.2. Definition 6.5. Let TM be the tangent bundle on a smooth manifold M. Then we may define the r r s ∗ tensor product bundle on M by Ts (TM) := (⊗ TM) ⊗ (⊗ TM ), and by defining π analogously ∗ 1 to the procedure in Lemma 6.4. Here TM is to be taken to mean Lalt(TM). Finally, tensor fields are defined as sections of tensor bundles on a smooth manifold. Definition 6.6. If we let TM denote the tangent bundle of a smooth manifold M, then the space r r of sections Γ Ts (TM) is denoted by Ts (M), and its elements are referred to as (r, s)-tensor 0 fields. We may let any zero fall away, thus denoting e.g. Ts (M) by simply Ts(M).

52 Although differential forms could be given a concrete pictorial representation (see page 36), tensor fields in general are perhaps best viewed as an extension of vector fields, allowing for more complex geometric properties assigned to each point. We do note, however, that differential forms are a particular class of tensor fields. Take for example a metric tensor field, which is in its basic form covariant of dimension two, that is a (symmetric) tensor product between two 1-forms. We could consider each point to be assigned a family of ellipsoids, such that each vector is measured by the number of ellipsoids it pierces, and the two vectors together sweep out a solid angle, which is then in turn measured according to some machination. Notation is simplified by introducing the Einstein summation convention, which dictates that all indices that appear once as subscript and once as superscript should be summed over. The limits of summation is to be understood implicitly. In tensor calculations it is often the dimensions of the underlying vector field. For example, we may write the local expression of a 1-form as

n i X i aiω := aiω , i=1

i where ω is a local frame ﬁeld, and ai are to be taken as smooth functions on M. To further facilitate notation and calculations in concrete examples it is common in physics to identify a tensor (ﬁeld) with its components. For an (r, s)-tensor τ we may thus write locally

τ = τ i1...ir e ⊗ · · · ⊗ e ⊗ ωj1 ⊗ · · · ⊗ ωjs j1...js i1 ir = τ i1...ir , j1...js

i ∗ k where ei is a local frame ﬁeld of TM and ω is the dual frame ﬁeld of TM = Lalt(TM). For a k-form η we may write locally

i1 ik η = η|i1,...,ik|e ∧ · · · ∧ e

= η|i1,...,ik|

= ηI~ where the absolute signs indicate that it is a form and that summation is to be performed only over indices such that i1 < i2 < ··· < ik, and correspondingly the arrow over the multi-index k I indicates that summation should occur over In. The choice of frame field is taken to be implicitly understood unless otherwise stated. For clarity we shall use greek indices µ, ν, . . . to denote coordinate indices, and latin indices i, j, . . . to denote frame field indices. Additionally we will use bold symbols to signal if we are not using index notation, thus writing e.g. v = vi. In practice, matters are made even simpler by the fact that general relativity often considers only local effects, i.e. effects such that they are contained in some chart, unless e.g. singularities are studied.

6.3 Lie Derivative µ Consider a map that carries a point p, with coordinates xp to a nearby point q, with coordinates µ µ xq , along a vector v . We describe it by

µ µ µ xq = xp + εvp ,

53 µ where v = v ∂µ is a vector representing a line element, and ε is taken to be some very small num- ν ber; in the inﬁnitesimal case we let ε → 0. The Jacobian, J µ, of this coordinate transformation µ and its inverse, K ν , are given by

ν ν ν µ µ µ 2 J µ = δ µ + εv ,µ,K ν = δ ν − εv ,ν + O(ε ),

µ µ where we used comma in index notation to denote partial diﬀerentiation, so v ,ν ≡ ∂ν v . Since we are interested ultimately in the inﬁnitesimal case we take the terms of order ε2 or higher to be identically zero. Now, tensor components are transformed by the Jacobian and its inverse:

µ µ µ σ τ µ T ν |q = Te ν |p + εv ,σTe ν |p − εv ,ν Te τ |p , where Te is to be taken to be T before the map. If we specialize this transformation to a vector u the last term vanishes and we ﬁnd

µ µ µ σ u − u = u | ν ν − u | ν − εv ,σu , e xp +εvp xp p so that

u − ue µ ν µ ν £vu := lim = u ,ν v − v ,ν u ε→0 ε = v ◦ u − u ◦ v.

We call it the Lie derivative, and it effectively allows us to describe how vectors change infinites- imally. However, the commutator expression that we end up with can also readily be verified to be a Lie bracket on the tangent bundle. Indeed, the tangent bundle is a Lie algebra with the bracket thus defined if we take scalars to be smooth functions on M, and we may alternatively i denote £vu by [v, u]. Similarly, we can of course take, for T = T j

i k k i i k £vT := T j,kv − T jv ,k + T kv ,j, but note that there is no natural way to define a bracket in this case. If £vT = 0 the change in T can be seen as a pure coordinate transformation. The Lie derivative of a function is defined as the ordinary vector derivative introduced earlier, and in this manner straightforward calculation verifies that for a tensor T and vectors or covectors u, and v, we have £ξ T (u, v) = £ξT (u, v) + T £ξu, v + T u, £ξv , i.e. the Lie derivative is natural with respect to contraction. This can be viewed as the Lie derivative obeying the Leibniz rule if we view contraction as a multiplication map.

6.4 Lorentz Manifolds We have previously mentioned that metric tensors are symmetric, and for the purposes of this section we will make the deﬁnition precise:

Deﬁnition 6.7. A covariant tensor t ∈ Ts(V ) is symmetric if

t(v1, . . . , vs) = t(vσ(1), . . . , vσ(s)) for all v1, . . . , vs ∈ V and all permutations σ of (1, . . . , s).

54 Since metric tensors are covariant tensors of dimension 2, this reduces to g(u, v) = g(v, u) for any metric tensor g ∈ T2(V ). For semi-Riemannian manifolds in general the metric must also be nondegenerate:

Deﬁnition 6.8. A symmetric tensor g ∈ T2(V ) is nondegenerate if and only if g(u, v) = 0 for all v ∈ V implies that u = 0.

If g is a nondegenerate metric on V , then upon choosing an orthonormal basis. e1, . . . , en, of V we must have either g(ei, ei) = 1 or g(ei, ei) = −1, which gives us a list of +1’s and −1’s. Permuting the basis so that all +1’s (or −1’s, as depending on the convention) come ﬁrst, we get a list (1,..., 1, −1,..., −1) that we refer to as the signature of the metric space (V, g). Lorentz manifolds are semi-Riemannian manifolds of a speciﬁc signature:

Deﬁnition 6.9. Let M be a smooth manifold and let g ∈ T2(M). If g is symmetric, nondegenerate, and of constant signature on M, we call g a semi-Riemannian metric on M, and we say that (M, g) is a semi-Riemannian manifold. If (the dimension of M is ≥ 2 and) the signature of the metric is (1, −1, −1, −1,...) we also say that (M, g) is a Lorentz manifold.

∗ Returning to vector spaces in general, exhibiting a linear isomometry g[ between V and V ] is easy: first define g[(u)(v) = g(u, v) for all u, v ∈ V . If we denote the inverse by g then we can ∗ ] ] ∗ force an isometry by defining g ∈ T2(V ) by g(α, β) = g g (α), g (β) . For v ∈ V and α ∈ V ] ] we will temporarily use the notation v[ ≡ g[(v) and α ≡ g (α). These isometries (called musical isomorphisms) must be represented by some matrices, say Aij for g[. Then for some basis e1, . . . , en we have

k gij = g(ei, ej) = ei[(ej) = Aike (ej) k = Aikδj = Aij.

k i i i j Thus it is clear that ei[ = gike , and if v = v ei then v[ = v ei[ = v gije , i.e. the components i j of v[ = v[ are vi = gijv . Note the convention of simply lowering the index, whence applying g[ i] ij ij is often referred to as index lowering. Similarly we can write e = g ej for some matrix g , ] −1 −1 ij and since g = (g[) we must have gij = g . It is easy to see that the components of ] ] i ij ] α = αi are α = g αj, and we may refer to applying g as index raising. We also say that v[ corresponds to v and that α] corresponds to α. Finally we note that

ij i j i] j] ik j` g = g(e , e ) = g(e , e ) = g(g ek, g e`) ik j` ik j ij = g g gk` = g δk = g .

ij When using index notation, it is common to denote g and g by gij and g respectively. For a r 0 r+1 tensor A ∈ Ts (V ) we can deﬁne a new tensor A ∈ Ts−1 (V ) by taking

0 1 r 1 r A (α , . . . , α , vk[, v1,..., vbk, . . . , vs) = A(α , . . . , α , v1, . . . , vs), for some k ∈ {1, . . . , s}. This is made clearer if we retain the position of indices (i.e. do not consolidate them), in which case it is easy to see that index notation gives

i1,...,ir 0 i1,...,ir jk A j1,...,js 7→ (A ) j1,...,jk−1 jk+1,...,js

jk` i1,...,ir = g A j1,...,jk−1,`,jk+1,...,js . It is standard to denote both A and A0 by the kernel letter A, and we view them as diﬀerent manifestations of the same tensor. The above process is then referred to as type changing, and

55 it becomes an isometry if we define a metric h·, ·i on two tensors, χ and τ, of the same type, say 1 i j T1 (V ) as hχ, τi = χ jτi (summing as required by the Einstein summation convention). An obvious globalization of index lowering and index raising, and the extension of type changing is frequently used in the context of general relativity. Before moving on to the connection, note that we may refer to a frame on TM as rigid if the components of the metric are constants. Such frames are in general non-holonomic, because if there is a rigid coordinate frame then there is a coordinate frame in which the metric is orthonormal. This corresponds to flat space. A rigid frame may be classified as e.g. Lorentz if 1 0 0 0  0 −1 0 0  gij =   . 0 0 −1 0  0 0 0 −1

6.5 Connection The connection on a smooth manifold is a mathematically rich topic, but general relativity limits us to a uniquely defined connection, called the Levi-Civita connection. We shall therefore limit ourselves to covering the absolute basics and the application of the covariant derivative, which is equivalent to the connection. We shall also describe how this gives a description of the curvature of a manifold, and briefly mention the conditions behind the Levi-Civita connection. To begin with a connection is something that exists on a vector bundle (mentioned earlier), and provides a structure to lift curves from the base manifold onto the bundle. This view is not important for us, although it is to some degree in general relativity, being the principle behind parallel transport. Rather, we consider specifically the connection on the tangent bundle TM, but note that it gives rise to a connection on the co-tangent bundle T ∗M by demanding that it acts naturally with respect to contraction. In this manner, the connection considered can be extended to tensor bundles in general. Secondly, we are interested in the connection primarily because it is equivalent to the covariant derivative (indeed, these terms are often used interchangeably) which allows us to measure infinitesimal changes in direction, as well as magnitude. Definition 6.10. The covariant derivative (on the tangent bundle) is a map ∇ : X(M)×X(M) → X(M), or more generally ∇ : X(M) × T (M) → T (M), where ∇(v, T ) is written ∇vT or in i c component form T j;cv , such that

(i) ∇fv+guT = f∇vT + g∇uT ;

(ii) ∇v (T + Q) = ∇vT + ∇vQ;

(iii) ∇v (fT ) = (vf) T + f∇vT . ∞ r r For scalar fields f, g ∈ C (M). Clearly this also defines a map ∇ : Ts (M) → Ts+1(M), which we may denote by ∇T (without a subscript for the vector argument). See further below. To differ between the two maps one often refers to the former one as the covariant derivative along a vector v, and the latter as alternatively simply the covariant derivative or the gradient. One important property that is obfuscated by definition 6.10 is the fact that the covariant derivative of a tensor along a vector, v, at a point, p, depends on the value of v at p only. See e.g. [7] for a complete proof. Because of the linearity in the first argument, we can define the so called connection forms. So suppose as usual that e1,..., en is a frame field, and let v be an arbitrary vector field. Then

i ∇vej = γ j(v)ei,

56 i for some 1-form γ j. This matrix of 1-forms are referred to as connection forms and the compo- i nents γ jk are referred to as Ricci rotation coefficents in rigid frames, because they in fact detail the rotation of any vector field as you move through the manifold. Here rotation is to be taken to be relative to parallel transported vectors (a topic that we will not delve into, for brevity). In i a coordinate frame the components γ jk are the Christoffel symbols. To see how the covariant derivative of vector fields can be extended to 1-forms by demanding that it acts naturally with respect to contraction, recall that it should act on functions as vector derivation and consider

i c ∇u α (v) = αiv u |c i c = αiv u ;c i c i c = αi;cv u + αiv ;cu i c i c = αi|cv u + αiv |cu , where we used the | in index notation to denote ordinary vector differentiation of the component functions, and the final expression is simply the first line expanded by the product rule. Imposing this rule as a demand, it is clear that we can define the covariant derivative of a 1-form using only vector differentiation and the covariant derivative of vector fields. We are now ready to write down the covariant derivative of vector fields and 1-forms, and thus tensors in general:

i c i c i j c v ;cu = v |cu + γ jcv u , c c j c αi;cu = αi|cu − γ icαju , where the ﬁrst expression follows from the deﬁnitions above, and the second from the contraction demand. To clarify how this applies to tensors in general we provide one example:

i c i c i k c k i c T j;cu = T j|cu + γ kcT ju − γ jcT ku .

From here one can go on to define the curvature of the manifold, giving rise to among others the Riemann and Ricci curvature tensors, the latter of which forms one half of the Einstein field equations. We shall however not dwell on this subject. The Levi-Civita connection, which is always used in general relativity is uniquely defined by requiring

(i) gij;c ≡ 0;

i c i c (ii) v ;cu − u ;cv = [u, v]. The proof is actually straight-forward, but we will omit it for brevity, and focus instead on the i i i i implications. The ﬁrst demand can easily be seen to imply that u vi |c ≡ u vi ;c = u ;cvi+u vi;c and is intuitively understood as requiring that the covariant derivative treat the metric as a product, thus obeying the Leibniz rule of standard derivatives. The second demand may be a bit more tricky, but [8] has gone so far as to say that any geometric theory of gravity obeying the equivalence principle must have a connection obeying this demand. It is referred to as having zero torsion, and can be seen to equate the Lie bracket with commuting covariant derivatives of vectors. This means that the Lie derivative can be taken with respect to the connection, for

i i i £u v αi = £uv αi + v (£uαi)

57 i c i c i = v ;cu αi − u ;cv αi + v (£uαi) i = ∇u v αi i c i c = v ;cu αi + v αi;cu .

Combining the two expressions we ﬁnd

j j £uαi = αi;ju + αjv ;i, which is exactly the formula for the Lie derivative, but with the covariant derivative taking the place of vector differentiation of the components. Since the result holds for vectors and 1-forms it holds for tensors in general. Qualitatively this can be understood to mean that the connection does not “twist” around paths in the manifold. First observe that the first demand gives gij|k = 2γ(ij)k and second demand may be equiva- i i i lently stated as 2γ [jk] = C kj, where C kj are defined by

k ei, ej = C ijek, for a given frame. Here parenthesis denote symmetrization and brackets denote anti-symmetrization. This is in fact a Lie algebra when we consider smooth functions to be scalars. We will use capital letter C to denote what are formally the structure constants of this specific algebra, but here take the shape of smooth functions, and refer to them instead as structure coefficients. Now note that the two demands together allow us to write down the rotation coefficents as 1 γ = g + g − g + C + C − C , ijk 2 ij|k ik|j jk|i kij jik ijk where we let the metric lower the indices on both structure coefficients and rotation coefficients. In particular, in a rigid frame 1 γ = C + C − C , ijk 2 kij jik ijk in which case we observe γijk = γ[ij]k. Now let us extend the exterior derivative to tensor-valued forms. A tensor-valued form is a differential form that produces a tensor field rather than a real number. Thus a tensor-valued form can be seen as an element of T (M) ⊗ Ω(M), and in particular all tensor fields are at least tensor valued 0-forms. For a tensor valued form we extend the exterior derivative with the help of the covariant derivative. For η = Q ⊗ ω we take

dη = ∇Q ∧ ω + Q ⊗ dω, where the gradient of the tensor Q is taken to be a tensor-valued 1-form, and

(Q ⊗ ω) ∧ α ≡ Q ⊗ (ω ∧ α) .

Thus we demand that the exterior derivative follows the same rules as the covariant derivative, but lose the property d2 ≡ 0 unless applied to a ordinary differential form (or flat space). Note also that if we choose to consider e.g. a 1-form to be a co-vector-valued 0-form we will get a different result than if we consider it a 1-form. To be rigorous it may therefore be important to specify to what bundle the exterior derivative is extended. Alternatively, one can demand that any such bundle is completely contravariant for clarity, thus demanding T ∈ T r(M) in the above example. We shall take this approach.

58 i i Now consider eiω ≡ n, by the deﬁnition of dual frames. Thus d eiω = 0, but expanding the product we ﬁnd

i i 0 = ∇ei ∧ ω + eidω i j i = ei γ j ∧ ω + dω , by relabeling of indices. Thus we can conclude

i j i dω = ω ∧ γ j.

i i j k This is the ﬁrst Cartan equation, and may alternatively be written dω = γ jkω ∧ ω , or in a i 1 i j k rigid frame dω = − 2 C jkω ∧ ω .

6.6 Hodge Duality The Hodge star operator, or Hodge dual operator, is usually defined on an oriented scalar product k n−k k space, as the unique map ∗ : Lalt(V ) → Lalt (V ) such that for α, β ∈ Lalt(V ) we have α ∧ ∗β = hα|βi vol, where vol is the appropriate volume element, and a scalar product has been defined k on Lalt(V ) such that n o ωI k I∈In i is orthogonal whenever e1, . . . , en is (here ω as usual is the dual base of ei). One can then show that Definition 6.11 defines the same operator on a scalar product space, and the definition may be extended to pseudo-Riemmanian manifolds since the metric tensor induces an inner product on each tangent space. We shall instead start with Definition 6.11 for a semi-Riemannian manifold. In this we will use the notation = 1,...,n . j1,...,jn j1,...,jn Definition 6.11. Given an oriented semi-Riemannian n-manifold (M, [$]), with the metric tensor g, let η ∈ Ωk(M). We then define the Hodge star operator ∗ :Ωk(M) → Ωn−k(M) locally by r |j1...jk| jk+1 jn ∗η = det gij η ω ∧ · · · ∧ ω j1...jk|jk+1...jn| r = det g ηJ~ ij J~I~

J~ ≡ η εJ~I~, 1 n for some local frame ﬁelds e1, . . . , en of TM such that ω ∧ · · · ∧ ω ∈ [$]. We say that ∗η is the Hodge dual of η. Note that for a local Lorentz frame r det gij = 1, simplifying the formula. As demonstrated above we will let εI~ denote the Levi-Civita tensor deﬁned by r ε := det g . I~ ij I~

59 In the case of a 1-form the Hodge dual is essentially a (n − 1)-form completely ortogonal to the original 1-form. For let αi be the 1-form, then contraction on some index with ∗α yields

i j α α εik1...kr−1jkr+1...kn = 0.

It therefore should come as no surprise that the Hodge star operator can be used when considering flows across a surface. To make matters more precise consider a hypersurface, Σ, defined by the ηi (note that an orientation of Σ can easily be seen to be equivalent to an orientation of the ambient manifold, given that ηi is defined), which we take at first to be non- null and normalized. Then the surface element of Σ is given by ∗η, and by abuse of notation we may designate the directed surface element by dΣi := ηi∗η. However, we can also write

J~ dΣi = εiJ~ω |Σ, where |Σ denotes projection onto the surface Σ. To see this note that contraction with ηi produces the same result for both expressions of dΣi, as does contraction with any 1-form or vector orthogonal to ηi. The latter expression is however also well deﬁned for null hypersurfaces, and by continuity also gives the directed surface element in these cases.

6.7 Conservation of Energy-Momentum and Killing Vectors The stress-energy tensor takes two arguments and is completely symmetric (this can be shown by a completely physical argument, see e.g. [8]), which in the context of type changing means that when T ∈ T2(M) it is symmetric (Definition 6.7). The stress-energy tensor is defined by j i requiring that Ti v describes the flux of the v component of energy-momentum. Now consider a vector, ξ, called a Killing vector, such that the metric is invariant along ξ. Thus £ξg = 0, but then

c c c £ξg = gij;cξ + gcjξ ;i + gicξ ;j

= 2ξ(i;j) = 0.

We call this the Killing equation, and it is useful for finding and testing isometries. Additionally, i i j if we define J := Tj ξ we find

i ij ji ji J ;j = T ξi;j + ξjT ;i = ξjT ;i, by the Killing equation and the symmetry of the stress-energy tensor. In particular, since the ji i Einstein ﬁeld equations guarantee that T ;i ≡ 0 we must also have J ;i ≡ 0. We shall not show this, but instead note that this result must hold by the equivalence principle, since it is well known that the stress-energy tensor must be divergence-free in ﬂat spacetime.

Theorem B. Let S be a Lorentz manifold with the Levi-Civita connection, with dim(S) = n, which we will refer to as spacetime, endowed with a stress-energy tensor, T . If ξ is a Killing i i j vector then J := Tj ξ is a conserved ﬂux, i.e. I i J dΣi ≡ 0, Σ for any suﬃciently nice closed hypersurface Σ.

60 Proof. Consider a sufficiently nice bounded region of S, such that it is a compact regular orientable n-dimensional submanifold with the boundary given by Σ. Denote it by M, whence Σ = ∂M, and select an appropriate orientation. By TheoremA we have I Z Z i i i J dΣi = d J dΣi = J ;idV = 0, Σ M M where dV is the volume element of M. We note that crucially T has compact support because M is compact. The second equality can readily be verified by straight-forward calculation in a rigid frame from the first Cartan equation, and since it is a covariant result it holds in general.

Note that this specializes to the conservation of energy, momentum, and angular momentum under the correct Killing vectors. These all appear in ﬂat spacetime and is one traditional method ij of showing that we must have T ;j ≡ 0 in ﬂat spacetime.

61 References

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