Master’s thesis

Integration on Manifolds, Volume of Geodesic Balls and Tubes, and Invariant Measure of Sl(n, ℤ)\Sl(n, ℝ)

Einar Gudfinnsson

Spring 2001

Abstract

In this Master’s thesis I will generalize integration from ℝ^{n} to smooth finite-dimensional manifolds. This generalization will be used to calculate the volume of some classical manifolds and give a proof of Weyl’s tube formula [We].

In chapter 1 a classical example for the volume of the n-dimensional Euclidean sphere and ball in ℝ^{n} is given. A few concepts from differential geometry is mentioned without definitions.

Chapter 2 is a theoretical chapter which summarizes the basics in the theory of differential and Riemannian geometry. Many theorems are stated without proofs, but detailed discussion is made when integration is extended from ℝ^{n} to manifolds. The proof of theorem 2.82 is my contribution, although a better proof can without a doubt be found.

Chapter 3 goes through the proof of Weyl’s Tube formula, which was first published in 1939 [We]. For a q-dimensional submanifold P of ℝ^{n}, the volume of a tube around P with small radius is deduced. The result is independent of the embedding of P. Moreover, the volume of geodesic balls is calculated for some special kinds of manifolds. The main source for this part of my thesis are chapters 2–4 and a part of chapter 6 in a book called Tubes by Alfred Gray. Many theorems will be stated without proofs, either because they are well known or I do not find the proofs instructive. In many cases proofs are written in much more detail than Gray does. The last section deals with Kähler manifolds.

In chapter 4 both number theory and differential geometry are used to calculate an invariant meassure of the quotient space Sl(n, ℤ)\Sl(n, ℝ) with respect to an invariant density on Sl(n, ℝ). My main source for this part are a couple of pages taken from [Vos]. The source [Vos] is extremely compact and many statements are stated without proof and without citation. My search for better proofs were unsuccessful in most cases so the majority of the proofs in this part of this thesis are my personal deductions. At the end of the chapter I will discuss some speculations which materialized during my writing and could easily be sufficient material for a small project or another Master’s thesis.

4

Acknowledgments

I am very grateful to my supervisor, Associate Professor Jesper Michael Møller, who generously devoted much of his time and effort to clarify numerous things for me. His patience, optimism and support has been very important to me.

I also want to give special thanks to Sigmundur Gudmundsson at the Centre for Mathematical Science at Lund University for allowing me to visit him in February. His good help and great hospitality will always be remembered.

I would like to give many thanks to the staff at the Icelandic University, especially Reynir Axelsson for introducing me to differential and Riemannian geometry.

I also want to thank Associate Professor Ian Kiming at the Copenhagen University for some nice suggestions and discussion on number theory and Associate Professor Henrik Schlichtkrull for discussion on measure theory.

I am extremely grateful to my father, Gudfinnur Palmar Sigurfinnsson, and my mother, Gudridur Einarsdottir, for all their love and support. My mother passed away last April after a long battle with cancer. I miss her very much.

Finally I want to thank my friends in Iceland and Denmark who have been very supportive and understanding.

Einar Gudfinnsson 6

Contents

1 Introduction 9 1.1 The meaning of area and volume in relations to manifolds . . . . . . 9 1.2 Formulas for μ(S_{p}^{n}(r)) and μ(B_{p}^{n+1}(R)) . . . . . . . . . . . . . . . . . 10

2 Differential and Riemannian geometry 13

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 The Permutation group . . . . . . . . . . . . . . . . . . . . . 13 2.2 Manifolds and tangent spaces . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Tensors, vector bundles and differential forms . . . . . . . . . . . . . 19 2.4 Integration of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Lie groups and related topics . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.1 Topological groups . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.2 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . 28 2.5.3 Some classical Lie-groups and their Lie algebras . . . . . . . . 29

2.6 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6.1 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6.2 Geodesics and the exponential map . . . . . . . . . . . . . . . 32 2.6.3 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 The Riemannian volume element . . . . . . . . . . . . . . . . . . . . 37

3 Tubes 41 3.1 Fermi coordinates and fields . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Definition of a general tube . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 The second fundamental form of submanifolds . . . . . . . . . 48 3.3 The infinitesimal change of volume . . . . . . . . . . . . . . . . . . . 51 3.4 The Bishop-Günther inequalities . . . . . . . . . . . . . . . . . . . . . 54 3.5 Weyl’s tube formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.1 Double forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5.3 Some definitions and lemmas . . . . . . . . . . . . . . . . . . 60 3.5.4 Calculating the volume of B_{p}^{n}(r) . . . . . . . . . . . . . . . . . 66

3.6 Kähler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6.1 A vital Lemma without proof . . . . . . . . . . . . . . . . . . 71 3.6.2 Volume of a geodesic ball in 𝕂_{hol}^{n}(λ) . . . . . . . . . . . . . . . 72 3.6.3 Complex projective space ℂP^{n}(λ) . . . . . . . . . . . . . . . . 73 4 Invariant measure on Sl(n, ℤ)\Sl(n, ℝ) 75

4.1 Topics from linear algebra . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Topics from number theory . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1 Dirichlet theorem . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2.2 Calculating ω_{∞}(Sl(n, ℤ)\Sl(n, ℝ)) . . . . . . . . . . . . . . . . 82

4.3 Volume of SO(n, ℝ) and further study . . . . . . . . . . . . . . . . . 87

Chapter 1

Introduction

1.1 The meaning of area and volume in relations to manifolds

A real n-dimensional manifold is a topological space which locally is “similar” to an open subset in ℝ^{n}. An (n − 1)-dimensional embedded submanifold in another n-dimensional manifold is called a hypersurface. The precise definitions of the these concepts will be given in chapter 2.

Curves and surfaces are examples of a 1-dimensional and 2-dimensional manifolds respectively. ℝ^{n} is an example of an n-dimensional manifold. A curve in ℝ^{2} and a surface in ℝ^{3} are examples of hypersurfaces. In the following definition an example of a manifold and a hypersurface is given. We will then deduce formulas for the volume of the manifold and for the area of the hypersurface.

1.1 Definition (1) Let R > 0. The set

B_{p}^{n}(R) := {x ∈ ℝ^{n} : |x − p| < R}

is called the n-dimensional open ball with center p and radius R. If R = 1 and p = 0 we write B^{n} instead of B_{p}^{n}(R). B^{n} is called the n-dimensional open unit ball. The n-dimensional open ball is an example of an n-dimensional manifold.

(2) Let r > 0. The set

S_{p}^{n}(r) := {x ∈ ℝ^{n+1} : |x − p| = r}

is called the n-dimensional sphere with center p and radius r. If r = 1 we write S^{n} instead of S_{p}^{n}(r). S^{n} is called the n-dimensional unit sphere. The n-dimensional sphere is an example of an n-dimensional manifold. Furthermore S_{p}^{n}(r) is a hypersurface in B_{p}^{n+1}(R) when 0 < r < R. S_{p}^{n}(r) is also a hypersurface in ℝ^{n+1}.

{1.2 Example \atop (1)} S_{p}^{1}(r) is a closed curve in ℝ^{2} and its length is 2πr.

(2) S_{p}^{2}(r) is a closed surface in ℝ^{3} and its area is 4πr^{2}.

(3) B_{p}^{1}(R) is an open interval and it’s length is 2R.

(4) B_{p}^{2}(R) is an open surface and it’s area is πR^{2}.

(5) B_{p}^{3}(R) is an open ball and it’s volume is 4πR^{3}/3.

Instead of the common words length, area and volume in the above definition we could have used Lebesgue measure using concepts from measure theory. In this thesis however the word volume will in general be used independent of n. Thus the first line in the above example would change to

(1) S_{p}^{1}(r) is a closed curve in ℝ^{2} and its volume is 2πr.

It’s convenient to use the word area instead of volume when we are working with manifolds as hypersurfaces in another manifold. This convention will be used in chapter 3. So if we for example are working with S_{p}^{1}(r) as a hypersurface in B_{p}^{2}(R) for 0 < r < R we wold say that the area of S_{p}^{1}(r) as a hypersurface in B_{p}^{2}(R) is 2πr. On the other hand the volume of B_{p}^{2}(R) is πR^{2}.

We denote by μ(S_{p}^{n}(r)) and μ(B_{p}^{n}(R)) the Lebesgue measure of S_{p}^{n}(r) and B_{p}^{n}(R) respectively. So here we’d call μ(S_{p}^{n}(r)) the n-dimensional area of S_{p}^{n}(r) (as a hypersurface in B_{p}^{n}(R)) and μ(B_{p}^{n+1}(R)) the (n+1)-dimensional volume of B_{p}^{n+1}(R) independent of n. In the following section we’ll deduce formulas for μ(S_{p}^{n}(r)) and μ(B_{p}^{n+1}(R)) [Cox].

1.2 Formulas for μ(S_{p}^{n}(r)) and μ(B_{p}^{n+1}(R))

We first observe that μ(S_{p}^{n}(r)) and μ(B_{p}^{n+1}(R)) are independent of p. We therefore can assume that p = 0. We have that μ(S_{p}^{n}(r)) = μ(S^{n})r^{n} and the relation

μ(B_{p}^{n+1}(R)) = ∫_{0}^{R} μ(S^{n})r^{n} dr = \frac{μ(S^{n})}{n + 1}R^{n+1}.  (1.1)

If f : ℝ^{n+1} → ℝ is a function of r = (∑_{i=1}^{n+1} x_{i}^{2})^{1/2}, we have

∫_{0}^{∞} f(r)μ(S^{n})r^{n} dr = ∫_{−∞}^{∞} ∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞} f(x_{1}, …, x_{n+1}) dx_{1} ⋯ dx_{n+1}.

This is a well known fact from functional analysis. We apply this formula to the function f(r) = e^{−r^{2}}. Then

\begin{aligned}∫_{0}^{∞} e^{−r^{2}}μ(S^{n})r^{n} dr & = ∫_{−∞}^{∞} ∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞} e^{−x_{1}^{2}− ··· −x_{n+1}^{2}} dx_{1} ⋯ dx_{n+1} \\ & = (∫_{−∞}^{∞} e^{−x^{2}} dx)^{n+1} = π^{(n+1)/2}. \end{aligned}

1.2. FORMULAS FOR μ(S_{P}^{N}(R)) AND μ(B_{P}^{N+1}(R)) 11

The substitution t = r^{2} yields

∫_{0}^{∞} e^{−r^{2}}r^{n} dr = \frac{1}{2} ∫_{0}^{∞} e^{−t}t^{(n+1)/2−1} dt = \frac{1}{2}Γ (\frac{n + 1}{2}) .

By combining these last two equations we obtain the area of S_{p}^{n}(r), namely

μ(S_{p}^{n}(r)) = μ(S^{n})r^{n} = \frac{2π^{(n+1)/2}r^{n}}{Γ (\frac{n+1}{2})} .  (1.2)

From this and (1.1), we deduce that

μ(B_{p}^{n+1}(R)) = \frac{2π^{(n+1)/2}R^{n+1}}{(n + 1)Γ (\frac{n+1}{2})} = \frac{π^{(n+1)/2}R^{n+1}}{Γ (\frac{n+3}{2})} .  (1.3)

Naturally we get the values in example 1.2 when we put a suitable value of n in the above formulas.

1.3 Remark We have used many undefined concepts in the last couple of pages. In the following chapters precise definitions of the terms mentioned are given and I’ll explain explicitly what is meant by integration on manifolds.

12 CHAPTER 1. INTRODUCTION

Differential and Riemannian geometry

In this chapter many basic definitions and theorems from differential and Riemannian geometry are stated. The material presented is often taught in an one-year graduate course. Most of the theorems will be stated without proofs. For more details and proofs see for example [Ax, Bre, Lee, War] or other books about differential geometry.

2.1 Notation

2.1 Definition (1) Let {A_{i}|i ∈ I} be a collection of sets. Then

∐_{i∈I} A_{i}  (2.1)

denotes the disjoint union of the sets A_{i} for i ∈ I.

(2) For i, j ∈ ℤ we define

δ_{i,j} := \begin{cases}1 & i = j; \\ 0 & i ≠ j. \end{cases}  (2.2)

δ_{i,j} is called the Kronecker symbol.

2.1.1 The Permutation group

2.2 Definition (1) Let i, j ∈ ℕ where i ⩽ j. Then we let

[[i, j]] := ℕ ∩ [i, j].

(2) Let s ∈ ℕ. A bijective map σ : [[1, s]] → [[1, s]] is called a permutation of s elements.

(3) The set of all permutations of s elements form a group. It is non-commutative for s > 2. It is called the permutation group of s elements and is denoted by

𝔖_{s}.

(4) Let i, j ∈ [[1, s]]. We denote by (i, j) ∈ 𝔖_{s} the element which satisfies the following conditions:

(a) (i, j)(i) = j and (i, j)(j) = i;

(b) (i, j)(k) = k if k ≠ i, j.

(i, j) is called a 2-cycle.

(5) For σ_{1}, …, σ_{r} ∈ 𝔖_{s} the element σ_{1} ⋯ σ_{r} := σ_{1} ∘ ⋯ ∘ σ_{r} ∈ 𝔖_{s} is called the product of σ_{1}, …, σ_{r}.

(6) Note that (i, j)^{1} = (i, j) and (i, j)^{0} = id_{𝔖_{s}}.

2.3 Remark It is well known that every permutation can be written as a product of 2-cycles, though of course not in a unique way. However, the numbers of two-cycles which product is the same as a given element σ ∈ 𝔖_{s} is either alway even or always odd. Furthermore it’s straightforward to show that card(𝔖_{s}) = s!

2.4 Definition Let σ ∈ 𝔖_{s}. Then we define

ε_{σ} := \begin{cases}1 & if σ is a product of even numbers of 2-cycles; \\ −1 & if σ is a product of odd numbers of 2-cycles. \end{cases}

We call ε_{σ} the sign of σ.

2.2 Manifolds and tangent spaces

2.5 Definition Let X be a topological space.

(1) Let U be a collection of subsets of X. If, for each x ∈ X there exists an open neighborhood of x which meets, non-trivially, only finitely many elements of U then U is said to be locally finite.

(2) Let U and V be open coverings of X. If

A ∈ U ⇒ ∃ B ∈ V : A ⊂ B

holds for every A ∈ U, then we say that U is a refinement of V.

(3) If every open covering of X has an open locally finite refinement then X is said to be paracompact.

2.6 Definition (1) Let 𝔽 stand for either ℝ or ℂ and let M be a Hausdorff space which has a countable basis. M, together with a collection of maps, called charts, is said to be an n-dimensional smooth manifold over 𝔽 if the following conditions holds:

(a) A chart is a homeomorphism ϕ : U → U′ ⊂ 𝔽^{n} where U is open in M and U′ is open in 𝔽^{n};

(b) Each point x ∈ M is in the domain of some chart;

(c) For charts ϕ : U → U′ ⊂ 𝔽^{n} and ψ : V → V′ ⊂ 𝔽^{n} the function ϕψ^{−1} : ψ(U ∩ V) → ϕ(U ∩ V) is infinitely often 𝔽-differentiable;

(d) The collection of charts is maximal with properties (a), (b) and (c).

A set of charts which satisfies the first three conditions above is called an atlas. If 𝔽 = ℝ we say that M is a real manifold. If 𝔽 = ℂ we say that M is a complex manifold.

(2) Let M and N be smooth manifolds over 𝔽. Then a map f : M → N is said to be smooth if, for all charts ϕ on M and ψ on N, the function ψ ∘ f ∘ ϕ^{−1} is infinitely often 𝔽-differentiable where it is defined. If N = ℝ and f : M → N is a smooth map then f is said to be a smooth function. We denote by C^{∞}(M) the algebra of all smooth functions on M.

2.7 Definition If we let 𝔽 = ℝ in definition 2.6(1) and let M satisfy the given conditions where we have replaced 𝔽^{n} by {(x_{1}, …, x_{n}) ∈ ℝ^{n}| x_{1} ⩽ 0} where it occurs, we say that M along with its collection of charts is a manifold with boundary. The (n−1)-dimensional manifold which consist of all points in M which get mapped to {(x_{1}, …, x_{n}) ∈ ℝ^{n}| x_{1} = 0} by a chart, is called the boundary of M.

General agreement

We’ll always let the letter M stand for a real smooth manifold of dimension n unless otherwise stated.

{2.8 Remark \atop (1)} There are other definitions of manifolds than given in definition 2.6 with weaker conditions. The countable basis axiom in the definition of a manifold guarantees that M is paracompact. Theorem 2.10 below shows how this can be useful.

(2) It can be shown that an atlas on a manifold has a unique maximal extension. This maximal extension is called a maximal atlas. If A is a maximal atlas for M, an atlas on M which has A as it’s maximal extension is called a subatlas of A on M.

2.9 Definition Let {U_{α}| α ∈ A} be an open covering of a space X. A collection of continuous maps {f_{β} : X → [0, 1]| β ∈ B} which satisfies:

(1) There exists a locally finite open refinement {V_{β}| β ∈ B} of {U_{α}| α ∈ A} such that supp(f_{β}) ⊂ V_{β} for all β ∈ B;

(2) ∑_{β} f_{β}(x) = 1 for all x ∈ X;

is called a partition of unity subordinate to {U_{α}| α ∈ A}.

2.10 Theorem Let {U_{α}| α ∈ A} be an open covering of a paracompact space X. Then there exists a partition of unity subordinate to {U_{α}| α ∈ A}. Since M is paracompact then for any open covering of M there exist a partition of unity subordinate to that covering. It can actually be shown that we can find a partition of unity for M such that the functions f_{β} in definition 2.9 are smooth. For proof see [Bre]

□

2.11 Definition Let M have a maximal atlas A. If there exist a subatlas B of A on M such that

det (\frac{∂ϕ_{i}}{∂ψ_{j}} ) > 0 on V ∩ U

for all charts ϕ : U → U′ ⊂ ℝ^{n} and ψ : V → V′ ⊂ ℝ^{n} in B then M is said to be orientable and M along with B is said to be an oriented manifold.

2.12 Remark Note that an oriented manifold is not a manifold in general since we don’t assume that it’s atlas is maximal. There are more equivalent definitions of an oriented manifold. See for example [War].

2.13 Definition (1) We denote by E(M, p) the set of all smooth functions which are defined in a neighborhood of p ∈ M. If f, g ∈ E(M, p) then we regard f = g if f agrees with g on some neighborhood of p.

(2) A map δ : E(M, p) → ℝ is called a differential operator if it is ℝ-linear and satisfies

δ(fg) = f(p)δ(g) + g(p)δ(f)  (2.3)

for all f, g ∈ E(M, p).

(3) Let γ be a smooth curve on M with γ(0) = p. A map D_{γ} : E(M, p) → ℝ defined by

D_{γ}(f) = \frac{d}{dt}f(γ(t))|_{t=0}  (2.4)

is called the directional derivative of f along γ at p and D_{γ} is called the tangent vector to γ at p. We regard D_{γ} = D_{γ′} if they have the same value at p for every f.

2.14 Definition Let p ∈ M.

(1) The set of all differential operators δ : E(M, p) → ℝ is called the tangent space of M at p.

(2) The set of all tangent vectors D_{γ} : E(M, p) → ℝ at p is called the tangent space of M at p.

2.15 Remark The two definitions of the tangent space are isomorphic, and we will not make a distinction between them. The tangent space at p ∈ M is a vector space over ℝ of same dimension as M.

□

2.16 Definition We will denote the tangent space of M at p by

T_{p}M.

2.17 Example Let V be an n-dimensional manifold which also is a vector space over ℝ. Then for a vector v ∈ V we let γ_{v} : I → V be a curve such that

γ_{v}(0) = p and \frac{d}{dt}|_{t=0}γ_{v}(t) = v.

An obvious choice for such curve is γ_{v}(t) = p + tv. Then the map ϕ : V → T_{p}V defined by

ϕ(v) = D_{γ_{v}}

is a isomorphism, and as before the precise choice of γ_{v} is not important as long as it satisfies the required conditions. γ_{v} is called a representation curve for the tangent vector v. We call this the natural isomorphism of T_{p}V with V. Note that if U ⊂ V is open in V, then T_{p}U = T_{p}V for all p ∈ U. The most classical examples are V = ℝ^{n} and V = T_{p}M.

2.18 Definition Let ϕ : M → N be a smooth map between manifolds. Then the function ϕ_{∗} : T_{p}M → T_{ϕ(p)}N defined by

ϕ_{∗}(D_{γ}) = D_{ϕ∘γ}  (2.5)

is called the differential of ϕ at p ∈ M.

2.19 Theorem ϕ_{∗} defined above is well defined and linear and satisfies

ϕ_{∗}(D_{γ})(g) = D_{γ}(g ∘ ϕ).  (2.6)

for all g ∈ E(N, ϕ(p)). Furthermore if ψ : N → P is also a smooth map between manifolds we have

ψ_{∗}ϕ_{∗} = (ψ ∘ ϕ)_{∗}.  (2.7)

Equation (2.7) is known as the chain rule.

□

2.20 Remark Let V and W be n-dimensional manifolds, which are also vector spaces, and ϕ : V → W be a smooth map. Then by the natural identifications in example 2.17 we have

ϕ_{∗}(v) = \frac{d}{dt}|_{t=0}ϕ ∘ γ_{v}(t).

2.21 Definition Let ϕ : M → N be a smooth map between manifolds.

(1) We say that ϕ is:

(a) an immersion if ϕ_{∗} is a monomorphism at all p ∈ M.

(b) a submersion if ϕ_{∗} is onto at all p ∈ M.

(2) (M, ϕ) is said to be a submanifold if ϕ is an injective immersion.

(3) We say that ϕ is an embedding if (M, ϕ) is a submanifold and ϕ is homeomorphic to it’s range for the relative topology on ϕ(M). If ϕ is an embedding, then we say that ϕ(M) is an embedded submanifold of N.

(4) If ϕ(M) is an embedded submanifold of N and dim(M) = dim(N) − 1 then we say that ϕ(M) is a hypersurface of N.

2.3 Tensors, vector bundles and differential forms

2.22 Definition Let V be an n-dimensional vector space over ℝ, and let V^{∗} denote the dual space of V.

(1) A (k, l)-tensor over V is an ℝ-multilinear mapping

F : \underbrace{V^{∗} × ⋯ × V^{∗}}_{l times} × \underbrace{V × ⋯ × V}_{k times} → ℝ.

(2) A (k, l)-tensor is said to be contravariant if k = 0, and covariant if l = 0.

(3) The vector space of all (k, l)-tensors over V is denoted by

T^{(k,l)}(V).

(4) A covariant (k, 0)-tensor F which satisfies

F (v_{σ(1)}, …, v_{σ(k)}) = ε_{σ}F (v_{1}, …, v_{k})

for all v_{1}, …, v_{k} ∈ V and every σ ∈ 𝔖_{k} is called an alternating covariant tensor.

(5) We denote by

Λ_{k}(V)

the vector space of all alternating covariant (k, 0)-tensors on V. An element in Λ_{k}(V) is called a k-form.

(6) We define

Λ(V) := ⨁_{k=0}^{∞} Λ_{k}(V)

We regard Λ_{k}(V) as a subspace of Λ(V) by identify it with the subspace

⨁_{l=0}^{k−1}{0} ⊕ Λ_{k}(V) ⊕ ⨁_{l=k+1}^{∞}{0}.

When we take an element η ∈ Λ(V) we generally assume that there exist a non-negative integer k such that η ∈ Λ_{k}(V).

2.23 Remark It can be shown that M is orientable if and only if there exist a non-vanishing n-form on M. (See for example [War]).

2.24 Definition Let E be smooth manifold, and π : E → M be a surjective map. The triple (E, M, π) is said to be a k-dimensional smooth vector bundle if the following conditions are satisfied:

(1) E_{p} := π^{−1}(p) is endowed with the structure of an k-dimensional vector space, for every p ∈ M. E_{p} is called the fiber of E over p.

(2) For every p ∈ M there exists a neighborhood U ⊂ M of p and a diffeomorphism ϕ : π^{−1}(U) → U × ℝ^{k} such that the following diagram commutes:

[figure: π^{−1}(U); ϕ; π; U × ℝ^{k}; π_{1}; U; U]

Here π_{1} is the projection onto the first factor. Moreover ϕ|_{E_{q}} : E_{q} → {q} × ℝ^{k} is a linear isomorphism for all q ∈ U. ϕ is called a local trivilization of E.

We say that E is the total space, M is the base and π is the projection.

2.25 Definition (1) T M := ∐_{p∈M} T_{p}M is called the tangent bundle of M.

(2) T^{∗}M := ∐_{p∈M}(T_{p}M)^{∗} is called the cotangent bundle of M.

(3) T^{(k,l)}M := ∐_{p∈M} T^{(k,l)}(T_{p}M) is called the bundle of (k, l)-tensor of M.

(4) Λ^{k}M := ∐_{p∈M} Λ_{k}(T_{p}M) is called the bundle of k-forms of M.

2.26 Example We define mappings

(1) π_{1} : T M → M such that (T M)_{p} = π_{1}^{−1}(p) = T_{p}M;

(2) π_{2} : T^{∗}M → M such that (T^{∗}M)_{p} = π_{2}^{−1}(p) = (T_{p}M)^{∗};

(3) π_{3} : T^{(k,l)}M → M such that (T^{(k,l)}M)_{p} = π_{3}^{−1}(p) = T^{(k,l)}(T_{p}M);

(4) π_{4} : Λ^{k}M → M such that (Λ^{k}M)_{p} = π_{4}^{−1}(p) = Λ_{k}(T_{p}M).

Then (T M, M, π_{1}), (T^{∗}M, M, π_{2}), (T^{(k,l)}M, M, π_{3}) and (Λ^{k}M, M, π_{4}) are all examples of vector bundles. The first two are n-dimensional, and the other ones are of dimension \binom{n}{k}.

2.27 Remark The only non-trivial thing we have to do to show that the triples in the above example are indeed vector bundles, is to show that the sets we propose are the total spaces are indeed manifolds which satisfies the desired conditions. The following lemma, proved for example in [Lee], gives a very helpful criterium which often makes this easy. It can for example be used in the above example.

2.28 Lemma Let E be a set and π : E → M be a surjective map. Let’s assume that we have a collection 𝔄 of bijective maps ϕ_{α} : π^{−1}(U_{α}) → U_{α} × ℝ^{k} where {U_{α}} is an open covering on M which satisfies the following conditions:

(1) π_{1} ∘ ϕ_{α} = π;

(2) If U_{α} ∩ U_{β} ≠ 0, where α, β ∈ 𝔄, then the composite map ϕ_{α} ∘ ϕ_{β}^{−1} : (U_{α} ∩ U_{β}) × ℝ^{k} → (U_{α} ∩ U_{β}) × ℝ^{k} is of the form

ϕ_{α} ∘ ϕ_{β}^{−1}(p, V) = (p, τ(p)V)

where τ : U_{α} ∩ U_{β} → Gl(k, ℝ) is a smooth map where Gl(k, ℝ) denotes the group of invertible (k × k)-matrices (see definition 2.56).

Then E can be made into a smooth k-dimensional manifold in a unique way such that (E, M, π) is a smooth k-dimensional vector bundle and ϕ_{α} are local trivializations for all α ∈ 𝔄.

□

2.29 Definition Let (E, M, π) be a vector bundle. A map F : M → E such that F (p) ∈ E_{p} for all p, is said to be a section of E. If F (p) = 0 ∈ E_{p} for all p ∈ M then F is called the zero section of E. We say that a section F is smooth if F is a smooth map between the manifolds M and E. We denote the set of all smooth sections on (E, M, π) by

E(M).

2.30 Definition (1) A smooth section of T M is called a smooth vector field on M. The set of all smooth vector fields on M is denoted by

X (M)

(2) A smooth section of T^{(k,l)}M is called a smooth (k, l)-tensor field on M. The set of all smooth (k, l)-tensor fields on M is denoted by

T^{(k,l)}(M)

(3) A smooth section of Λ^{k}M is called a differential k-form on M. The vector space of all differential k-form on M is denoted by

Ω^{k}(M).

{2.31 Remark \atop (1)} For X ∈ X (M) and f : M → ℝ we define X(f) : M → ℝ by

X(f)(p) := X(p)(f).

(2) By definition we have that Ω^{k}(M) ⊂ Λ^{k}M.

2.32 Definition We define

Ω(M) := ⨁_{k=0}^{∞} Ω^{k}(M).

We make the same assumptions for Ω(M) as was done for Λ(V) in definition 2.22(6).

A well known lemma called the Tensor Characterization Lemma states the following:

{2.33 Lemma \atop (1)} A map

τ : \underbrace{Ω^{1}(M) × ⋯ × Ω^{1}(M)}_{l times} × \underbrace{X (M) × ⋯ × X (M)}_{k times} → C^{∞}(M)

is induced by a smooth (k, l)-tensor field in a natural way if and only if it is multilinear over C^{∞}(M). By abuse of language we say that the map τ is also a smooth (k, l)-tensor field if it is multilinear over C^{∞}(M).

(2) A map

τ′ : \underbrace{Ω^{1}(M) × ⋯ × Ω^{1}(M)}_{l times} × \underbrace{X (M) × ⋯ × X (M)}_{k times} → X (M)

is induced by a smooth (k, l + 1)-tensor field in a natural way if and only if it is multilinear over C^{∞}(M). We also say that the map τ′ is a smooth (k, l + 1)-tensor field if it is multilinear over C^{∞}(M).

For more details see for example [Lee].

□

2.34 Definition

The map ∧ : Λ(V) × Λ(V) → Λ(V), (ω, η) ↦ ω ∧ η defined by

ω ∧ η(X_{1}, …, X_{k+k′}) = \frac{1}{k!k′!} ∑_{σ∈𝔖_{k+k′}} ε_{σ} ω(X_{σ(1)}, …, X_{σ(k)})η(X_{σ(k+1)}, …, X_{σ_{(}k+k′)}), (2.8)

is called the exterior (or wedge) product. Here we have assumed that ω ∈ Λ_{k}(V) and η ∈ Λ_{k′}(V).

2.35 Remark We can define a wedge product ∧ : Ω(M) × Ω(M) → Ω(M) by (2.8) where X_{i} are now smooth vector fields, instead of vectors. We have that ∧(Λ_{k}(V) × Λ_{k′}(V)) ⊂ Λ_{k+k′}(V) and ∧(Ω^{k}(M) × Ω^{k′}(M)) ⊂ Ω^{k+k′}(M). If f ∈ Ω^{0}(M) = C^{∞}(M) and ω ∈ Ω^{k}(M) we write fω instead of f ∧ ω.

{2.36 Theorem \atop (1)} The wedge product is bilinear and associative.

(2) If ω_{1}, …, ω_{k} are 1-forms then

(ω_{1} ∧ ⋯ ∧ ω_{p})(X_{1}, …, X_{k}) = det (ω_{i}(X_{j})).  (2.9)

(3) If ω_{1}, …, ω_{n} is a basis for V^{∗} then {ω_{i_{1}} ∧ ⋯ ∧ ω_{i_{k}}| i_{1} < ⋯ < i_{k}} is a basis of Λ_{k}(V) over ℝ.

□

2.37 Corollary Let V be an n-dimensional vector space. Then

dim Λ_{k}(V) = \binom{n}{k};  (2.10)
ω ∈ Λ_{k}(V), η ∈ Λ_{k′}(V) ⇒ ω ∧ η = (−1)^{kk′}η ∧ ω;  (2.11)
ω ∈ Λ_{k}(V), k odd ⇒ ω ∧ ω = 0.  (2.12)
□

2.38 Remark ω is a differential k-form if and only if for every local coordinates x = (x_{1}, …, x_{n}), ω can be written as

ω = ∑_{i_{1},...,i_{k}=1}^{n} f_{i_{1},...,i_{k}}dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{k}}  (2.13)

where all the functions f_{i_{1},...,i_{k}} are smooth.

2.39 Definition The operator d : Ω^{k}(M) → Ω^{k+1}(M), defined by

(1) If k = 0, and X ∈ X (M) then df(X) := X(f);

(2) dω = d(fdx_{i_{1}} ∧⋯∧dx_{i_{k}}) := df ∧dx_{i_{1}} ∧⋯∧dx_{i_{k}} for k > 0 and then we extend by linearity to all forms;

is called the exterior derivative.

The following properties for the exterior derivative are easy to prove:

{2.40 Theorem \atop (1)} d is independent of the choice of coordinates, so d is well defined.

(2) df = ∑_{i} \frac{∂f}{∂x_{i}} dx_{i}.

(3) d^{2} = 0.

(4) For ω ∈ Ω(M), η ∈ Ω^{k}(M) we have d(ω ∧ η) = dω ∧ η + (−1)^{k}ω ∧ dη.

2.41 Definition Let ϕ : M → N be a smooth map between manifolds. We define an operator ϕ^{∗} : Ω(N) → Ω(M) by letting

ϕ^{∗}(ω)(X_{1}, …, X_{k}) = ω(ϕ_{∗}(X_{1}), …, ϕ_{∗}(X_{k})),  (2.14)

where X_{1}, …, X_{k} ∈ X (M) and then extend by linearity. Here we have assumed that ω ∈ Ω^{k}(N). Note that ϕ^{∗}(Ω^{k}(N)) ⊂ Ω^{k}(M) for all k. If k = 0 we set

ϕ^{∗}(f) = f ∘ ϕ.  (2.15)

In the following theorem many basic features of ϕ^{∗} are stated.

2.42 Theorem Let θ and ϕ be smooth mapping between manifolds. Then

(1) (θ ∘ ϕ)^{∗} = ϕ^{∗} ∘ θ^{∗};

(2) θ^{∗}(ω ∧ η) = θ^{∗}(ω) ∧ θ^{∗}(η);

(3) θ^{∗}(f dy_{1} ∧ ⋯ ∧ dy_{k}) = (f ∘ θ) d(y_{1} ∘ θ) ∧ ⋯ ∧ d(y_{k} ∘ θ);

(4) θ^{∗}(dω) = d(θ^{∗}(ω)).

2.43 Definition Let V be an n-dimensional vector space. By (2.10) we have that Λ_{n}(V) is 1dimensional and thus Λ_{n}(V)\{0} has two components.

(1) An orientation of V is a choice of a component of Λ_{n}(V)\{0}.

(2) V along with an orientation of V is said to be an oriented vector space.

(3) Let V be an oriented vector space. An element which is in the orientation of V is said to lie in the orientation.

2.44 Remark There are close relation between orientable manifolds and oriented vector spaces. For details see for example [War] The following lemma is obvious.

2.45 Lemma Let V be an oriented vector space, and let λ, μ lie in the orientation of V. Then there exist c > 0 such that λ = cμ.

□

2.4 Integration of forms

2.46 Definition Let X be a topological space, and f : X → ℝ be a function. Then we define the support of f to be

supp(f) = \overline{f^{−1}(ℝ\\{0\})}.

Let x_{1}, …, x_{n} be the coordinates in ℝ^{n} and ω be an n form on ℝ^{n} whose support is compact an lies in in some open set U ⊂ ℝ^{n}. We can thus write

ω = f(x_{1}, …, x_{n}) dx_{1} ∧ ⋯ ∧ dx_{n},

where supp(f) ⊂ U is compact. We define the integral of ω over U to be

∫_{U} ω := ∫∫ ⋯ ∫_{ℝ^{n}} f(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n}

where the right hand side is the ordinary Riemannian (or Lebesgue) integral. By this definition we see that

∫_{ℝ^{n}} ω = ∫_{U} ω.

If W ⊂ ℝ^{n} is another open set and θ : W → U is a diffeomorphism, we have

θ^{∗}(ω) = (f ∘ θ)d(x_{1} ∘ θ) ∧ ⋯ ∧ d(x_{n} ∘ θ)

by theorem 2.42(3). Furthermore by theorem 2.40(2), we have

d(x_{i} ∘ θ) = ∑_{j=1}^{n} \frac{∂(x_{i} ∘ θ)}{∂x_{j}} dx_{j} = ∑_{j=1}^{n} J_{i,j}(θ) dx_{j},

where J_{i,j}(θ) is the (i, j)-th entry of the Jacobian matrix of θ. By simple computation we have (see for example [Bre, Ru] for details):

d(x_{1} ∘ θ) ∧ ⋯ ∧ d(x_{n} ∘ θ) = det(J_{i,j}(θ)) dx_{1} ∧ ⋯ ∧ dx_{n}.  (2.16)

Hence we deduce that

θ^{∗}(ω) = (f ∘ θ) det(J_{i,j}(θ)) dx_{1} ∧ ⋯ ∧ dx_{n}  (2.17)

and thus by the standard Riemann change of variables we have (see for example [Ru])

\begin{aligned}∫_{W} θ^{∗}(ω) & = ∫_{ℝ^{n}} θ^{∗}(ω) = ∫∫ ⋯ ∫_{ℝ^{n}} f(θ(x_{1}, …, x_{n})) det(J_{i,j}(θ)) dx_{1} ⋯ dx_{n} \\ & = ± ∫∫ ⋯ ∫_{ℝ^{n}} f(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n} = ± ∫_{U} ω. \end{aligned} (2.18)

The sign is determined by the sign of det(J_{i,j}(θ)) where we assume the same sign on all components of U if U is not connected. We say that θ is orient preserving if det(J_{i,j}(θ)) > 0.

Let’s assume that M along with an atlas A is an oriented manifold. We let ϕ : U → U′ ⊂ ℝ^{n} be a chart in A and ω be an n-form on M such that supp(ω) is compact and contained in U. Then (ϕ^{−1})^{∗}ω is an n-form on U′ which we extend by 0 to the whole ℝ^{n}. Thus we can look at (ϕ^{−1})^{∗}ω as a n-form in ℝ^{n} which has compact support in U′. We define the integral of ω over U to be

∫_{U} ω := ∫_{U′}(ϕ^{−1})^{∗}ω.  (2.19)

This is clearly an extension of integrals of forms defined on U ⊂ ℝ^{n}. We’ll have to show though that our definition is independent of which chart in A we choose. Let ψ : V → V′ ⊂ ℝ^{n} be another chart in A such that supp(ω) ⊂ V, and let θ = ψ ∘ϕ^{−1}. Since ψ^{−1} ∘ θ = ϕ^{−1} we have (ϕ^{−1})^{∗} = θ^{∗} ∘ (ψ^{−1})^{∗} by theorem 2.42(1) and therefore

∫_{U′}(ϕ^{−1})^{∗}ω = ∫_{ℝ^{n}}(ϕ^{−1})^{∗}ω = ∫_{ℝ^{n}} θ^{∗}(ψ^{−1})^{∗}ω = ∫_{ℝ^{n}}(ψ^{−1})^{∗}ω = ∫_{V′}(ψ^{−1})^{∗}ω

by (2.18) so (2.19) is well defined. Note that we have used that (M, A) is an oriented manifold.

Now let ω be an arbitrary n-form on M with compact support K. Let {f_{i}} be a smooth partition of unity subordinate to a locally finite covering {U_{i}} on M such that U_{i} is in the domain of some chart on (M, A) for all i ∈ I. Let I′ = {i ∈ I| U_{i} intersects K}. Since K is compact, K intersects only finitely many U_{α} so I′ is a finite set. Now we have that supp(f_{i}ω) ⊂ U_{i} for all i ∈ I′ and we define

∫_{M} ω = ∑_{i∈I} ∫_{U_{i}} f_{i}ω.  (2.20)

This is also clearly an extension of integrals of forms defined on U ⊂ M. Let {g_{j}} be another smooth partition of unity subordinate to another locally finite covering {V_{j}} on M such that V_{j} is in the domain of some chart on (M, A) for all j ∈ J. Let J′ = {j ∈ J| V_{j} intersects K}. Since ∑_{i} ∑_{j} f_{i}g_{j} = 1 we have

∫_{U_{i}} f_{i}ω = ∫_{U_{i}} f_{i} ∑_{j∈J} g_{j}ω = ∫_{U_{i}} ∑_{j∈J′} f_{i}g_{j}ω = ∑_{j∈J′} ∫_{U_{i}} f_{i}g_{j}ω = ∑_{j∈J′} ∫_{V_{j}} g_{j}f_{i}ω (2.21)

where we have changed the integral and the finite sum. The last equality follows from (2.19). By combining (2.20) and (2.21) we get

∑_{i∈I} ∫_{U_{i}} f_{i}ω = ∑_{i∈I′} ∑_{j∈J′} ∫_{V_{j}} g_{j}f_{i}ω = ∑_{j∈J′} ∫_{V_{j}} g_{j} ∑_{i∈I} f_{i}ω = ∑_{j∈J} ∫_{V_{j}} g_{j}ω.

So we’ve shown that the definition (2.20) is independent of which smooth partition of unity we take.

Thus if θ : M → N is an orient preserving diffeomorphism between oriented ndimensional manifolds and ω is an n-form on N which has a compact support we have that

∫_{N} ω := ∫_{M} θ^{∗}ω,  (2.22)

by the above definitions and observations. The above definitions are very useful and can be used to give a rather simple proof of the general Stoke’s Theorem. The theorem states the following:

2.47 Theorem (Stoke’s Theorem) Let N be an n-dimensional oriented manifold with boundary ∂N and assume that ω ∈ Ω^{n−1}(N) has a compact support. Then

∫_{N} dω = ∫_{∂N} ω.  (2.23)

2.48 Remark Many classical theorems in 2- and 3- dimensional calculus are obtained as a corollary from the above theorem (see for example [Bre, Fl]).

2.5 Lie groups and related topics

2.5.1 Topological groups

2.49 Definition A topological group is a Hausdorff space G together with a group structure on G such that

(1) group multiplication m : G × G → G, (g, h) ↦ gh is continuous;

(2) group inversion inv : G → G, g ↦ g^{−1} is continuous.

2.50 Definition (1) Let G be a topological group, and X be a space. We denote the identity element of G by e. Then an action of G on X is a map A : G × X → X, (g, x) ↦ g(x), which satisfies

(a) gh(x) = g(h(x));

(b) e(x) = x;

for all g, h ∈ G, and all x ∈ X.

(2) Let x ∈ X. Then G(x) = {g(x) ∈ X| g ∈ G} is called the orbit of x. x, y ∈ X are in the same orbit with respect to G if and only if there exits an element g ∈ G such that x = gy. Since G is a group we see that orbits define an equivalence relation on X. Sometimes we write G · x instead of G(x).

2.5.2 Lie groups and Lie algebras

2.51 Definition (1) A topological group G which also has a manifold structure is called a Lie group.

(2) Let 𝔤 be a real vector space over ℝ and [·, ·] : 𝔤 × 𝔤 → 𝔤 be a bilinear form that satisfies

[x, y] = −[y, x]; (anti-commutativity)
[[x, y], z] + [[y, z], x] + [[z, x], y] = 0.  (Jacobi identity)

(2.24)

Then the pair (𝔤, [·, ·]) is said to be a Lie algebra.

(3) Let G be a Lie group and σ ∈ G. The function l_{σ} : G → G defined by

l_{σ}(τ) = στ  (2.25)

is called the left translation by σ.

(4) Let X be a vector field on G. If

(l_{σ})_{∗} ∘ X = X ∘ l_{σ}  (2.26)

for every σ ∈ G then X is said to be a left invariant vector field on G.

(5) Let G be a Lie group. We denote by the letter 𝔤 the vector space of all left invariant vector fields on G.

(6) Let G be a n-dimensional Lie group. A form ω ∈ Ω^{k}(G) such that

l_{g}^{∗}ω = ω.  (2.27)

for all g ∈ G is said to be a left invariant k-form on G.

In the following theorem we get one of the main property of 𝔤. I’ll go through the proof since I find it instructive.

2.52 Theorem 𝔤 is a real vector space and the map α : 𝔤 → T_{e}G defined by α(X) = X(e) is an isomorphism. Hence dim 𝔤 = dim T_{g}G = dim G.

Proof: It’s easy to show the first part and α is linear by definition of a vector field. Assume α(X) = α(Y). For σ ∈ G we have

X(σ) = X ∘ l_{σ}(e) = ((l_{σ})_{∗} ∘ X)(e)
= (l_{σ})_{∗}(X(e)) = (l_{σ})_{∗}(α(X)) = (l_{σ})_{∗}(α(Y))
= (l_{σ})_{∗}(Y (e)) = ((l_{σ})_{∗} ∘ Y)(e) = (Y ∘ l_{σ})(e) = Y (σ),

so α is injective. Let x ∈ T_{e}G. We want to find X ∈ 𝔤 such that α(X) = x. Define X on G by X(σ) = (l_{σ})_{∗}(x). Now

α(X) = X(e) = (l_{e})_{∗}(x) = x

and since by the chain rule (see (2.7)) we have

(X ∘ l_{τ})σ = X(τσ) = (l_{τσ})_{∗}(x) = (l_{τ}l_{σ})_{∗}(x) = (l_{τ})_{∗}(l_{σ})_{∗}(x) = (l_{τ})_{∗}(X(σ)),

for all σ ∈ G and thus X lies in 𝔤 and satisfies the required property. Hence α is surjective. The proof of the theorem is complete.

□

2.53 Remark It can be shown that 𝔤 forms a Lie algebra under the Lie bracket operation (see definition 2.63(3)) on vector fields [War]. This explains the common notation of 𝔤 in definition 2.51(2) and 2.51(5).

2.54 Definition The Lie algebra 𝔤 of left invariant vector fields on a Lie group G is said to be the Lie algebra of the Lie group G.

2.55 Remark By theorem 2.52, we can identify the Lie algebra of the Lie group G with T_{e}G. We will make use of this in chapter 4.

2.5.3 Some classical Lie-groups and their Lie algebras

In this section I will give some examples of Lie groups and their Lie algebras. These examples are well known and more details can be found in books about Lie groups and differential geometry.

2.56 Definition Let R̃ be a ring.

(1) We denote by M(n, R̃) the vector space of all (n×n)-matrices, with coefficients in R̃.

(2) We let Gl(n, R̃) = {A ∈ M(n, R̃)| det(A) ≠ 0}. Gl(n, R̃) is a Lie group.

(3) We let M_{m}(n, R̃) = {A ∈ M(n, R̃)| det(A) = m} where m ∈ R̃.

(4) We let O(n, R̃) = {A ∈ Gl(n, R̃)| A^{t} = A^{−1}}, where A^{t} denotes the transpose of A. O(n, R̃) is a Lie group and is called the orthogonal group.

(5) We let SO(n, R̃) = {A ∈ O(n, R̃)| det(A) = 1}. SO(n, R̃) is a Lie group and is called the special orthogonal group.

(6) We let Sl(n, R̃) = M_{1}(n, R̃). Sl(n, R̃) is a Lie group called the special linear group.

2.57 Theorem Let 𝔽 stand for either ℝ or ℂ.

(1) The Lie algebra of Gl(n, 𝔽) is

𝔤𝔩(n, 𝔽) := M(n, 𝔽)  (2.28)

(2) The Lie algebra of O(n, 𝔽) is

𝔬(n, 𝔽) := {A ∈ 𝔤𝔩(n, 𝔽)| A^{t} = −A}.  (2.29)

Hence dim_{𝔽} 𝔬(n, 𝔽) = n(n − 1)/2. 𝔬(n, 𝔽) is also the Lie algebra of SO(n, 𝔽).

(3) The Lie algebra of Sl(n, 𝔽) is

𝔰𝔩(n, 𝔽) := {A ∈ 𝔤𝔩(n, 𝔽)| traceA = 0}.  (2.30)

Hence dim_{𝔽} 𝔰𝔩(n, 𝔽) = n^{2} − 1.

□

2.6 Riemannian manifolds

2.58 Definition Let M be a smooth manifold and g : X (M) × X (M) → C^{∞}(M) be a smooth (2, 0)-tensor field.

(1) g is said to be symmetric if g(X, Y) = g(Y, X) for all X, Y ∈ X (M).

(2) g is said to be positive definite if g(X, X) > 0 for all X ∈ X (M), X ≠ 0.

(3) g is said to be a Riemannian metric if it is symmetric and positive definite, and then M is said to be a Riemannian manifold with respect to g.

2.59 Remark If g is a Riemannian metric on a manifold M then it induces an inner product ⟨·, ·⟩ on each tangent space T_{p}M, by setting ⟨X(p), Y (p)⟩ := g(X, Y)|_{p} for all X, Y ∈ X (M). By using partition of unity it can be shown that there exist a Riemannian metric on every real manifold. We often write ⟨X, Y ⟩ instead of g(X, Y) for X, Y ∈ X (M).

2.60 Definition Let p ∈ M and U ⊂ M be an open neighborhood of p and (E_{1}, …, E_{n}) be n smooth vector fields defined on U. If (E_{1}|_{q}, …, E_{n}|_{q}) is a basis for T_{q}M for every q ∈ U, then (E_{1}, …, E_{n}) is said to be a local frame for T M on U.

2.61 Remark We often say that a local frame for T M defined on U is a local frame on M.

2.62 Definition (1) Let (E_{1}, …, E_{n}) be a local frame for T M. If we construct 1-forms ϕ_{1}, …, ϕ_{n}, defined by letting ϕ_{i}(E_{j}) = δ_{i,j}, then (ϕ_{1}, …, ϕ_{n}) is a local frame for T M^{∗}, called the dual coframe.

(2) Let x : U → U′ ⊂ ℝ^{n} be a chart on M. If we write x = (x_{1}, …, x_{n}) then

(\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} )

is a local frame on U for T M called a coordinate frame. It’s dual coframe is denoted by

(dx_{1}, …, dx_{n}) .

When it’s clear which chart we’re working with we often write (∂_{i}) for the coordinate frame and (dx_{i}) for it’s dual coframe.

2.6.1 Connections

2.63 Definition (1) Let (E, M, π) be a vector bundle. A map ∇ : X (M) × E(M) → E(M), (X, Y) ↦ ∇_{X}Y such that for all X, X_{1}, X_{2}, Y, Y_{1}, Y_{2} ∈ X (M) we have

∇_{fX_{1}+gX_{2}}Y = f∇_{X_{1}}Y + g∇_{X_{2}}Y, for all f, g ∈ C^{∞}(M); (2.31)
∇_{X}(aY_{1} + bY_{2}) = a∇_{X}Y_{1} + b∇_{X}Y_{2}, for all a, b ∈ ℝ;  (2.32)
∇_{X}(fY) = f∇_{X}Y + (Xf)Y, for all f ∈ C^{∞}(M); (2.33)

is called a connection in E.

(2) A connection in T M is called a linear connection.

(3) A linear connection ∇ which satisfies

[X, Y] := XY − Y X = ∇_{X}Y − ∇_{Y} X  (2.34)

for all vector fields X, Y is said to be symmetric. [X, Y] is called the Lee brackets of (X, Y) ∈ X (M) × X (M). [X, Y] acts on functions via

[X, Y]_{p}f = X_{p}(Y f) − Y_{p}(Xf).

where p ∈ M.

(4) If ⟨·, ·⟩ is a Riemannian metric on M then a linear connection ∇ which satisfies

∇_{X}⟨Y, Z⟩ = ⟨∇_{X}Y, Z⟩ + ⟨Y, ∇_{X}Z⟩  (2.35)

for all vector fields X, Y, Z ∈ X (M) is said to be compatible with ⟨·, ·⟩.

2.64 Theorem (The fundamental theorem of Riemannian geometry) Let (M, g) be a Riemannian manifold. There exist a unique linear connection that is compatible with g and symmetric.

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2.65 Definition The unique connection in the above theorem is called the Levi-Civita connection of g. It is also known as the Riemannian connection of g.

2.6.2 Geodesics and the exponential map

2.66 Definition Let γ : I → M, t ↦ γ(t) be a map, where I ⊂ ℝ is an interval.

(1) We define γ′(t) as γ_{∗}(d/dt).

(2) If γ is smooth, then γ is called a curve.

(3) If I = [a, b], and γ is a curve defined on I, then γ is said to be a curve segment.

(4) If γ is a curve and γ′(t) ≠ 0 for all t ∈ I then γ is said to be a regular curve.

(5) If γ is a regular curve such that |γ′(t)| = 1 for all t ∈ I we say that γ is a unit speed curve.

(6) If γ : [a, b] → M is continuous and there exist real numbers a_{0}, a_{1}, …, a_{k} such that

(a) a_{i} < a_{j} if i < j;

(b) γ|_{[a_{i−1},a_{i}]} is a regular curve for i = 1, …, k; then γ is said to be an admissible curve. The numbers a_{0}, a_{1}, …, a_{k} are said to be a subdivision for the admissible curve γ.

(7) If γ is an admissible curve with subdivision a_{0}, a_{1}, …, a_{k}, then we define the length of γ to be

L(γ) := ∑_{i=1}^{k} ∫_{a_{i−1}}^{a_{i}} |γ′(t)| dt.

It’s easy to show that this is well defined.

(8) Let γ : I → M be an admissible curve such that, for every admissible curve γ̃ : [c, d] → M with γ(a) = γ̃(c) and γ(b) = γ̃(d) we have that L(γ) ⩽ L(γ̃). Then γ is said to be a minimizing curve.

(9) If γ : I → M is an admissible curve such that for every t in the interior of I there exist an open neighborhood ]a, b[⊂ I of t with the property that γ|_{[a,b]} is a minimizing curve then γ is said to be a locally minimizing curve.

2.67 Definition (1) Let γ :]a, b[→ M be a curve. A vector field along the curve γ is a mapping Y, defined on ]a, b[ which assigns every t ∈]a, b[ a vector Y_{t} from the tangent space T_{γ(t)}M such that, the mapping ]a, b[→ ℝ, t ↦ Y_{t}f is smooth for every smooth function f on M.

(2) Let Y be a vector field along the curve γ and ∇ be a linear connection on M. We can define a new vector field

∇_{γ′}Y

along the curve γ by the following:

(a) If γ′(t) = 0, then we put (∇_{γ′}Y)_{t} := 0.

(b) If γ′(t) ≠ 0, then there exist an open interval J around t, a neighborhood U around the point γ(t) in M, such that γ(J) is a 1-dimensional submanifold in U and γ|_{J} : J → γ(J) is a homeomorphism. Moreover the vector field Y may be extended from γ(J) to a smooth tangent fields Ŷ on U such that Ŷ_{γ(t)} = Y_{t}. We put

(∇_{γ′}Y)_{t} := ∇_{γ′(t)}Ŷ

{2.68 Remark \atop (1)} It can be shown that we can indeed find J, U and Ŷ and that the definition does not depend on the choice we make. Furthermore if γ : I → M is a curve, then the derivative γ′ : I → T_{γ(t)}(M), t ↦ γ′(t) is a smooth tangent field along the curve γ.

(2) Another equivalent method for defining vector field along curves can be found in [Lee].

2.69 Definition Let ∇ be a linear connection on M. A geodesic on M with respect to γ is a smooth curve γ : I → M on M, such that

∇_{γ′}γ′ = 0.

A geodesic with respect to the Riemannian connection is called a Riemannian geodesic.

2.70 Definition (1) We denote by Γ_{p,q} the set of all admissible curves γ : [a, b] → M such that γ(a) = p and γ(b) = q.

(2) We define a map d : M × M → ℝ ∪ {∞} by letting d(p, q) = ∞ if p and q lies in different connected components and by letting

d(p, q) = inf_{γ∈Γ_{p,q}}{L(γ)},

otherwise. If M is connected it can be shown that d is a metric on M.

(3) If M is connected and (M, d) is a complete metric space then we say that M is a complete Riemannian manifold.

(4) Let M be connected and let V, U ⊂ M. Then we define

dist(V, U) := inf_{p∈V,q∈U} d(p, q)

2.71 Remark There are more equivalent definitions of a complete Riemannian manifold.

The following two theorems describe the importance of geodesics and gives of a intuitive way to think about them. We’ll not go through the proofs but they can be found in most books about Riemannian geometry (see for example [Lee]).

2.72 Theorem Let ∇ be a linear connection on M, p ∈ M, V ∈ T_{p}M and t_{0} ∈ ℝ. Then there exist an open interval around t_{0} and a geodesic γ : I → M such that γ(t_{0}) = p and γ′(t_{0}) = V. Every two such geodesics agree on their common domains.

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{2.73 Theorem \atop (1)} Every minimizing regular curve is a geodesic.

(2) All Riemannian geodesics are constant speed curves.

(3) Every Riemannian geodesic is locally minimizing.

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2.74 Definition Let v ∈ T_{p}M. Then we denote by

ξ_{v}

the unique maximal unit speed geodesic which satisfies ξ_{v}(0) = p and ξ′_{v}(0) = v.

2.75 Definition Let p ∈ M.

(1) The set

E_{p} := {v ∈ T_{p}M| ξ_{v} is defined on an interval containing [0, 1]}

is called the domain for the exponential map at p.

(2) The map exp_{p} : E_{p} → M defined by

exp_{p}(v) = ξ_{v}(1)  (2.36)

is called the exponential map at p.

(3) Let ε > 0. If

exp_{p} : B_{ε}^{n}(0) → exp_{p}(B_{ε}^{n}(0))

is a diffeomorphism then exp_{p}(B_{ε}^{n}(0)) is called a geodesic ball in M with center p and radius ε.

In the following theorem some basic features of geodesics and the exponential map at p is stated.

{2.76 Theorem \atop (1)} exp_{p} is smooth;

(2) ξ_{v}(t) = exp_{p}(tv) for all t ∈ ℝ;

(3) ξ_{cv}(t) = ξ_{v}(ct), for all t ∈ ℝ and all c ∈ ℝ\{0}.

□

In the following theorem we get one of the most essential property of the exponential map. I’ll write down the proof since I find it instructive.

2.77 Theorem Let p ∈ M. Then there are neighborhoods V of 0 ∈ T_{p}M and U of p ∈ M such that exp_{p} : V → U is a diffeomorphism.

Proof: We have a natural isomorphism between T_{0}(T_{p}M) and T_{p}M. Let v ∈ T_{p}M and let τ : ℝ → T_{p}M be defined by letting τ(t) = tv. Then τ is a curve in T_{p}M such that τ(0) = 0 ∈ T_{p}M and τ′(0) = v ∈ T_{0}(T_{p}M) = T_{p}M. By theorem 2.76(2) we have

(exp_{p})_{∗}(v) = \frac{d}{dt}|_{t=0}(exp_{p} ∘τ)(t)

= \frac{d}{dt}|_{t=0} exp_{p}(tv) = \frac{d}{dt}|_{t=0}ξ_{v}(t) = v.

(2.37)

Hence (exp_{p})_{∗} is the identity map, which is invertible so exp_{p} is a diffeomorphism by the inverse function theorem.

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2.6.3 Curvatures

2.78 Definition Let X, Y ∈ X (M) and ∇ be the Riemannian connection on X (M) × X (M). The map R_{XY} : X (M) → X (M) defined by

R_{XY} := ∇_{[X,Y]} − [∇_{X}, ∇_{Y}]  (2.38)

is called the curvature transformation with respect to (X, Y).

2.79 Theorem The curvature transformation of M satisfies

R_{XY} = −R_{Y X}  (2.39)
⟨R_{W X}Y, Z⟩ = −⟨R_{W X}Z, Y ⟩  (2.40)
⟨R_{W X}Y, Z⟩ = ⟨R_{Y Z}W, X⟩  (2.41)
𝔖_{XY Z}R_{XY} Z = 0  (2.42)

where 𝔖_{XY Z} denotes the cyclic sum. (2.42) is known as the first Bianchi identity.

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2.80 Definition We let X, Y, Z, W ∈ X (M).

(1) The function R_{W XY Z} := ⟨R_{W X}Y, Z⟩ is called the curvature tensor. It is a (4, 0)-tensor field.

(2) Let (E_{1}, …, E_{n}) be a local orthonormal frame for T M defined on U ⊂ M. The function ρ : X (U) × X (U) → ℝ defined by

ρ(X, Y) = ∑_{a=1}^{n} R_{XE_{a}Y E_{a}}  (2.43)

is called the Ricci curvature. Note that ρ is symmetric due to (2.41). By the scalar curvature we mean the function τ defined by

τ = ∑_{a=1}^{n} ρ(E_{a}, E_{a}) = ∑_{a,b=1}^{n} R_{E_{a}E_{b}E_{a}E_{b}}.  (2.44)

(3) The map K^{M} : X (M) × X (M) → ℝ, X, Y ↦ K_{XY} defined for linearly independent X, Y ∈ X (M) by

K_{XY} = \frac{R_{XY XY}}{∥X∥^{2}∥Y ∥^{2} − ⟨X, Y ⟩^{2}} .  (2.45)

is called the sectional curvature of M. If M is 2-dimensional there is only one sectional curvature which we call the Gaussian curvature.

{2.81 Remark \atop (1)} It can be shown that both the Ricci and the scalar curvature are independent of the choice of orthonormal local frame. There are other equivalent definitions of the Gaussian curvature, which can be found in most books about differential geometry.

(2) We often use lowercase letters to indicate the value of a vector field at a given point. For example we would write R_{xyzw} for R_{XY ZW} evaluated at some point p ∈ M.

2.7 The Riemannian volume element

2.82 Theorem Let (M, g) be an oriented Riemannian manifold. Then there exist a unique differential n-form dV on M such that dV (E_{1}, …, E_{n}) = 1 for every oriented orthonormal basis (E_{1}, …, E_{n}) for T_{p}M for all p ∈ M. Moreover dV can be written in terms of any oriented local frame {Ê_{i}} as

\sqrt{det(ĝ_{ij})}ϕ̂_{1} ∧ ⋯ ∧ ϕ̂_{n},  (2.46)

where (ĝ_{ij}) = (⟨Ê_{i}, Ê_{j}⟩) and (ϕ_{i}) is the dual coframe.

Proof:

Uniqueness: In the proof we’ll use Einstein summation convention, namely

α^{i}E_{i} := ∑_{i=1}^{n} α_{i}E_{i}  (2.47)

where α_{i} ∈ ℝ and E_{i} is a smooth vector field on M, for i = 1, …, n. Assume that dV and dV′ are n-forms that satisfies

dV (E_{1}, …, E_{n}) = dV′(E_{1}, …, E_{n}) = 1

for every oriented orthonormal basis (E_{1}, …, E_{n}) for T_{p}M. Now let X_{1}, …, X_{n} be vectors in T_{p}M. Now for every n-form dW we have

\begin{aligned}dW (X_{1}, …, X_{n}) & = dW (α_{1}^{i_{1}}E_{i_{1}}, …, α_{n}^{i_{n}}E_{i_{n}}) = α_{1}^{i_{1}} ⋯ α_{n}^{i_{n}}dW (E_{i_{1}}, …E_{i_{n}}) \\ & = ∑_{σ∈𝔖_{n}} ε_{σ}a_{σ(1)} ⋯ a_{σ(n)} dW (E_{1}, …E_{n}) \end{aligned}

From this is follows that dV = dV′.

Existence: We show that 2.46 applied to oriented orthonormal basis (E_{1}, …, E_{n}) gives 1. Let

A := [E_{1} ; ⋮ ; E_{n}] [Ê_{1} ⋯ Ê_{n}] = [\begin{matrix}⟨E_{1}, Ê_{1}⟩ & ⋯ & ⟨E_{1}, Ê_{n}⟩ \\ ⋮ & ... & ⋮ \\ ⟨E_{n}, Ê_{1}⟩ & ⋯ & ⟨E_{n}, Ê_{n}⟩ \end{matrix}]

and observe that

[E_{1} ⋯ E_{n}] · A = [⟨E_{1}, Ê_{1}⟩E_{1} + ⋯ + ⟨E_{n}, Ê_{1}⟩E_{n} ; ⋮ ; ⟨E_{1}, Ê_{n}⟩E_{1} + ⋯ + ⟨E_{n}, Ê_{n}⟩E_{n}]′ = [Ê_{1} ⋯ Ê_{n}] . (2.48)

We’ll define |A| := det(A) for a (n × n)-matrix A. Since for matrices A, B we have |A · B| = |A| · |B| we see that

\sqrt{det(g_{ij})}ϕ̂_{1} ∧ ⋯ ∧ ϕ̂_{n}(E_{1}, …, E_{n})

= |[Ê_{1} ; ⋮ ; Ê_{n}] [Ê_{1} ⋯ Ê_{n}]|^{1/2} |A^{−1} · [ϕ̂_{1} ; ⋮ ; ϕ̂_{n}] [E_{1} ⋯ E_{n}] · A|

= |[Ê_{1} ; ⋮ ; Ê_{n}] [Ê_{1} ⋯ Ê_{n}]|^{1/2} |A^{−1}| (by 2.48).

Since A · A is equal to

\begin{matrix}[ & \begin{matrix}⟨E_{1}, Ê_{1}⟩⟨E_{1}, Ê_{1}⟩ + ⋯ + ⟨E_{1}, Ê_{n}⟩⟨E_{n}, Ê_{1}⟩ & … & ⟨E_{1}, Ê_{1}⟩⟨E_{1}, Ê_{n}⟩ + ⋯ + ⟨E_{1}, Ê_{n}⟩⟨E_{n}, Ê_{n}⟩ \\ ⋮ & ... & ⋮ \\ ⟨E_{n}, Ê_{1}⟩⟨E_{1}, Ê_{1}⟩ + ⋯ + ⟨E_{n}, Ê_{n}⟩⟨E_{n}, Ê_{1}⟩ & … & ⟨E_{n}, Ê_{1}⟩⟨E_{1}, Ê_{n}⟩ + ⋯ + ⟨E_{n}, Ê_{n}⟩⟨E_{n}, Ê_{n}⟩ \end{matrix} & ] \\ = [Ê_{1} ; ⋮ ; Ê_{n}] [Ê_{1} … Ê_{n}] \end{matrix}

we have that \sqrt{det(g_{ij})}ϕ̂_{1} ∧ … ∧ ϕ̂_{n}(E_{1} … E_{n}) = \sqrt{|A|^{2}}|A^{−1}| = 1, due to the orientation of {Ê_{i}}. So dV has the desired properties.

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2.83 Definition The unique form dV in the last theorem is called the volume form of (M, g).

2.84 Theorem Let V be an oriented n-dimensional vector space. Then there exist a linear operator ∗ : Λ(V) → Λ(V) called the Hodge star operator which is well-defined by requiring that for any orthonormal basis (e_{1}, …, e_{n}) of V, it satisfies

\begin{matrix}∗(1) = ±e_{1} ∧ ⋯ ∧ e_{n}, & ∗ (e_{1} ∧ ⋯ ∧ e_{n}) = ±1, \\ ∗(e_{1} ∧ ⋯ ∧ e_{k}) & = ±e_{k+1} ∧ ⋯ ∧ e_{n}, \end{matrix}

and then extend it linearly to whole Λ(V). Here we choose + if e_{1} ∧ ⋯ ∧ e_{n} lies in the orientation and − otherwise. Note that ∗(Λ_{k}(V)) ⊂ Λ_{n−k}(V).

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2.85 Remark Like with the wedge product we have a natural extension of the Hodge star operators to differential forms. Let (ϕ_{1}, …, ϕ_{n}) be a local orthonormal frame for T M^{∗}. Then we define the Hodge star operator ∗ : Ω(M) → Ω(M) by

\begin{matrix}∗(1) = ±ϕ_{1} ∧ ⋯ ∧ ϕ_{n}, & ∗ (ϕ_{1} ∧ ⋯ ∧ ϕ_{n}) = ±1, \\ ∗(ϕ_{1} ∧ ⋯ ∧ ϕ_{k}) & = ±ϕ_{k+1} ∧ ⋯ ∧ ϕ_{n}, \end{matrix}

and then extend linearly to all differential forms. We choose the sign as before.

{2.86 Remark \atop (1)} Assume that M is oriented and ω is a non-vanishing differential k-form on M. Then ω ∧ ∗ω is the volume form on M.

(2) For more details and proofs about the Hodge star operator see for example [Bis, War].

40 CHAPTER 2. DIFFERENTIAL AND RIEMANNIAN GEOMETRY

Tubes

General agreement

In this chapter we assume that M is Riemannian and P is a qdimensional embedded submanifold of M unless otherwise stated.

To describe the geometry of a neighborhood around P in M, it is convenient to use the so-called Fermi coordinates and Fermi fields. We’ll define these concepts in the following section and name some of their basic properties. When P is a point (0dimensional manifold) Fermi coordinates and fields coincide with the better known normal coordinates and normal fields. Fermi fields are also closely related to Jacobi fields which are commonly used. For more details see [Gr] and for example [Lee].

3.1 Fermi coordinates and fields

3.1 Definition The normal bundle of P in M is given by

ν := {(p, v)| p ∈ P, v ∈ (T_{p}P)^{⊥}}.  (3.1)

Here (T_{p}P)^{⊥} stands for the orthogonal complement of T_{p}P in T_{p}M.

3.2 Remark Let π : ν → P be defined by letting π(p, v) = p. Then by lemma 2.28 it’s not difficult to show that (ν, P, π) is a (n − q)-dimensional vector bundle. We’ll identify P with the zero section of ν. Thus P can be regarded as a submanifold of ν as well as a submanifold of M. By this it follows that

T_{(p,0)}ν = T_{p}P ⊕ (T_{p}P)^{⊥}.

3.3 Definition (1) A vector u ∈ (T_{p}P)^{⊥} is called a radial vector.

(2) A vector x ∈ T_{p}P is called a spherical vector.

3.4 Definition (1) The set

E_{ν} := {(p, v) ∈ ν| v ∈ E_{p}}

is called the domain for the exponential map of the normal bundle.

(2) The map exp_{ν} : E_{ν} → M defined by

exp_{ν}(p, v) = exp_{p}(v),

is called the exponential map of the normal bundle ν.

3.5 Lemma Let p ∈ P. Then ((exp_{ν})_{∗})_{(p,0)}|_{T_{0}((T_{p}P)^{⊥})} is the canonical identification of T_{0}((T_{p}P)^{⊥}) with (T_{p}P)^{⊥} and ((exp_{ν})_{∗})_{(p,0)}|_{T_{0}(T_{p}P)} is the canonical inclusion of T_{p}P in T_{p}M.

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3.6 Lemma There exists a neighborhood U of P in ν such that exp_{ν} |_{U} : U → exp_{ν}(U) is a diffeomorphism.

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3.7 Definition We denote by O_{P} the largest neighborhood of the zero section of ν for which exp_{ν} : O_{P} → exp_{ν}(O_{P}) is a diffeomorphism. Remark 3.2 and the last lemma guarantees the existence of O_{P}.

3.8 Definition Let p ∈ V ⊂ P where y = (y_{1}, …, y_{q}) : V → V′ ⊂ ℝ^{q} is a chart on P, and let E_{q+1}, …, E_{n} be a orthonormal local frame for ν. The Fermi coordinates (x_{1}, …, x_{n}) of P ⊂ M centered at p relative to the given coordinate system (y_{1}, …, y_{q}) on P and the given orthonormal local frame E_{q+1}, …, E_{n} of ν are defined by

x_{a} (exp_{ν} ( ∑_{j=q+1}^{n} t_{j}E_{j}(p′))) = y_{a}(p′) (a = 1, …, q),  (3.2)

x_{i} (exp_{ν} ( ∑_{j=q+1}^{n} t_{j}E_{j}(p′))) = t_{i}  (i = q + 1, …, n),  (3.3)

for p′ ∈ V, provided that ∑_{j=q+1}^{n} t_{j}E_{j}(p′) ∈ O_{P}. We make the notational convention that x_{a} (1 ⩽ a ⩽ q) are given by (3.2) and x_{i} (q + 1 ⩽ i ⩽ n) are given by (3.3).

The figure above shows how the normal bundle ν and Fermi coordinates could look like when dim M = 2 and dim P = 1. The figure should look a little bit like a tip of a finger. U ⊂ exp_{ν}(O_{P}) is a neighborhood of p ∈ P in M. y_{1} : V → y_{1}(V) ⊂ ℝ is a chart on P in a neighborhood of p. (E_{2}) is a local (orthonormal) frame for the normal bundle ν and is indicated by several vectors on the figure. m ∈ U ⊂ M is a point which we want to describe. There exist a unique elements p′ ∈ P and t_{2} ∈ ℝ such that m = exp_{ν}(t_{2}E_{2}(p′)) if we assume that exp_{ν}(tE_{2}(p′)) ∈ U for all t > 0 with t_{2}/t ⩾ 1 and t = 0. On the figure ξ stands for ξ_{E_{2}(p′)}. The Fermi coordinates of m is thus given by x_{1}(m) = y_{1}(p′) ∈ ℝ and x_{2}(m) = t_{2} ∈ ℝ.

3.9 Lemma Let (x_{1}, …, x_{n}) be a system of Fermi coordinates centered at p ∈ P. Then the coordinate vector fields

\frac{∂}{∂x_{q+1}} , …, \frac{∂}{∂x_{n}}

are orthonormal when restricted to P.

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The following lemma is very useful when dealing with Fermi coordinates.

3.10 Lemma Let (p, u) ∈ ν. So ξ := ξ_{u} is a geodesic normal to P at p. Then there exist a system of Fermi coordinates (x_{1}, …, x_{n}), centered at p such that for all t with (p, tu) ∈ O_{P} we have

\frac{∂}{∂x_{q+1}} |_{ξ(t)} = ξ′(t)  (3.4)

and

\frac{∂}{∂x_{a}} |_{p} ∈ T_{p}P,  \frac{∂}{∂x_{i}} |_{p} = (T_{p}P)^{⊥}  (3.5)

for 1 ⩽ a ⩽ q and q + 1 ⩽ i ⩽ n. Moreover,

(x_{α} ∘ ξ)(t) = tδ_{α,q+1}  (3.6)

for all α.

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3.11 Definition Let p ∈ P. Assume that U ⊂ exp_{ν}(O_{P}) ⊂ M and we have a system of Fermi coordinates (x_{1}, …, x_{n}) centered at p.

(1) If A ∈ X (U) can be written as

A = ∑_{a=1}^{q} c_{a} \frac{∂}{∂x_{a}}

where c_{a} ∈ ℝ for 1 ⩽ a ⩽ q then A is called a tangential Fermi field. The q-dimensional vector space of all tangential Fermi fields centered at p ∈ P is denoted by

X (P, p)^{⊤}.  (3.7)

(2) If X ∈ X (U) can be written as

X = ∑_{i=q+1}^{n} d_{i} \frac{∂}{∂x_{i}}

where d_{i} ∈ ℝ for q + 1 ⩽ i ⩽ n then X is called a normal Fermi field. The (n − q)-dimensional vector space of all normal Fermi fields centered at p ∈ P is denoted by

X (P, p)^{⊥}.  (3.8)

(3) We denote by

X (P, p) = X (P, p)^{⊤} ⊕ X (P, p)^{⊥}  (3.9)

the space of all Fermi fields centered at p ∈ P.

3.12 Definition Let (x_{1}, …, x_{n}) be a system of Fermi coordinates of P. We define

σ := \sqrt{∑_{i=q+1}^{n} x_{i}^{2}} and N := ∑_{i=q+1}^{n} \frac{x_{i}}{σ} \frac{∂}{∂x_{i}}  (3.10)

3.13 Remark It’s noteworthy to observe that when P is a point, σ and N are the familiar radial distance function and radial vector field (See for example [Lee]). N is also known as the Gauss map when P is a point. It can be shown that the definitions of σ and N are independent of Fermi coordinates.

3.14 Lemma Let m ∈ M. If there exist a unique unit speed geodesic ξ from P to m which is orthogonal to P then

σ(m) = dist(P, m) and N_{ξ(s)} = ξ′(s).

Hence we have that σ is defined on exp_{ν}(O_{P}) and N is defined on exp_{ν}(O_{P})\P.

Proof: We may assume that ξ(0) = p. Let b ⩾ 0 be the number which satisfies ξ(b) = m. We choose the Fermi coordinates described in lemma 3.10. Therefore by (3.4) and (3.6) we have

σ(m) = \sqrt{∑_{i=q+1}^{n} x_{i}^{2}(ξ(b))} = \sqrt{x_{q+1}^{2}(ξ(b))} = b = dist(m, P)

and, if b ≠ 0

N_{ξ(s)} = ∑_{i=q+1}^{n} \frac{x_{i}(ξ(s))}{σ(ξ(s))} \frac{∂}{∂x_{i}} |_{ξ(s)} = \frac{s}{s} \frac{∂}{∂x_{q+i}} |_{ξ(s)} = ξ′(s)

□

In the following lemma some basic properties of σ and N are stated.

3.15 Lemma Let X, Y ∈ X (P, p)^{⊥} and A, B ∈ X (P, p)^{⊤}. Then on exp_{ν}(O_{P}) we have

∇_{N}N = 0  (3.11)
[X, A] = [N, A] = [X, Y] = [A, B] = 0  (3.12)
[N, σX] = X(σ)N  (3.13)
∇_{N}∇_{N}U + R_{NU}N = 0  (for U = A + σX).  (3.14)

3.16 Definition Let f ∈ C^{∞}(M). The vector field grad f ∈ X (M) defined by setting

⟨grad f, X⟩ = Xf  (3.15)

for all X ∈ X (M) is called the gradient vector field of f.

3.17 Remark The well known Gauss lemma (see for example [Lee]) and it’s corollaries has an extension for submanifolds. I’ll let it be sufficient to state these generalizations without proof.

3.18 Theorem (The generalized Gauss lemma)

N = grad σ on exp_{ν}(O_{P})\P.  (3.16)
□

3.19 Corollary N is the unit normal to each of the tubular hypersurfaces σ = constant about P of M.

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3.20 Corollary exp_{ν} : E_{ν} → M preserves the lengths of radial vectors and orthogonality between radial and spherical vectors.

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3.2 Definition of a general tube

General agreement

For the rest of this chapter P is allowed to have a boundary.

3.21 Definition (1) Let m ∈ M. The set of all geodesics ξ : [a, 0] → M where a ⩽ 0 such that ξ(0) = m is denoted by

Ξ_{m}

(2) Let r > 0. The set

T (P, r) := {m ∈ M| ∃ ξ ∈ Ξ_{m}, L(ξ) ⩽ r : ξ meets P orthogonally} (3.17)

is called a tube of radius r about P.

3.22 Remark We will assume that exp_{ν} : {(p, v) ∈ ν| ∥v∥ ⩽ r} → T (P, r) ⊂ exp_{ν}(O_{P}) is a diffeomorphism. If M is complete and the closure of P is compact then this can always be achieved for sufficiently small r > 0. We can then write

T (P, r) = ⋃_{p∈P}{exp_{p}(v)| v ∈ (T_{p}P)^{⊥}, ∥v∥ ⩽ r}.  (3.18)

3.23 Definition The hypersurface

P_{t} = {m ∈ T (p, r)| dist(m, P) = t}  (3.19)

is called the tubular hypersurface at a distance t from P.

3.24 Definition Let ∇ be the Levi-Civita connection on M and R be the curvature transformation on M.

(1) We denote by R_{N} : X (exp_{ν}(O_{P})\P) → X (exp_{ν}(O_{P})\P) the operator which is

defined by

R_{N}U = R_{NU}N.  (3.20)

It’s easy to see that R_{N} is a tensor field.

(2) The operator S : X (exp_{ν}(O_{P})\P) → X (exp_{ν}(O_{P})\P) defined by

SU = −∇_{U}N  (3.21)

is called the shape operator.

(3) We define the covariant derivative of S by

∇_{U}(S)V = ∇_{U}SV − S∇_{U}V  (3.22)

for U, V ∈ X (exp_{ν}(O_{P})\P). For more details about covariant derivative of a tensor field, see for example [Lee].

{3.25 Lemma \atop (1)} S(fU) = f(SU) for all U ∈ X (exp_{ν}(O_{P})\P) and f ∈ C^{∞}(exp_{ν}(O_{P})\P).

Thus S is a tensor field.

(2) SN = 0;

(3) If U, V ∈ X (exp_{ν}(O_{P})\P) satisfies ⟨U, N⟩ = ⟨V, N⟩ = 0 then ⟨SU, V ⟩ = ⟨SV, U⟩.

Proof:

(1) S(fU) = −∇_{fU}N = −f∇_{U}N = f(SU).

(2) SN = −∇_{N}N = 0 by (3.11).

(3) We have ⟨U, N⟩ = ⟨V, N⟩ = 0. Hence

⟨SU, V ⟩ − ⟨SV, U⟩

= −⟨∇_{U}N, V ⟩ + ⟨∇_{V} N, U⟩
= ⟨N, ∇_{U}V ⟩ − U⟨N, V ⟩ − ⟨N, ∇_{V} U⟩ + V ⟨N, U⟩ (by def. 2.63(4))
= ⟨N, [U, V]⟩  (by def 2.63(3))
= ⟨grad σ, [U, V]⟩  (by the gen. Gauss lemma)
= ([U, V])σ  (by definition of grad)
= U(V σ) − V (Uσ) = U⟨N, V ⟩ − V ⟨N, U⟩ = 0

and the lemma follows.

□

3.26 Lemma

∇_{N}(S) = S^{2} + R_{N} on exp_{ν}(O_{P})\P.  (3.23)

Proof: For every point p ∈ exp_{ν}(O_{P})\P it suffices to show that (3.23) is valid for vector fields of the form U = A + σX, where A ∈ X (P, p)^{⊤} and X ∈ X (P, p)^{⊥} since every vector field on exp_{ν}(O_{P})\P is locally spanned by vector fields of this form. Since by (3.12) and (3.13), [N, A] = 0 and [N, σX] = X(σ)N, we have

[N, U] = [N, A + σX] = [N, A] + [N, σX] = X(σ)N.  (3.24)

Now

∇_{N}(S)U
= ∇_{N}SU − S∇_{N}U  (definition of covariant derivative)
= −∇_{N}∇_{U}N − S([N, U] + ∇_{U}N)  (definition of S and def. 2.63(3))
= −∇_{N}∇_{U}N + ∇_{U}∇_{N}N − S[N, U] + S^{2}U (by (3.11) and def. of S)
= R_{NU}N − ∇_{[N,U]}N − S[N, U] + S^{2}U  (by (2.38))
= R_{N}U − ∇_{X(σ)N}N − SX(σ)NU + S^{2}U (by def. of R_{N} and (3.24))
= R_{N}U − X(σ)∇_{N}N − X(σ)SNU + S^{2}U (by lemma 3.25(1) and (2.31))
= R_{N}U + S^{2}U  (by (3.11) and 3.25(2)),

and the proof is complete.

□

3.27 Corollary For t > 0 we define S(t), R(t) and S′(t) to be the restriction to P_{t} of S, R_{N} and ∇_{N}(S) respectively. Then we have

S′(t) = S(t)^{2} + R(t).  (3.25)

A differential equation of this form is called a Riccati differential equation.

□

3.2.1 The second fundamental form of submanifolds

3.28 Definition Let A, B ∈ X (P, p)^{⊤} and ∇ be the Levi-Civita connection on M. We define the map L : X (P, p)^{⊤} × X (P, p)^{⊤} → X (P, p)^{⊥}, L(A, B) ↦ L_{A}B to be the vector field normal to P which satisfies

⟨L_{A}B, X⟩ = ⟨∇_{A}B, X⟩  (3.26)

for all X ∈ X (P, p)^{⊥}.

3.29 Remark L is clearly linear with respect to smooth functions on P in the first coordinate. Since by (2.33) we have

⟨L_{A}(fB), X⟩ = ⟨∇_{A}(fB), X⟩
= ⟨(Af)B, X⟩ + ⟨f∇_{A}B, X⟩
= ⟨fL_{A}B, X⟩

L is also linear with respect to smooth functions on P in the second coordinate. Thus L is a tensor field on P.

3.30 Definition (1) The map T : X (P, p)^{⊤} × X (P, p)^{⊤} × X (P, p)^{⊥} → C^{∞}(P) defined by

T (A, B, X) = ⟨L_{A}B, X⟩

is called the second fundamental form of P. Since L is a tensor field, T is also a tensor field.

(2) Let p ∈ P, A, B ∈ X (P, p)^{⊤}, and X ∈ X (P, p)^{⊥}. We denote by a, b ∈ T_{p}P and u ∈ (T_{p}P)^{⊥} the vectors of A, B and X at the point p respectively. Then we define

T_{abu} := T (A, B, X)(p).

Furthermore we define T_{ab} = L_{A}B(p).

(3) The map T_{u} : T_{p}P → T_{p}P defined by

⟨T_{u}(a), b⟩ = T_{abu}

is called the Weingarten map.

3.31 Remark Let’s assume that P is a point. Then N is the familiar Gauss map, as already mentioned. Furthermore we have that S is the ordinary Weingarten map. Since we have

\begin{matrix}⟨L_{A}B, N⟩ = & ⟨∇_{A}B, N⟩ \\ = & − ⟨B, ∇_{A}N⟩ + A⟨B, N⟩ \\ = & ⟨−∇_{A}N, B⟩ = ⟨SA, B⟩ \end{matrix}  (3.27)

we see that T_{u} and S coincide when P is a point so the definitions above are generalizations of well known concepts.

3.32 Remark The Weingarten map is clearly linear and by using lemma 3.25(3) and (3.27) it’s easy to show that it is also symmetric.

The following theorem is taken from spectral theory and is a basic theorem in the field. It’s proof can be found in many books about linear algebra.

3.33 Theorem (Spectral theorem) Let V be a n-dimensional Euclidean inner product space over ℝ and let A : V → V be a linear map which is symmetric with respect to the inner product. Then there exists an orthonormal basis for V such that the matrix of A with respect to this basis has a diagonal form. The elements of this basis are eigenvectors for A and the elements on the diagonal are the corresponding eigenvalues. We say that the orthonormal basis diagonalizes A.

□

3.34 Remark Let p ∈ P and u ∈ (T_{p}P)^{⊥}. By the spectral theorem and remark 3.32 we can find an orthonormal basis (f_{1}, …, f_{q}) of T_{p}P that diagonalizes T_{ξ′_{u}(0)} : T_{p}P → T_{p}P. We extend these vectors to unit vector fields (F_{1}(t), …, F_{q}(t)) along ξ_{u} such that F_{a}(t) is an eigenvector for S(t) for all a and all t. Then we have eigenvalues κ_{a}(t) corresponding to these eigenvectors, and we can write

S(t)F_{a}(t) = κ_{a}(t)F_{a}(t)  (3.28)

for a = 1, …, q. We denote the remaining eigenvalues of S(t) by κ_{q+2}(t), …, κ_{n}(t). Hence there are unit vector fields F_{q+2}(t), …, F_{n}(t) along ξ_{u} such that

S(t)F_{i}(t) = κ_{i}(t)F_{i}(t)  (3.29)

for i = q + 2, …, n. By letting F_{q+1}(t) = ξ′_{u}(t) we have that {F_{1}(t), …, F_{n}(t)} is a local frame along ξ_{u}.

Note that

trace S(t) = ∑_{α=1,...,q,q+2,...,n} κ_{α}(t).  (3.30)

3.35 Definition (1) κ_{1}, …, κ_{q}, κ_{q+2}, …, κ_{n} in the above remark are called the principal curvature functions of S and F_{1}, …, F_{q}, F_{q+2}, …, F_{n} are called the principal curvature vector fields of S.

(2) The eigenfunctions of S of any orientable hypersurface are called the principal curvatures of the hypersurface.

From the definition we have the following lemma.

3.36 Lemma Let t > 0 and assume that P_{t} is orientable. Then the restriction of the principal curvature functions to P_{t} are the principal curvatures of P_{t}.

□

3.37 Corollary If F_{α} (1 ⩽ α ⩽ n) are differentiable at t then

κ′_{α}(t) = κ_{α}(t)^{2} + R_{ξ′(t)F_{α}(t)ξ′(t)F_{α}(t)}.

Proof: By (3.28) and (3.29) we have that κ_{α} is differentiable for all α. By the generalized Gauss lemma ξ′(t) has the same properties as N. Furthermore since ∇_{ξ′}F_{α} = F′ and ⟨F′_{α}, F_{α}⟩ = 0 we have

R_{ξ′F_{α}ξ′F_{α}}
= ⟨R_{ξ′F_{α}}ξ′, F_{α}⟩
= ⟨R_{ξ′}F_{α}, F_{α}⟩  (by defininiton of R_{N})
= ⟨∇_{ξ′}(S)F_{α}, F_{α}⟩ − ⟨S^{2}F_{α}, F_{α}⟩  (by lemma 3.26)
= ⟨∇_{ξ′}SF_{α}, F_{α}⟩ − ⟨S∇_{ξ′}F_{α}, F_{α}⟩ − κ_{α}^{2}  (def. of covariant derivative)
= ⟨∇_{ξ′}SF_{α} + SF′_{α}, F_{α}⟩ − 2S⟨F′_{α}, F_{α}⟩ − κ_{α}^{2}
= ⟨SF_{α}, F_{α}⟩′ − κ_{α}^{2} = κ′_{α} − κ_{α}^{2}

and the proof is complete.

□

3.3 The infinitesimal change of volume

General agreement

For the rest of the chapter we assume that M and P are oriented unless otherwise stated.

Let ω be the Riemannian volume form on M and ω_{ν} be the Riemannian volume form on ν. Since dim Ω^{n}(ν) = 1 we have that exp_{ν}^{∗}(ω) = fω_{ν} where f : ν → ℝ is a positive real valued function on ν. With this in mind we’ll make the following definition:

3.38 Definition (1) The function chvol : exp_{ν}(O_{P}) → ℝ is defined such that the following equation is satisfied:

exp_{ν}^{∗}(ω) = (chvol ∘ exp_{ν})ω_{ν}.  (3.31)

We note that chvol is everywhere non-negative.

(2) For a point p ∈ P and a unit vector u ∈ (T_{p}P)^{⊥} the function

ϑ_{u}(t) = (chvol ∘ exp_{ν})(p, tu)  (3.32)

defined for all (p, tu) ∈ O_{P} is called the infinitesimal change of volume function of P in the direction u.

3.39 Lemma Let (x_{1}, …, x_{n}) be a system of Fermi coordinates for P centered at p ∈ P which has

the same orientation as ω, i.e. ω (\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} ) > 0. If (y_{1}, … y_{q}) are system of Fermi coordinates on P such that

\{\frac{∂}{∂x_{1}} |_{p}, …, \frac{∂}{∂x_{q}} |_{p}\}

forms an orthonormal basis of T_{p}P then

ϑ_{u}(t) = ω (\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} ) (exp_{ν}(p, tu)).  (3.33)

Proof: By lemma 3.9 we have that

\{\frac{∂}{∂x_{q+1}} |_{p}, …, \frac{∂}{∂x_{n}} |_{n}\}

is orthonormal basis for (T_{p}P)^{⊥}. Hence

\frac{∂}{∂x_{1}} |_{p}, …, \frac{∂}{∂x_{n}} |_{n}

is an orthonormal basis for T_{(p,0)}ν = (T_{p}P) ⊕ (T_{p}P)^{⊥} and since ω_{ν} is a volume form on ν, its value on this basis is equal to 1. We’ll evaluate both sides of equation (3.31) on the basis above at the point (p, tu). The left hand side yields:

exp_{ν}^{∗}(ω) (\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} ) (p, tu) = ω ((exp_{ν})_{∗} \frac{∂}{∂x_{1}} , …, (exp_{ν})_{∗} \frac{∂}{∂x_{n}} ) exp_{ν}(p, tu)

= ω (\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} ) exp_{ν}(p, tu);

(3.34)

by lemma 3.5, and the right hand side yields:

(chvol ∘ exp_{ν})ω_{ν} (\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} ) (p, tu) = (chvol ∘ exp_{ν})(p, tu) = ϑ_{u}(t), (3.35)

where we have used the remarks above. Combining (3.34) and (3.35) we get (3.33).

□

The following theorem will be vital for our further study. The proof is a little technical and I choose to omit it.

3.40 Theorem Let (p, u) ∈ ν. Then along the unit speed geodesic ξ_{u} we have

\frac{ϑ′_{u}(t)}{ϑ_{u}(t)} = − (\frac{n − q − 1}{1} + trace S(t))  (3.36)

for all t > 0 if (p, tu) ∈ O_{P}.

□

3.41 Definition We define

V_{P}^{M}(r) := the n-dimensional volume of T (P, r),
A_{P}^{M}(r) := the (n − 1)-dimensional volume of P_{r},

3.42 Remark We often refer to A_{P}^{M}(r) as the area of P_{r}. This terminology was used in chapter 1.

3.43 Lemma We assume as before that the exponential map exp_{ν} : {(p, v) ∈ ν|∥v∥ ⩽ r} → T (P, r) is a diffeomorphism. Then

A_{P}^{M}(r) = r^{n−q−1} ∫_{P} ∫_{S^{n−q−1}} ϑ_{u}(r)du dP.  (3.37)

Proof: Let V be a neighborhood of the zero section of ν. Define s : V → ℝ by

s(p, v) = ∥v∥.  (3.38)

By the generalized Gauss Lemma and lemma 3.14 we have that s = σ ∘ exp_{ν}. We let ∗ denote the Hodge star operator either on M or on ν. By remark 2.86 we have that ds ∧ ∗ ds is the volume form of ν and thus ∗ ds is a volume form for the hypersurface {(p, v) ∈ ν| ∥v∥ = t, p ∈ P } on ν. Now dσ ∧ ∗ dσ is the volume form of M in a neighborhood of P. We thus have (by corollaries of the generalized Gauss lemma) that ∗ dσ is the volume form of P_{t} for each t. We have that the map exp_{ν}^{∗} : Ω^{n}(M) → Ω^{n}(ν) preserves wedge product (by theorem 2.42(2)) and we can thus calculate that

exp_{ν}^{∗}(dσ) ∧ exp_{ν}^{∗}(∗ dσ) = exp_{ν}^{∗}(dσ ∧ ∗ dσ)

= \{ω (\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} ) ∘ exp_{ν}\} ds ∧ ∗ ds.

Since σ is a smooth function we have by theorem 2.42(4) and (2.15) that

exp_{ν}^{∗}(dσ) = d(exp_{ν}^{∗}(σ)) = d(σ ∘ exp_{ν}) = ds  (3.39)

and thus

exp_{ν}^{∗}(∗ dσ) = \{ω (\frac{∂}{∂x_{1}} ∧ ⋯ ∧ \frac{∂}{∂x_{n}} ) ∘ exp_{ν}\} ∗ ds.

P_{r} has a compact closure, so the integral of ∗ dσ over P_{r} is A_{P}^{M}(r). Hence we get by definition of integration of form on manifolds (see (2.22)) that

A_{P}^{M}(r) = ∫_{P_{r}} ∗ dσ = ∫_{exp_{ν}^{−1}(P_{r})} exp_{ν}^{∗}(∗ dσ)

= ∫_{P} ∫_{S^{n−q−1}(r)} \{ω (\frac{∂}{∂x_{1}} ∧ ⋯ ∧ \frac{∂}{∂x_{n}} ) ∘ exp_{ν}\} dũ dP

= r^{n−q−1} ∫_{P} ∫_{S^{n−q−1}} \{ω (\frac{∂}{∂x_{1}} ∧ ⋯ ∧ \frac{∂}{∂x_{n}} ) ∘ exp_{ν}\} du dP

(3.40)

Here dP is the volume form of P, dũ the volume form on S^{n−q−1}(r) and du is the volume form of S^{n−q−1}. Now the lemma follows from lemma 3.39.

□

3.44 Lemma Assuming the same assumptions as in the previous lemma we have

\frac{d}{dr}V_{P}^{M}(r) = A_{P}^{M}(r).  (3.41)

Proof:

V_{P}^{M}(r)
= ∫_{T (P,r)} dσ ∧ ∗ dσ
= ∫_{exp_{ν}^{−1}(T (P,r))} exp_{ν}^{∗}(dσ) ∧ exp_{ν}^{∗}(∗ dσ) (by (2.22))
= ∫_{0}^{r} ∫_{exp_{ν}^{−1}(P_{r})} ds ∧ exp_{ν}(∗ dσ)  (by (3.39))
= ∫_{0}^{r} A_{P}^{M}(s) ds  (by (3.40))

and the lemma follows from the fundamental theorem of the calculus.

□

The following lemma is easy to prove:

3.45 Lemma For M = ℝ^{n} we have

ϑ_{u}(t) = det(δ_{a,b} − tT_{abu})  (3.42)

if the right hand side is non-negative.

□

3.46 Remark (3.42) is not valid for M in general. We’ll see that the previous lemma will be a key concept in deriving Weyl’s Tube formula later in this chapter.

3.4 The Bishop-Günther inequalities

In this section the volume of tubes around P will be calculated when P is a point in some special kind of manifold M. In that case the volume of T (P, r) is the volume of a geodesic ball. The following lemma is easily verified.

3.47 Lemma Let k : (0, t_{0}] → ℝ be a differentiable function which satisfies

k′ ⩾ k^{2} + λ,  \xymatrix{k(t) \ar[r]_{t→0+} & −∞,}  (3.43)

where λ ∈ ℝ. Then

k(t) ⩾ \frac{−\sqrt{λ}}{tan(t\sqrt{λ})}  (3.44)

for 0 < t ⩽ t_{0}. If the inequality in (3.43) in the assumption is reversed then the inequality in (3.44) is also reversed. Note that λ = 0 is a removable singularity.

□

3.48 Definition (1) Let m ∈ M and ξ be a geodesic with ξ(0) = m. Let t_{0} > 0 which satisfies

There exists an admissible curve γ on M with γ(0) = m and γ(t) = ξ(t) such that L(γ(t)) < L(ξ(t));

and assume that for all t such that 0 < t < t_{0}, the curve ξ|_{[0,t]} is minimizing, then t_{0} is said to be a cut point of m along the geodesic ξ.

(2) The set of all cut points of m is called the cut locus of m.

3.49 Theorem Let M be a complete Riemannian manifold. Assume that r is less than or equal to the distance between m and its cut locus. For λ ∈ ℝ we have

K^{M} ⩾ λ ⇒ V_{m}^{M}(r) ⩽ \frac{2π^{n/2}}{Γ(n/2)} ∫_{0}^{r} (\frac{sin(t\sqrt{λ})}{\sqrt{λ}})^{n−1} dt;

K^{M} ⩽ λ ⇒ V_{m}^{M}(r) ⩾ \frac{2π^{n/2}}{Γ(n/2)} ∫_{0}^{r} (\frac{sin(t\sqrt{λ})}{\sqrt{λ}})^{n−1} dt.

Proof: We shall first prove the latter statement. Then R_{ξ′(t)F_{i}(t)ξ′(t)F_{i}(t)} ⩽ λ. Thus the functions k_{i}(t) satisfies the Riccati differential inequality

k′_{i} ⩽ k_{i}^{2} + λ.

by corollary 3.37. By lemma 3.47 we have

k_{i}(t) ⩽ \frac{−\sqrt{λ}}{tan(t\sqrt{λ})},

and thus

trace S(t) ⩽ \frac{−(n − 1)\sqrt{λ}}{tan(t\sqrt{λ})}.

By theorem 3.40 and the above observation we obtain the equality

\frac{ϑ′_{u}(t)}{ϑ_{u}(t)} ⩾ \frac{−(n − 1)}{t} + \frac{(n − 1)\sqrt{λ} cos(t\sqrt{λ})}{sin(t\sqrt{λ})}.

The integral of 1/ tan(at) where a ∈ ℝ is ln(sin(at))/a for 0 < t < π/(2a) so we get

ln ϑ_{u}(t) ⩾ C − (n − 1) ln(t) + (n − 1) ln(sin(t\sqrt{λ}))

where C ∈ ℝ or, equivalently

ϑ_{u}(t) = e^{C} (\frac{sin(t\sqrt{λ})}{t})^{n−1}

Since ϑ_{u}(0) = 1 we see that e^{C} = 1/λ^{(n−1)/2}. Making use of lemma 3.43 and lemma 3.44 we have shown the desired inequality. The proof of the former statement is exactly the same, where we reverse all inequalities.

□

3.50 Definition If K^{M} = λ ∈ ℝ then M is called a space of constant sectional curvature and is denoted by

K^{n}(λ).

The following corollary follows immediately from the last theorem.

3.51 Corollary If M = K^{n}(λ) is a space of constant sectional curvature, then

V_{m}^{M}(r) = \frac{2π^{n/2}}{Γ(n/2)} ∫_{0}^{r} (\frac{sin(t\sqrt{λ})}{\sqrt{λ}})^{n−1} dt,  (3.45)

for all m ∈ M.

□

{3.52 Remark \atop (1)} It is well known that K^{n}(λ) is locally isometric to one of three classical spaces depending on the value of λ. Those spaces are

(a) The Euclidean space, when λ = 0;

(b) A sphere, when λ > 0;

(c) A hyperbolic space λ < 0.

(2) For a great survey of hyperbolic spaces, I recommend [Rat]. For more about spaces of constant sectional curvature, see for example [Wo].

3.53 Corollary It can be shown that S^{n}(r) is a space of constant sectional curvature λ where λ = π/r. By setting s = t\sqrt{λ} in (3.45) and evaluate the volume at r = π we get a formula for the volume of S^{n}(r), since the geodesic ball exactly covers S^{n}(r). The result is

V_{m}^{S^{n}(r)}(π) = \frac{2π^{n/2}}{Γ(n/2)λ^{n/2}} ∫_{0}^{π}(sin s)^{n−1} ds.  (3.46)

□

3.54 Remark The integral in the last corollary can easily be calculated explicitly and is done so in most calculus books. It is left to the reader to show that the result for the volume of S^{n}(r) in (3.46) is the same as the area of S^{n}(r) as a hypersurface given in chapter 1. This should not come as a surprise, since this is exactly the same thing we’re doing, though we use different words for it, and different method to find the answer.

3.55 Remark In this section we have seen how we could get an estimate of the volume of a geodesic ball in terms of the sectional curvature K^{M}, and that we could get some nice equalities when K^{M} is constant. In the general theory of tubes often an estimate for the volume is all we can hope to have and many articles in the field today gives some estimates of the volume of a tube for more general manifolds.

3.5 Weyl’s tube formula

3.5.1 Double forms

3.56 Definition (1) A smooth covariant (p+q, 0)-tensor field α : X (M)^{p}×X (M)^{q} → C^{∞}(M) which is antisymmetric in the first p and the last q variables is called a double form of type (p, q) on M. We write α(X_{1}, …, X_{p})(Y_{1}, …, Y_{q}) to denote the value on vector fields X_{1}, …, X_{p}, Y_{1}, …, Y_{q} ∈ X (M).

(2) Let X_{1}, …, X_{p} ∈ X (M). Then we define a mapping α(X_{1}, …, X_{p}) : X (M)^{q} → C^{∞}(M), (Y_{1}, …, Y_{q}) ↦ α(X_{1}, …, X_{p})(Y_{1}, …, Y_{q}). Then α(X_{1}, …, X_{p}) is a q-form by definition.

(3) Let α be a double form of type (p, q) and β be a double form of type (r, s). We define an exterior product ∧ of the double forms α and β by

∧(α, β)(X_{1}, …, X_{p+r})(Y_{1}, …, Y_{q+s}) := (α ∧ β)(X_{1}, …, X_{p+r})(Y_{1}, …, Y_{q+s})

= \frac{1}{p!q!r!s!} ∑_{ρ∈𝔖_{p+r}} ∑_{σ∈𝔖_{q+s}} ε_{ρ} ε_{σ} α(X_{ρ(1)}, …, X_{ρ(p)})(X_{σ(1)}, …, X_{σ(q)})

·β(X_{ρ(p+1)}, …, X_{ρ(p+r)})(X_{σ(q+1)}, …, X_{σ(q+s)}).

(4) If p = q and α(X_{1}, …, X_{p})(Y_{1}, …, Y_{p}) = α(Y_{1}, …, Y_{p})(X_{1}, …, X_{p}) for all X_{1}, Y_{1}, …, X_{p}, Y_{p} ∈ X (M) we say that α is a symmetric double form.

{3.57 Remark \atop (1)} The exterior product of double forms is associative and satisfies α ∧ β = (−1)^{pr+qs}β ∧ α, for α and β as in the previous definition. The double forms of type (p, 0) along with the exterior product agrees with ordinary differential forms and the ordinary exterior product of differential forms.

(2) The Riemannian metric tensor field g = ⟨·, ·⟩ is a symmetric double form of type (1, 1) and the curvature tensor R is a symmetric double form of type (2, 2). Thus g^{c} := \underbrace{g ∧ ⋯ ∧ g}_{c times} is a double form of type (c, c) and R^{c} := \underbrace{R ∧ ⋯ ∧ R}_{c times} is

a double form of type (2c, 2c).

3.58 Definition Let (E_{1}, …, E_{n}) be an orthonormal local frame on M. For a double form α of type (p, q) we let C^{0}(α) = α and define C^{c}(α) inductively for c > 0 by

\begin{matrix}C^{c}(α)(X_{1}, …, X_{p−c})(Y_{1}, …, Y_{q−c}) \\ = & ∑_{a=1}^{n} C^{c−1}(α)(X_{1}, …, X_{p−c}, E_{a})(Y_{1}, …, Y_{q−c}, E_{a}). \end{matrix}

The operators C^{c} are called contraction operators. It’s easy to show that C^{c} are independent of the choice of the orthonormal local frame.

3.59 Definition Let R be the curvature tensor of P. We note that C^{2c}(R^{c}) is a smooth function on P. We’ll assume that ∫_{P} C^{2c}(R^{c}) dP converges. Then the real number

k_{2c}(P) = \frac{1}{c!(2c)!} ∫_{P} C^{2c}(R^{c}) dP  (3.47)

is called the (2c)^{th} integrated mean curvature of P.

3.60 Remark It’s easy to calculate k_{2c} for small c. For example we have k_{0}(P) = Vol(P) (i.e. the volume of P) and k_{2}(P) = \frac{1}{2} ∫_{P} τ dP. For large numbers of c it seems non-trivial to calculate k_{2c} explicitly.

3.5.2 Invariants

3.61 Definition Let V be an n-dimensional inner product space.

(1) We let W_{k} = \underbrace{V^{∗} ⊗ ⋯ ⊗ V^{∗}}_{k times}.

(2) For g ∈ O(n, ℝ) we define a map g : W_{k} → W_{k} by letting

g(ϕ)(v_{1} ⊗ ⋯ ⊗ v_{k}) = ϕ(g^{−1}v_{1} ⊗ ⋯ ⊗ g^{−1}v_{k}).

O(n, ℝ) acts on W_{k} by this definition.

(3) For σ ∈ 𝔖_{k} we define a map σ : W_{k} → W_{k} by letting

σ(ϕ)(v_{1} ⊗ ⋯ ⊗ v_{k}) = ϕ(v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(k)}).

𝔖_{k} acts on W_{k} by this definition.

(4) A map P : W_{k} → ℝ which has a symmetric multilinear extension to the tensor product \underbrace{W_{k} ⊗ ⋯ ⊗ W_{k}}_{h times} is called a polynomial of degree h on W_{k}.

We denote the space of all such polynomials by

P_{h}(W_{k}).  (3.48)

(5) For g ∈ O(n, ℝ) we define a map g : P_{h}(W_{k}) → P_{h}(W_{k}) by letting

g(P)(ϕ) = P (g^{−1}ϕ).

O(n, ℝ) acts on P_{h}(W_{k}) by this definition.

(6) Let P ∈ P_{h}(W_{k}). If g(P) = P for all g ∈ O(n) then we say that P is invariant.

(7) Let (e_{1}, …, e_{n}) be an orthonormal basis for V. We set

P_{σ}(ϕ) = ∑_{a_{1},...,a_{m}=1}^{n} σ(ϕ)(e_{a_{1}}, e_{a_{1}}, …, e_{a_{m}}, e_{a_{m}})

for ϕ ∈ W_{2m} and σ ∈ 𝔖_{2m}. It is straightforward to check that the definition is independent of which orthonormal basis we choose.

(8) We define a map ⊗^{h}ϕ ∈ W_{hk} by letting

(⊗^{h}ϕ)(v_{1} ⊗ ⋯ ⊗ v_{hk}) = ϕ(v_{1} ⊗ ⋯ ⊗ v_{k}) ⋯ ϕ(v_{k(h−1)+1} ⊗ ⋯ ⊗ v_{hk})

for ϕ ∈ W_{k} and v_{1}, …, v_{hk} ∈ V.

3.62 Definition Let hk = 2m, where m ∈ ℕ and let σ ∈ 𝔖_{2m}. The polynomial P_{σ} ∈ P_{h}(W_{k}) defined by

P_{σ}(ϕ) = ∑_{a_{1},...,a_{m}=1}^{n} σ(⊗^{h}ϕ)(e_{a_{1}}, e_{a_{1}}, …, e_{a_{m}}, e_{a_{m}})  (3.49)

where ϕ ∈ W_{k} is called the elementary invariant corresponding to σ.

The following theorem is a key theorem in the theory of invariants. I’ll not go through its proof.

3.63 Theorem Every invariant polynomial is a sum of products of the elementary invariants.

□

3.64 Remark Invariant theory has many applications and is used for example in algebraic topology. For some discussion see for example [Spi]. [Gilk] is also interesting.

3.5.3 Some definitions and lemmas

3.65 Definition Let V be an n-dimensional positive definite inner product space. We denote by S^{n−1} the unit sphere in V, where we use the metric obtained from the inner product on V. We define a function I_{s} : W_{s} → ℝ by

I_{s}(ϕ) = ∫_{S^{n−1}} ϕ(v, …, v) dv.  (3.50)

\begin{matrix}3.66 Remark \\ (1) \end{matrix} ∫_{S^{n−1}} dv is independent of which inner product space V we use.

(2) I_{s} is a polynomial of degree 1 and by spherical symmetry it’s easy to see that it is invariant.

3.67 Definition For an even positive number q we define

q) = 1 · 3 ⋯ (q − 1)  (3.51)

We let 0) = 1.

3.68 Definition We define

𝔔_{s} = {σ ∈ 𝔖_{2s}| σ(2t − 1) < σ(2t + 1) for t = 1, …, s − 1 and
σ(2t − 1) < σ(2t) for t = 1, …, s}.

3.69 Lemma The cardinality of 𝔔_{s} is

card(𝔔_{s}) = 2s) = 1 · 3 ⋯ (2s − 1) = \frac{(2s)!}{2^{s}s!}.  (3.52)

Proof: We’ll prove (3.52) by induction. It’s easy to check the formula directly for small numbers of s. Assume that the formula is valid for s = t − 1, where t > 3. We’ll show that then (3.52) is also valid for s = t. Clearly σ(1) = 1 for all σ ∈ 𝔔_{t}. We then have two choices for σ(2) ∈ 𝔔_{t}, namely σ(2) = 2 or σ(2) = 3. We’ll calculate the numbers of elements in 𝔔_{t} separately depending on their value of σ(2) and then add the numbers together.

(1) σ(2) = 2: Elements in 𝔔_{t} of this form clearly has the same cardinality as 𝔔_{t−1} which is

1 · 3 ⋯ (2t − 3)  (3.53)

according to our induction hypothesis.

(2) σ(2) = 3: We first choose i, j ∈ [[2, 2s]]\{3}, i ≠ j, such that σ(i) = 2 and σ(j) = 4. We can do this in (2t − 2) · (2t − 3) ways. For every such choice we have card(𝔔_{t−2}) possibilities for the rest of the elements in [[1, 2t]] which is 1 · 3 ⋯ (2t − 5) by our induction hypothesis. Multiplying these numbers we get the whole cardinality of elements in 𝔔_{t} of this form, namely

(2t − 2) · (2t − 3) · 1 · 3 ⋯ (2t − 5).  (3.54)

By adding (3.53) and (3.54) we get that card(𝔔_{t}) = 2t) which proves (3.52).

□

3.70 Remark For ϕ ∈ W_{2s} and σ ∈ 𝔖_{2s} we have

\begin{matrix}∑_{σ∈𝔖_{2s}} ∑_{a_{1},...,a_{s}=1}^{n} σ(ϕ)(e_{a_{1}}, e_{a_{1}}, …, e_{a_{s}}, e_{a_{s}}) \\ = & 2^{s}s! ∑_{σ∈𝔔_{s}} ∑_{a_{1},...,a_{s}=1}^{n} σ(ϕ)(e_{a_{1}}, e_{a_{1}}, …, e_{a_{s}}, e_{a_{s}}), \end{matrix}

since the inner sum is independent of orthonormal basis.

3.71 Definition Let F ∈ W_{s} be an integrable function. Then we define the average of F over the unit sphere to be the real number

⟨F ⟩ = \frac{∫_{S^{n−1}} F du}{∫_{S^{n−1}} du}.  (3.55)

3.72 Theorem Let (e_{1}, …, e_{n}) be an orthonormal basis of V and denote by (u_{1}, …, u_{n}) its dual basis. For c = ∑_{k=1}^{n} i_{k}, we have

⟨u_{j_{1}}^{i_{1}} ⋯ u_{j_{n}}^{i_{n}}⟩ = \frac{i_{1}) ⋯ i_{n})}{n(n + 2) ⋯ (n + c − 2)},  (3.56)

where i_{1}, …, i_{n} ∈ ℕ.

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3.73 Remark The proof of the above theorem when V = ℝ^{n} can be found in [Gr] but I see nothing wrong with applying the proof to general positive definite inner product spaces.

3.74 Remark Let ρ ∈ 𝔖_{s}, and f : 𝔖_{s} → ℝ be a function. Then

∑_{σ∈𝔖_{s}} f(ρσ) = ∑_{σ∈𝔖_{s}} f(σρ) = ∑_{σ∈𝔖_{s}} f(σ).  (3.57)

{3.75 Lemma \atop (1)} If s is odd, then I_{s} = 0;

(2) Let ϕ ∈ W_{2s}. Then I_{2s} is given by

I_{2s}(ϕ) = \frac{2π^{n/2}}{n(n + 2) ⋯ (n + 2s − 2)Γ(n/2)} ∑_{σ∈𝔔_{s}} ∑_{i_{1},...,i_{s}=1}^{n} σ(ϕ)(e_{i_{1}}, e_{i_{1}}, …, e_{i_{s}}, e_{i_{s}}) (3.58)

where (e_{1}, …, e_{n}) is any orthonormal basis of V.

Proof: The former statement follows trivially from theorem 3.63. By the same theorem there are constants b_{σ} such that

I_{2s}(ϕ) = ∑_{σ∈𝔖_{2s}} b_{σ} ( ∑_{i_{1},...,i_{s}=1}^{n} σ(ϕ)(e_{i_{1}}, e_{i_{1}}, …, e_{i_{s}}, e_{i_{s}})) .  (3.59)

We denote by (u_{1}, …, u_{n}) the dual basis of (e_{1}, …, e_{n}) as before. Let ϕ = u_{j_{1}}^{2} ⋯ u_{j_{s}}^{2}. By (1.2) and theorem 3.72 we have that

I_{2s}(u_{j_{1}}^{2} ⋯ u_{j_{s}}^{2}) = \frac{2π^{n/2}}{n(n + 2) ⋯ (n + 2s − 2)Γ(n/2)},  (3.60)

independent of j_{1}, …, j_{s} as long as j_{l} ≠ j_{k} for l ≠ k. Now let ρ ∈ 𝔖_{2s}. Then we have

I_{2s}(u_{j_{1}}^{2} ⋯ u_{j_{s}}^{2})
=I_{2s}(ρ^{−1}(u_{j_{1}}^{2} ⋯ u_{j_{s}}^{2})) (since I_{2s} is an invariant polynomial)
= ∑_{σ∈𝔖_{2s}} b_{σ} ( ∑_{i_{1},...,i_{s}=1}^{n} σ(ρ^{−1}(u_{j_{1}}^{2} ⋯ u_{j_{s}}^{2}))(e_{i_{1}}, e_{i_{1}}, …, e_{i_{s}}, e_{i_{s}}))  (by (3.59))

= ∑_{σ∈𝔖_{2s}} b_{σρ} ( ∑_{i_{1},...,i_{s}=1}^{n} σ(u_{j_{1}}^{2} ⋯ u_{j_{s}}^{2})(e_{i_{1}}, e_{i_{1}}, …, e_{i_{s}}, e_{i_{s}}))  (by (3.57))

=2^{s}s! ∑_{σ∈𝔔_{s}} b_{σρ} ( ∑_{i_{1},...,i_{s}=1}^{n} σ(u_{j_{1}}^{2} ⋯ u_{j_{s}}^{2})(e_{i_{1}}, e_{i_{1}}, …, e_{i_{s}}, e_{i_{s}})) . (by remark 3.70)

The inner sum is always zero unless σ = id_{𝔖_{s}} but then it is equal to one. Thus the last line is equal to 2^{s}s!b_{ρ}. We deduce that b_{σ} are the same for all σ, namely

b_{σ} = \frac{2π^{n/2}}{n(n + 2) ⋯ (n + 2s − 2)Γ(n/2)2^{s}s!}.  (3.61)

Since card(𝔖_{2s})/card(𝔔_{s}) = 2^{s}s! the lemma now follows from lemma 3.69.

□

3.76 Corollary Assume that ϕ ∈ W_{2s} is symmetric. Then (3.58) simplifies to

I_{2s}(ϕ) = \frac{1 · 3 ⋯ (2s − 1) · 2π^{n/2}}{n(n + 2) ⋯ (n + 2s − 2)Γ(n/2)} ∑_{i_{1},...,i_{s}=1}^{n} ϕ(e_{i_{1}}, e_{i_{1}}, …, e_{i_{s}}, e_{i_{s}}).

Proof: This follows immediately from the previous lemma and lemma 3.69.

□

3.77 Theorem Let M = ℝ^{n}, and assume that P has a second fundamental form T and curvature tensor R^{P}. Let S_{p}^{n−q−1} denote the unit sphere in (T_{p}P)^{⊥}. Then

∫_{S_{p}^{n−q−1}} det(δ_{a,b} − tT_{abu}) du

=\frac{2π^{(n−q)/2}}{Γ((n − q)/2)} ∑_{c=0}^{[ \frac{q}{2} ]} \frac{C^{2c}((R^{P})^{c})t^{2c}}{c!(2c)!(n − q)(n − q + 2) ⋯ (n − q + 2c − 2)}

=2π^{(n−q)/2)} ∑_{c=0}^{[ \frac{q}{2} ]} \frac{C^{2c}((R^{P})^{c})t^{2c}}{c!(2c)!Γ((n − q)/2 + c)2^{c}} .

Note that we define (n − q)(n − q + 2) ⋯ (n − q + 2c − 2) = 1 if c = 0.

Proof : We define the mapping ψ_{c} be setting

ψ_{c}(u_{1}, …, u_{c}) = \frac{1}{(c!)^{2}} ∑_{a_{1},...,a_{c}=1}^{q} ∑_{σ,π∈𝔖_{c}} ε_{σ}ε_{π}T_{a_{σ(1)}a_{π(1)}u_{1}} ⋯ T_{a_{σ(c)}a_{π(c)}u_{c}}  (3.62)

for u_{1}, …, u_{c} ∈ (T_{p}P)^{⊥}. By remark 3.57 it’s straightforward to show that ψ_{c} is symmetric and by methods from linear algebra it can be shown that we can expand det(δ_{a,b} − tT_{abu}) by minors which gives us:

det(δ_{a,b} − tT_{abu}) = ∑_{c=0}^{q} ψ_{c}(u, …, u)(−t)^{c},  (3.63)

for u ∈ (T_{p}P)^{⊥}. We want to integrate ψ_{c} over S^{n−q−1}. We note that due to symmetry the integral of ψ_{2c+1} is equal to 0. Let (e_{q+1}, …, e_{n}) be an orthonormal basis for (T_{p}P)^{⊥}. By corollary 3.76 we have that

\begin{aligned}I_{2c}(ψ_{2c}) & = \frac{1 · 3 ⋯ (2c − 1) · 2π^{(n−q)/2} ∑_{i_{1},...,i_{c}=q+1}^{n} ψ_{2c}(e_{i_{1}}, e_{i_{1}}, …, e_{i_{c}}, e_{i_{c}})}{(n − q)(n − q + 2) ⋯ (n − q + 2c − 2)Γ((n − q)/2)} \\ & = \frac{(2c)!2π^{(n−q)/2}}{c!4^{c}Γ((n − q)/2 + c)} ∑_{i_{1},...,i_{c}=q+1}^{n} ψ_{2c}(e_{i_{1}}, e_{i_{1}}, …, e_{i_{c}}, e_{i_{c}}). \end{aligned} (3.64)

We will have use for the following lemma which is a special case of an equality known as the Gauss equation. It’s proof can be found for example in [On].

64 CHAPTER 3. TUBES 3.78 Lemma For a submanifold P ⊂ ℝ^{n} the following equation is valid:

R_{abcd}^{P} = ∑_{i=q+1}^{n}(T_{ace_{i}}T_{bde_{i}} − T_{ade_{i}}T_{bce_{i}}) = ⟨T_{ac}, T_{bd}⟩ − ⟨T_{ad}, T_{bc}⟩.

□

From this we obtain

\begin{matrix}∑_{i_{1},...,i_{c}=q+1}^{n} ψ_{2c}(e_{i_{1}}, e_{i_{1}}, …, e_{i_{c}}, e_{i_{c}}) \\ = \frac{1}{((2c)!)^{2}} ∑_{i_{1},...,i_{c}=q+1}^{n} ∑_{a_{1},...,a_{c}=1}^{q} ∑_{σ,π∈𝔖_{2c}} ε_{σ}ε_{π}\{T_{a_{σ(1)}a_{π(1)}e_{i_{1}}}T_{a_{σ(2)}a_{π(2)}e_{i_{1}}} \\ ⋯ T_{a_{σ(2c−1)}a_{π(2c−1)}e_{ic}}T_{a_{σ(2c)}a_{π(2c)}e_{ic}}\}. \end{matrix}  (3.65)

We have that

⟨T_{a_{σ(s)}a_{π(t)}}, T_{a_{σ(u)}a_{π(v)}}⟩ = ⟨T_{a_{σ(s)}a_{π(t)}}, ∑_{i=q+1}^{n} (⟨T_{a_{σ(u)}a_{π(v)}}, e_{i}⟩e_{i})⟩
= ∑_{i=q+1}^{n}⟨T_{a_{σ(s)}a_{π(t)}}, ⟨T_{a_{σ(u)}a_{π(v)}}, e_{i}⟩e_{i}⟩ = ∑_{i=q+1}^{n}⟨T_{a_{σ(s)}a_{π(t)}}, e_{i}⟩⟨T_{a_{σ(u)}a_{π(v)}}, e_{i}⟩

= ∑_{i=q+1}^{n} T_{a_{σ(s)}a_{π(t)}e_{i}}T_{a_{σ(u)}a_{π(v)}e_{i}}

since (e_{q+1}, …, e_{n}) is an orthonormal basis for (T_{p}P)^{⊥}. Hence we can simplify the right hand side of (3.65) to

\frac{1}{((2c)!)^{2}} ∑_{a_{1},...,a_{c}=1}^{q} ∑_{σ,π∈𝔖_{2c}} ε_{σ}ε_{π}\{⟨T_{a_{σ(1)}a_{π(1)}}, T_{a_{σ(2)}a_{π(2)}}⟩ ⋯ ⟨T_{a_{σ(2c−1)}a_{π(2c−1)}}, T_{a_{σ(2c)}a_{π(2c)}}⟩\} (3.66)

3.79 Lemma The expression (3.66) above is the same as

\begin{matrix}\frac{1}{2^{c}((2c)!)^{2}} ∑_{a_{1},...,a_{c}=1}^{q} ∑_{σ,π∈𝔖_{2c}} ε_{σ}ε_{π}\{ (⟨T_{a_{σ(1)}a_{π(1)}}, T_{a_{σ(2)}a_{π(2)}}⟩ − ⟨T_{a_{σ(1)}a_{π(2)}}, T_{a_{σ(2)}a_{π(1)}}⟩) \\ ⋯ & (⟨T_{a_{σ(2c−1)}a_{π(2c−1)}}, T_{a_{σ(2c)}a_{π(2c)}}⟩ − ⟨T_{a_{σ(2c−1)}a_{π(2c)}}, T_{a_{σ(2c)}a_{π(2c−1)}}⟩) & \} \end{matrix} (3.67)

3.5. WEYL’S TUBE FORMULA 65 Proof: Let j_{1}, …, j_{c} ∈ {0, 1}. We define ρ_{j_{1},...,j_{c}} ∈ 𝔖_{2c} to be (1, 2)^{j_{1}} ⋯ (2c − 1, 2c)^{j_{c}}. We note that

ε_{ρ_{j_{1},...,jc}}(−1)^{∑_{i=1}^{c} j_{i}} = 1.  (3.68)

Now (3.67) can be written as

\frac{1}{2^{c}((2c)!)^{2}} ∑_{a_{1},...,a_{c}=1}^{q} ∑_{σ,π∈𝔖_{2c}} ε_{σ}ε_{π} ∑_{j_{1},...,j_{c}=0}^{1}(−1)^{j_{1}}⟨T_{a_{σ(1)}a_{π(1+j_{1})}}, T_{a_{σ(2)}a_{π(2−j_{1})}}⟩

⋯ (−1)^{j_{c}}⟨T_{a_{σ(2c−1)}a_{π(2c−1+jc)}}, T_{a_{σ(2c)}a_{π(2c−jc)}}⟩

= \frac{1}{2^{c}((2c)!)^{2}} ∑_{a_{1},...,a_{c}=1}^{q} ∑_{j_{1},...,j_{c}=0}^{1} ∑_{σ,π∈𝔖_{2c}} ε_{σ}ε_{πρ_{j_{1},...,jc}}

(−1)^{j_{1}}⟨T_{a_{σ(1)}a_{πρ_{j_{1},...,jc} (1+j_{1})}}, T_{a_{σ(2)}a_{πρ_{j_{1},...,jc} (2−j_{1})}}⟩ ⋯ (−1)^{j_{c}}⟨T_{a_{σ(2c−1)}a_{πρ_{j_{1},...,jc} (2c−1+jc)}}, T_{a_{σ(2c)}a_{πρ_{j_{1},...,jc} (2c−jc)}}⟩

which gives us (3.66) since the last expression is independent of j_{1}, …, j_{c} by (3.68).

□

We now use the Gauss equation and obtain

∑_{i_{1},...,i_{c}=q+1}^{n} ψ_{2c}(e_{i_{1}}, e_{i_{1}}, …, e_{i_{c}}, e_{i_{c}})

= \frac{1}{2^{c}((2c)!)^{2}} ∑_{a_{1},...,a_{c}=1}^{q} ∑_{σ,π∈𝔖_{2c}} ε_{σ}ε_{π} \{R_{a_{σ(1)}a_{σ(2)}a_{π(1)}a_{π(2)}}^{P} ⋯ R_{a_{σ(2c−1)}a_{σ(2c)}a_{π(2c−1)}a_{π(2c)}}^{P}\}

= \frac{2^{c}}{((2c)!)^{2}} ∑_{a_{1},...,a_{c}=1}^{q}(R^{P})^{c}(a_{1}, …, a_{2c})(a_{1}, …, a_{2c})

= \frac{2^{c}}{((2c)!)^{2}} C^{2c}((R^{P})^{c}).

By this and (3.64) we get

I_{2s}(ψ_{2c}) = \frac{2π^{(n−q)/2}C^{2c}((R^{P})^{c})}{c!(2c)!2^{c}Γ((n − q)/2 + c)}.

Hence by (3.63) we obtain

∫_{S^{n−q−1}} det(δ_{a,b} − tT_{abu}) du = ∑_{c=0}^{q}(−t)^{c} ∫_{S^{n−q−1}} ψ_{c}(u, …, u) du

= \begin{matrix}[ & \frac{q}{2} & ] \\ ∑ \\ c=0 \end{matrix} I_{2s}(ψ_{2c})t^{2c} = 2π^{(n−q)/2} \begin{matrix}[ & \frac{q}{2} & ] \\ ∑ \\ c=0 \end{matrix} \frac{C^{2c}((R^{P})^{c})t^{2c}}{c!(2c)!2^{c}Γ((n − q)/2 + c)}

which was what was wanted.

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Now we’re ready to prove Weyl’s Tube formula. In order to prevent misunderstanding I’ll write the theorem exactly as it is written in [Gr].

3.80 Theorem (Weyl’s Tube Formula) Let P be a q-dimensional topologically embedded submanifold in Euclidean space ℝ^{n}. Assume that P has compact closure, and that every point in the tube T (P, r) has a unique shortest geodesic connecting it with P. Then the volume V_{P}^{ℝ^{n}}(r) of T (P, r) is given by

V_{P}^{ℝ^{n}}(r) = \frac{(πr^{2})^{(n−q)/2}}{Γ((n − q)/2 + 1)} \begin{matrix}[ & \frac{q}{2} & ] \\ ∑ \\ c=0 \end{matrix} \frac{k_{2c}(P)r^{2c}}{(n − q + 2)(n − q + 4) ⋯ (n − q + 2c)}. (3.69)

Proof: By lemma 3.43 we have that

A_{P}^{ℝ^{n}}(t) = t^{n−q−1} ∫_{P} ∫_{S^{n−q−1}} ϑ_{u}(r)du dP,  (3.70)

where ϑ_{u}(r) = det(δ_{a,b} − tT_{abu}) by lemma 3.45. By theorem 3.77 and definition 3.59 we therefore have

\begin{aligned}A_{P}^{ℝ^{n}}(r) & = \frac{2π^{(n−q)/2}}{Γ((n − q)/2)} \begin{matrix}[ & \frac{q}{2} & ] \\ ∑ \\ c=0 \end{matrix} \frac{k_{2c}(P)t^{n−q−1+2c}}{(n − q)(n − q + 2)(n − q + 4) ⋯ (n − q + 2c − 2)} \\ & = \frac{π^{(n−q)/2}}{Γ((n − q)/2 + 1)} \begin{matrix}[ & \frac{q}{2} & ] \\ ∑ \\ c=0 \end{matrix} \frac{k_{2c}(P)t^{n−q−1+2c}}{(n − q + 2)(n − q + 4) ⋯ (n − q + 2c − 2)}. \end{aligned}

Now, since V_{P}^{ℝ^{n}}(r) = ∫_{0}^{r} A_{P}^{ℝ^{n}}(t) dt by lemma 3.44, Weyl’s Tube Formula follows.

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3.81 Corollary The coefficients in (3.69) depends only on the curvature of P and thus the tube volume V_{P}^{ℝ^{n}}(r) depends only on P and r and not on the particular way that P is embedded in ℝ^{n}.

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3.5.4 Calculating the volume of B_{p}^{n}(r)

I’ll now calculate the volume of B_{p}^{n}(r) in two ways. We have already seen one way how to do this in chapter 1. Now we’ll use other methods. We observe that B_{p}^{n}(r) is just a geodesic ball with center p and radius r.

(1) Using corollary 3.51: By remark 3.52 we have that the sectional curvature of ℝ^{n} is 0. Thus we have by corollary 3.51 that

B_{p}^{n}(r) = \frac{2π^{n/2}}{Γ(n/2)} lim_{λ→0} ∫_{0}^{r} (\frac{sin(t\sqrt{λ})}{\sqrt{λ}})^{n−1} dt

= \frac{2π^{n/2}}{Γ(n/2)} ∫_{0}^{r} t^{n−1} lim_{λ→0} (\frac{sin(t\sqrt{λ})}{t\sqrt{λ}})^{n−1} dt

= \frac{2π^{n/2}}{Γ(n/2)} ∫_{0}^{r} t^{n−1}dt = \frac{2π^{n/2}r^{n}}{nΓ(n/2)},

which agrees with (1.3).

(2) Using Weyl’s tube formula: As mentioned in remark 3.60, k_{0}(P) = Vol(P). So when P is a point we have Vol(P) = 1. From Weyl’s tube formula we have

B_{p}^{n}(r) = \frac{(πr^{2})^{(n−0)/2}}{Γ((n − 0)/2 + 1)} \begin{matrix}[ & \frac{0}{2} & ] \\ ∑ \\ c=0 \end{matrix} \frac{k_{2c}(P)r^{2c}}{(n − q + 2)(n − q + 4) ⋯ (n − q + 2c)}

= \frac{(πr^{2})^{n/2}}{Γ(n/2 + 1)}k_{0}(P)r^{0}

= \frac{2π^{n/2}r^{n}}{nΓ(n/2)},

which is the same result as before.

3.6 Kähler manifolds

3.82 Remark In this section I’ll introduce some concept from the theory of complex manifolds. Some of the lemmas and theorems in this section will be stated without proof. For more details see [Gr]. For some introduction to complex manifolds, see for example [1].

3.83 Definition Let N be a smooth manifold. A (1,1)-tensor field J : X (N) → X (N) which satisfies J^{2} = −I is called an almost complex structure on N. N together with this structure is called an almost complex manifold.

3.84 Remark If J and N are as in the above definition, then J is a field of linear transformations which sends every m ∈ N to a linear map J_{m} : T_{m}N → T_{m}N such that J_{m}^{2} = −I_{m}, where I_{m} is the identity map on T_{m}N. If dim N = n, then we have

(−1)^{n} = det(−I_{m}) = det(J_{m}^{2}) = (det(J_{m}))^{2},

for all m ∈ N, so N is even dimensional.

3.85 Definition Let ⟨·, ·⟩ denote the Riemannian metric on M, and assume that M has an almost complex structure J.

(1) ⟨·, ·⟩ is said to be an almost Hermitian metric and M to be an almost Hermitian manifold if ⟨JX, JY ⟩ = ⟨X, Y ⟩ for all X, Y ∈ X (M).

(2) Let M be an almost Hermitian manifold. The 2-form F : X (M) × X (M) → ℝ defined by setting F (X, Y) = ⟨JX, Y ⟩ for all X, Y ∈ X is called a Kähler form of M with respect to J.

3.86 Lemma An almost complex n-dimensional manifold N is orientable and can be made into an almost Hermitian manifold.

Proof: N has a Riemannian metric (·, ·) by remark 2.59. We define another Riemannian metric ⟨·, ·⟩ : X (N) × X (N) → ℝ by

⟨X, Y ⟩ = (X, Y) + (JX, JY).

Now

⟨JX, JY ⟩ = (JX, JY) + (−X, −Y) = (JX, JY) + (−1)^{2}(X, Y) = ⟨X, Y ⟩,

so ⟨·, ·⟩ is an almost Hermitian metric on M. By remark 3.84 dim N is even, so F^{n/2} is well defined. It is straightforward to show that F^{n/2} is a non-vanishing n-form and thus N is orientable by remark 2.23.

□

3.87 Definition Let M be an almost Hermitian manifold, and let m ∈ M.

(1) A real orthonormal basis of the form

(1, 1^{∗}, …, n, n^{∗}) := (e_{1}, Je_{1}, …, e_{n}, Je_{n})

is called a holomorphic orthonormal frame. Note that Ji = i^{∗}, Ji^{∗} = −i for all i = 1, …, n.

(2) Let X ∈ X (M). Then we define the covariant derivative of J as

∇_{X}(J) = ∇_{X}J − J∇_{X}.

(See also definition 3.22). If ∇_{X}(J)Y = 0 for all X, Y ∈ X then J is said to be parallel.

(3) If J is parallel then M is called a Kähler manifold.

The following lemma shows the naturality of J. I’ll go through the proof since I find it instructive.

3.88 Lemma Let M be a Kähler manifold, and let X, Y ∈ X (M). Then

R_{JXJY}  = R_{XY}  (Kähler identidy)
ρ(JX, JY) = ρ(X, Y)

Proof : We have for all X, Y, Z, W ∈ X (M) that

⟨R_{JXJY} Z, W ⟩
= ⟨R_{ZW} JX, JY ⟩  (by (2.41))
= ⟨(∇_{[Z,W]} − [∇_{Z}, ∇_{W}])JX, JY ⟩  (by (2.38))
= ⟨(∇_{[Z,W]} − ∇_{Z}∇_{W} + ∇_{W} ∇_{Z})JX, JY ⟩ (def. of Lie brackets)
= ⟨J(∇_{[Z,W]} − ∇_{Z}∇_{W} + ∇_{W} ∇_{Z})X, JY ⟩ (since J is parallel)
= ⟨JR_{ZW} X, JY ⟩  (by (2.38))
= ⟨R_{ZW} X, Y ⟩  (M is almost Hermitian)
= ⟨R_{XY} Z, W ⟩  (by (2.41))

so this proves the first part. Let (E_{1}, …, E_{2n}) be a real local orthonormal frame for T M. Then (JE_{1}, …, JE_{2n}) is also a local orthonormal frame for T M. By using the same definitions and theorems as before we have

ρ(JX, JY) = ∑_{a=1}^{2n} R_{JXE_{a}JY E_{a}} = ∑_{a=1}^{2n} R_{JXJE_{a}JY JE_{a}} = ∑_{a=1}^{2n}⟨R_{JXJE_{a}}JY, JE_{a}⟩

= ∑_{a=1}^{2n}⟨R_{XE_{a}}JY, JE_{a}⟩ = ∑_{a=1}^{2n}⟨R_{JY JE_{a}}X, E_{a}⟩ = ∑_{a=1}^{2n}⟨R_{Y E_{a}}X, E_{a}⟩

= ρ(Y, X) = ρ(X, Y).

□

3.89 Remark For a holomorphic orthonormal frame we have by (2.39)–(2.42)

\begin{aligned}ρ(X, Y) & = ∑_{i=1}^{n}(R_{xiyi} + R_{xi^{∗}yi^{∗}}) = ∑_{i=1}^{n}(R_{xiy^{∗}i^{∗}} − R_{xi^{∗}y^{∗}i}) \\ & = ∑_{i=1}^{n}(−R_{ixy^{∗}i^{∗}} − R_{y^{∗}ixi^{∗}}) = ∑_{i=1}^{n} R_{xy^{∗}ii^{∗}}. \end{aligned}

3.90 Definition (1) Let N be an almost complex manifold. A two dimensional subspace T_{m}Π of T_{m}N which can be written on the form span{x, Jx} where x ∈ T_{m}N is called a holomorphic section.

(2) Let M be a Kähler manifold. The sectional curvature (see definition 2.45) restricted to holomorphic sections of tangent spaces is called the holomorphic sectional curvature and is denoted by

K_{hol}.

{3.91 Remark \atop (1)} If M is Hermitian then ⟨JX, X⟩ = 0 for all X ∈ X (M).

(2) K_{hol} can be regarded as a function that assigns each unit tangent vector x to M a real number K_{hol}(x). We extend the definition to all tangent vectors x to M by

K_{hol}(x) = R_{xJxxJx}.

3.92 Definition Let M be an almost Hermitian manifold with an almost Hermitian structure J.

(1) If there exist a constant λ such that

K_{hol}(x) = 4λ∥x∥^{4}  (3.71)

for all x tangent to M, then M is said to have a constant sectional curvature 4λ.

(2) If N is a Kähler manifold of dimension 2n which satisfies (3.71) we say that N is a space of constant holomorphic sectional curvature 4λ and denote it by

𝕂_{hol}^{n}(λ).

Note that if we regard N as a complex manifold, its complex dimension is n.

The following theorem is well known in the theory of complex manifolds.

3.93 Theorem Let 𝕂_{hol}^{n}(λ) be a Kähler manifold of constant holomorphic sectional curvature 4λ. Then it’s sectional curvature of linearly independent x and y is given by

K_{xy} = λ (1 + \frac{3⟨Jx, y⟩^{2}}{∥x∥^{2}∥y∥^{2} − ⟨x, y⟩^{2}} ) .

Thus

λ < K < 4λ.
□

3.94 Definition Let R denote the curvature tensor of M.

(1) As before we define the covariant derivate of R as a tensor field by

\begin{matrix}∇_{V} (R) & _{W XY Z} = V (R_{W XY Z}) − R_{∇_{V} W XY Z} \\ − R_{W ∇_{V} XY Z} − R_{W X∇_{V} Y Z} − R_{W XY ∇_{V} Z} \end{matrix}

for V, X, Y, Z, W ∈ X (M). (For more details see for example page 53 in [Lee]).

(2) M is said to be locally symmetric if ∇R = 0.

3.95 Definition Let P and ν be as before.

(1) Let (p, u) ∈ ν. Then we define a map R_{u} : T_{p}M → T_{p}M by

R_{u}x = R_{ux}u

for x ∈ T_{p}M. We note the similarities between R_{u} and R_{N}. By (2.41) it follows that R_{u} is symmetric, i.e. ⟨R_{u}x, y⟩ = ⟨x, R_{u}y⟩ for all x, y ∈ T_{p}M.

(2) Let ξ be a unit speed geodesic on M. Then we define a map R(t) : T_{ξ(t)}M →

T_{ξ(t)}M by

R(t)x = R_{ξ′(t)x}ξ′(t).

This generalizes R(t) defined in corollary 3.27.

(3) Let (p, u) ∈ ν, T_{u} : T_{p}P → T_{p}P be the Weingarten map and I : (T_{p}P)^{⊥} → (T_{p}P)^{⊥} be the identity map.. Since we have the ordinary identification T_{p}M = T_{p}P ⊕ (T_{p}P)^{⊥} we have a natural map T_{u} ⊕ I : T_{p}M → T_{p}M, defined in an obvious way.

(4) If for all (p, u) ∈ ν, where u is a unit vector we have

T_{u} ⊕ I is diagonalizable ⇔ R_{u} is diagonalizable

then P is said to be compatible with M.

3.6.1 A vital Lemma without proof

{3.96 Lemma \atop (1)} K^{n}(λ) is locally symmetric.

(2) 𝕂_{hol}^{n}(λ) is locally symmetric.

(3) Any submanifold of K^{n}(λ) is compatible with K^{n}(λ).

(4) Any complex submanifold of 𝕂_{hol}^{n}(λ) is compatible with 𝕂_{hol}^{n}(λ).

(5) If M is locally symmetric, and P ⊂ M is a submanifold of M and compatible with M then the eigenvectors of the shape operator of the tubular hypersurface P_{t} can be chosen parallel along geodesics normal to P.

□

3.6.2 Volume of a geodesic ball in 𝕂_{hol}^{n}(λ)

3.97 Lemma The volume of a geodesic ball in 𝕂_{hol}^{n}(λ) of constant holomorphic sectional curvature 4λ is given by

V_{m}^{𝕂_{hol}^{n}(λ)}(r) = \frac{1}{n!} (\frac{π}{λ})^{n} (sin(r\sqrt{λ}))^{2n}.

Proof: Let u ∈ T_{m}𝕂_{hol}^{n}(λ), with ∥u∥ = 1. Let e_{1} = u and extend e_{1} to a holomorphic orthonormal frame (e_{1}, Je_{1}, …, e_{n}, Je_{n}) for T_{m}𝕂_{hol}^{n}(λ) which diagonalizes R_{u}. Let (E_{1}(t), …, JE_{n}(t)) be a holomorphic frame along ξ which coincides with (e_{1}, Je_{1}, …, e_{n}, Je_{n}) at m. Clearly m is compatible with M and thus we have by theorem 3.96 that E_{1}(t), …, JE_{n}(t) are eigenvectors both for R(t) and S(t). Here S(t) is the shape operator at ξ(t) of the geodesic ball

{m′ ∈ 𝕂_{hol}^{n}(λ)| dist(m, m′) = t}.  (3.72)

Thus by theorem 3.93 we have the following Riccati differential equations for the principal curvature functions on (3.72):

\begin{aligned}κ′_{2} & = κ_{2}^{2} + 4λ \\ κ′_{i} & = κ_{i}^{2} + λ (3 ⩽ i ⩽ 2n) \end{aligned}

where

\xymatrix{k_{i}(t) \ar[r]_{t→0+} & −∞,}

for i = 2, …, 2n. By theorem 3.47 we have

κ_{2}(t) = \frac{−2\sqrt{λ}}{tan(2t\sqrt{λ})},  κ_{i}(t) = \frac{−\sqrt{λ}}{tan(t\sqrt{λ})}, (3 ⩽ i ⩽ 2n)  (3.73)

By theorem 3.40 we have

\frac{ϑ′_{u}(t)}{ϑ_{u}(t)} = −\frac{2n − 1}{t} + \frac{2\sqrt{λ}}{tan(2t\sqrt{λ})} + \frac{(2n − 2)\sqrt{λ}}{tan(t\sqrt{λ})}.

We solve this in a similar fashion as in the proof of theorem 3.49 and get

ln ϑ_{u}(t) = −(2n − 1) ln(t) + ln(sin(2t\sqrt{λ})) + (2n − 2) ln(sin(t\sqrt{λ})) + C

where C ∈ ℝ^{n}, or equivalently

t^{2n−1}ϑ_{u}(t) = e^{C} sin(2t\sqrt{λ})(sin(t\sqrt{λ}))^{2n−2}.

Due to our initial condition ϑ_{u}(0) = 1, we can easily calculate that e^{C} = \frac{1}{2}λ^{−n+1/2}. Since \frac{1}{2} sin(2t\sqrt{λ}) = sin(t\sqrt{λ}) cos(t\sqrt{λ}) we have that

t^{2n−1}ϑ_{u}(t) = (sin(t\sqrt{λ}))^{2n−1} cos(t\sqrt{λ})λ^{−n+1/2}.  (3.74)

Now we’ll use (1.2), lemmas (3.43) and (3.44) along with the above to obtain the desired volume, namely

V_{m}^{𝕂_{hol}^{n}(λ)}(r) = ∫_{0}^{r} ∫_{S^{2n−1}}(sin(t\sqrt{λ}))^{2n−1} cos(t\sqrt{λ})λ^{−n+1/2} du dt

= \frac{2π^{n}}{Γ(n)} ∫_{0}^{r}(sin(t\sqrt{λ}))^{2n−1} cos(t\sqrt{λ})λ^{−n+1/2} dt

= \frac{1}{n!} (\frac{π}{λ})^{n} (sin(r\sqrt{λ}))^{2n}.

□

3.98 Definition If ϑ_{u}(t) is independent of which unit tangent vector u on M we choose, then M is said to be a harmonic space.

3.6.3 Complex projective space ℂP^{n}(λ)

ℂP^{n}(λ) is a complex manifold, defined to be the space of complex lines in ℂ^{n+1}. As the notation indicates, then ℂP^{n}(λ) is a harmonic space. If we define the following equivalence relation on S^{2n+1}

(z_{0}, …, z_{n}) ∼ (w_{0}, …, w_{n}) ⇔ ∃ a ∈ ℂ, |a| = 1 : ∀i : z_{i} = aw_{i},

where z_{1}, …, z_{n}, w_{1}, …, w_{n} ∈ ℂ then it is can be show that ℂP^{n}(1) is the set of equivalence classes of ∼. So ℂP^{n}(1) is a manifold and the projection π : S^{2n+1} → ℂP^{n}(1) is clearly differentiable and surjective.

More important is that ℂP^{n}(1) can be made into a Kählerian manifold, by using the so-called Fubini-Study metric. For details of its construction and references see [Gr]. Thus it follows that ℂP^{n}(λ) is ℂP^{n}(1) with the Fubini-Study metric divided by λ^{2}.

3.99 Theorem The volume of ℂP^{n}(λ) is given by

\frac{1}{n!} (\frac{π}{λ})^{n}

Proof: We have that

A_{m}^{ℂP^{n}(λ)}(t) = \frac{2π^{n}}{(n − 1)!λ^{n−1/2}} (sin(t\sqrt{λ}))^{2n−1} cos(t\sqrt{λ})  (3.75)

by lemmas 3.97 and 3.43. Since ϑ_{u}(t) is independent of u so is the distance from m to the first zero of ϑ_{u}(t). So for the first t > 0 such that A_{m}^{ℂP^{n}(λ)}(t) = 0 is when the geodesic ball

{m′ ∈ ℂP^{n}(λ)| dist(m, m′) < t}

74 CHAPTER 3. TUBES exactly covers ℂP^{n}. This apparently happens the first time when cos(t\sqrt{λ}) = 0, i.e. when t = π/(2\sqrt{λ}). By lemma 3.44 we obtain the volume of ℂP^{n}, namely

V_{m}^{ℂP^{n}}(π/(2\sqrt{λ})) = ∫_{0}^{π/(2\sqrt{λ})} A_{m}^{ℂP^{n}(λ)}(t) dt

= \frac{2π^{n}}{(n − 1)!λ^{n−1/2}} ∫_{0}^{π/(2\sqrt{λ})}(sin(t\sqrt{λ}))^{2n−1} cos(t\sqrt{λ}) dt

= \frac{2π^{n}}{(n − 1)!λ^{n−1/2}} [\frac{1}{2n\sqrt{λ}} sin(t\sqrt{λ})^{2n}]_{0}^{π/(2\sqrt{λ})} = \frac{1}{n!} (\frac{π}{λ})^{n} ,

which gives the desired volume.

□

Invariant measure on Sl(n, ℤ)\Sl(n, ℝ)

The main aim of this chapter is to calculate an invariant measure of the quotient space

Sl(n, ℤ)\Sl(n, ℝ) = {Sl(n, ℤ) · x| x ∈ Sl(n, ℝ)}.

I’ll also make a few speculations about fibre bundles and their relation to volume.

4.1 Topics from linear algebra

4.1 Definition An element in M(n, ℝ) which has 1 in the (i, j)-th entry and 0 elsewhere is denoted by E_{i,j}. The elements E_{i,j} ∈ M(n, ℝ) where i ≠ j and E_{i,i} − E_{i+1,i+1} where i ∈ [[1, n − 1]] form a basis for 𝔰𝔩(n, ℝ) [Hum]. This basis is called the zero trace basis.

4.2 Definition Let X be a matrix. We will denote the i-th row by L_{i}. The following procedures

(1) L_{i} → αL_{i}, meaning multiply row i by a non-zero scalar α;

(2) L_{i} ↔ L_{j}, meaning interchange rows i and j;

(3) L_{j} → αL_{i} + L_{j}, meaning linear combination αL_{i} + L_{j} replaces row j;

are called row operations and we write r(X) for a row operator r which is applied to X. For two row operators r, s we define sr(X) := s(r(X)). Two (n×m)-matrices X, X′ are said to be row equivalent if there exist a finite sequence of row operations (r_{1}, …, r_{t}) such that r_{t}r_{t−1} ⋯ r_{1}(X) = X′.

4.3 Remark It’s well known that every non-singular (n × n)-matrix is row-equivalent to the identity matrix I_{n}. Furthermore if (r_{1}, …, r_{t}) is a sequence of row operations such that r_{t}r_{t−1} ⋯ r_{1}(X) = I_{n} then

r_{t}r_{t−1} ⋯ r_{1}(I_{n}) = X^{−1}.

76 CHAPTER 4. INVARIANT MEASURE ON SL(N, ℤ)\SL(N, ℝ)

4.4 Definition (1) We define S_{i,j} := I_{n} − E_{i,i} − E_{j,j} + E_{i,j} − E_{j,i} ∈ Sl(n, ℤ) for all i, j such that

i ≠ j.

(2) Let x ∈ ℝ. Then we define the functions (a) floor(x) which rounds x to the nearest integer towards minus infinity;

(b) fix(x) which rounds x to the nearest integer towards zero.

(3) Let A be a (n × n)-matrix. For k ∈ [[1, n]] we define

Q_{k}(A) := {a_{k,k}, a_{k+1,k}, …a_{n,k}}.

If the number of non-zero elements in Q_{k}(A) equals 1 we let q(Q_{k}(A)) be the line number i of the non-zero element in Q_{k}(A).

(4) Let N,N'∈ ℕ, where N⩽N'and denote by P(i) an expression (or a finite sequence of expressions) which depends on i. The following

for i=N:N'
   P(i)
end

means that we apply P(i) for all integer values i between N and N' in an increasing order including N and N'.

4.5 Theorem Let x ∈ M_{m}(n, ℤ) and let Sl(n, ℤ)·x be it’s orbit. Then there exist a unique element

y = [ \begin{matrix}y_{1,1} & ⋯ & y_{1,n} \\ ⋮ & ... & ⋮ \\ y_{n,1} & ⋯ & b_{n,n} \end{matrix} ] ∈ Sl(n, ℤ) · x

which satisfies the following conditions:

(1) y_{j,j} > 0 for all j;

(2) y_{i,j} = 0 for all i, j where i > j;

(3) 0 ⩽ y_{i,j} < y_{j,j} for all i, j where i < j. Proof: Existence: Let

A = [ \begin{matrix}a_{1,1} & ⋯ & a_{1,n} \\ ⋮ & ... & ⋮ \\ a_{n,1} & ⋯ & a_{n,n} \end{matrix} ] ∈ Sl(n, ℤ) · x.

For a ∈ ℝ we define

sgn(a) := \begin{cases}1 & if a > 0; \\ 0 & if a = 0; \\ −1 & if a < 0. \end{cases}

We define

sgn(A) = [ \begin{matrix}sgn(a_{1,1}) & ⋯ & sgn(a_{1,n}) \\ ⋮ & ... & ⋮ \\ sgn(a_{n,1}) & ⋯ & sgn(a_{n,n}) \end{matrix} ]

In the following diagram we show how we can multiply A from the left side with elements in Sl(n, ℤ) to obtain a matrix which satisfies conditions (1) and (2):

[figure: \begin{matrix}START \\ k = 1 \end{matrix}; k = n; false; true; \begin{matrix}A → sgn(A) · A \\ STOP \end{matrix}; \begin{matrix}∃ i_{1}, i_{2} ⩾ k : i_{1} ≠ i_{2} \\ a_{i_{1},k} ≠ 0, a_{i_{2},k} ≠ 0 \end{matrix}; true; false; k → k + 1; i := q(Q_{k}(A)) = k; false; true; |_{\frac{a_{i_{1},k}}{a_{i_{2},k}}} | ⩾ 1; false; true; \begin{matrix}a_{i_{1},k} ≠ 0, \\ a_{i_{2},k} ≠ 0 \end{matrix}; true; false; A → S_{i,k} · A; A → S_{i_{1},i_{2}} · A; \begin{matrix}A → \\ (I − E_{i_{1},i_{2}} · fix\frac{a_{i_{1},k}}{a_{i_{2},k}} ) · A \end{matrix}]

A rounded box means that we perform what is written in the box. If the box is squared then we check if the statement in the box is true or false and then follow a suitable arrow. Since every cycle through the lowest three or four boxes brings one number in Q_{k}(A) closer to zero (though maybe reversing the order of terms), this diagram eventually comes to an end. In the last rounded box we have that sgn(A) ∈ Sl(n, ℤ) since det M_{m}(n, ℤ) ⩾ 1. After this procedure we apply the following to A.

for j=2:n
   for i=1:l-1
      A=(I-E_{i,j}floor(a_{i,j}/a_{j,j}))*A
   end
end

This gives the existence.

Uniqueness: Let’s assume that A, A′ ∈ Sl(n, ℤ) · x which satisfies the given conditions. Then there exist C ∈ Sl(n, ℤ) such that A′ = CA or, equivalently C = A′A^{−1}. We will first prove the following lemma 4.6 Lemma If we write

B = [ \begin{matrix}b_{1,1} & ⋯ & b_{1,n} \\ & ... & ⋮ \\ 0 & & b_{n,n} \end{matrix} ] := [ \begin{matrix}a_{1,1} & ⋯ & a_{1,n} \\ & ... & ⋮ \\ 0 & & a_{n,n} \end{matrix} ]^{−1}

then b_{i,i} = a_{i,i}^{−1} for all i and

b_{l,k} = − ∑_{j=0}^{k−l−1} \frac{a_{l,k−j}}{a_{l,l}} · b_{k−j,k},  (4.1)

for all l, k such that l < k.

Proof: We define the following row operations

(1) r_{i,i} : L_{i} → a_{i,i}^{−1}L_{i} for all i;

(2) r_{l,k} : L_{l} → −a_{l,k}a_{l,l}^{−1}L_{k} + L_{l} for all l, k such that l < k.

We also define the following combinations of row operators

(1) R = r_{1,1}r_{2,2} ⋯ r_{n,n}

(2) R_{k} = r_{1,k}r_{2,k} ⋯ r_{k−1,k}, (for all k ⩾ 2).

Then R_{2}R_{3} ⋯ R_{n}R(A) = I_{n} and thus B = R_{2}R_{3} ⋯ R_{n}R(I_{n}) by remark 4.3. This is just a Gauss elimination process. We clearly have that b_{i,i} = a_{i,i}^{−1} for all i. We want to calculate b_{l,k} for l < k. We shall look at how R_{2} ⋯ R_{k} reacts on R_{k+1} ⋯ R_{n}R(I_{n}), namely:

B = R_{2} ⋯ R_{k} [ \begin{matrix}b_{1,1} & & & & & 0 & & ⋮ & & · & & · & & · \\ & & & ... & & & & ⋮ & & · & & · & & · \\ 0 & & & & & b_{k,k} & & ⋮ & & b_{k,k+1} & & ⋯ & & b_{k,n} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ & & & & & & & ⋮ & & b_{k+1,k+1} & & ⋯ & & b_{k+1,n} \\ & & & 0 & & & & ⋮ & & & & ... & & ⋮ \\ & & & & & & & ⋮ & & & & & & b_{n,n} \end{matrix} ] .

We note that the (l, k)-th entry in R_{k+1} ⋯ R_{n}R(I_{n}) is 0 for l < k. Furthermore the (j, k)-th entry in R_{j+1} ⋯ R_{n}R(I_{n}) is b_{j,k}. Now R_{j} = r_{1,j}r_{2,j} ⋯ r_{l,j} ⋯ r_{j−1,j} for all j with l < j ⩽ k. The only terms in R_{j} which alters the (l, k)-th entry when R_{j} is applied to R_{j+1} ⋯ R_{n}R(I_{n}) is r_{l,j}, namely r_{l,j} : L_{l} → −a_{l,j}a_{l,l}^{−1}L_{j} + L_{l}. Thus the term

−\frac{a_{l,j}}{a_{l,l}} · b_{j,k}

is added to the (l, k)-th entry when we apply R_{j} to R_{j+1} ⋯ R_{n}R(I_{n}). Evidently we obtain (4.1) by summing over all j with l < j ⩽ k.

□

We’ll continue with the proof of the theorem. Now

C = [ \begin{matrix}c_{1,1} & ⋯ & c_{1,n} \\ & ... & ⋮ \\ 0 & & c_{n,n} \end{matrix} ] = [ \begin{matrix}a′_{1,1} & ⋯ & a′_{1,n} \\ & ... & ⋮ \\ 0 & & a′_{n,n} \end{matrix} ] · [ \begin{matrix}b_{1,1} & ⋯ & b_{1,n} \\ & ... & ⋮ \\ 0 & & b_{n,n} \end{matrix} ] .

and thus c_{l,k} can be written as

c_{l,k} = ∑_{j=0}^{k−l} a′_{l,k−j} · b_{k−j,k}.  (4.2)

By previous lemma we have that c_{i,i} = a_{i,i}^{−1} · a′_{i,i}. Since a_{i,i}, a′_{i,i}, c_{i,i} ∈ ℕ for all i we deduce that a_{i,i}|a′_{i,i} for all i. But since

∏_{i} a_{i,i} = det A = m = det A′ = ∏_{i} a′_{i,i}

we conclude that a_{i,i} = a′_{i,i} for all i. We’ll now show that a_{l,k} = a′_{l,k} for l < k. To do that we’ll use induction. Let’s first assume that a_{l,k} = a′_{l,k} for all l > r. We’ll then show that a_{r,r+1} = a′_{r,r+1}. Now we have by (4.2) and (4.1) that

\begin{aligned}c_{r,k} & = ∑_{j=0}^{k−r} a′_{r,k−j} · b_{k−j,k} = ∑_{j=0}^{k−r−1} a′_{r,k−j} · b_{k−j,k} + a_{r,r} · b_{r,k} \\ & = ∑_{j=0}^{k−r−1} (a′_{r,k−j} · b_{k−j,k} − a_{r,k−j} · b_{k−j,k}) \end{aligned}

For k = r + 1 this reduces to

c_{r,r+1} = \frac{a′_{r,r+1} − a_{r,r+1}}{a_{r+1,r+1}}

Since 0 ⩽ a_{r,r+1}, a′_{r,r+1} < a_{r+1,r+1} and c_{r,r+1} ∈ ℕ_{0} we deduce that a_{r,r+1} = a′_{r,r+1}. Now assume that a_{r,k} = a′_{r,k} for all l > r and all k < s, where l < s ⩽ n. We’ll show that a_{r,s} = a′_{r,s}. We have

\begin{aligned}c_{r,s} & = ∑_{j=0}^{s−r−1} (a′_{r,s−j} · b_{s−j,s} − a_{r,s−j} · b_{s−j,s}) \\ & = \frac{a′_{r,s} − a_{r,s}}{a_{s,s}} + ∑_{j=1}^{s−r−1} (a′_{r,s−j} · b_{s−j,s} − a_{r,s−j} · b_{s−j,s}) = \frac{a′_{r,s} − a_{r,s}}{a_{s,s}} , \end{aligned}

where the last equality signs follows from our induction assumption. As before we can conclude that a_{r,s} = a′_{r,s}. This observations shows that a_{l,k} = a′_{l,k} for all l, k hence A = A′ proving the uniqueness. The proof of the theorem is complete.

□

4.2 Topics from number theory

4.2.1 Dirichlet theorem

4.7 Definition (1) Let D be a subset in ℝ^{n}\{0}. Let C be the set of all half-lines from the origin which intersects D. The set of all points in C is called a cone with section D.

(2) Let (e_{1}, …, e_{n}) be a basis for ℝ^{n}. Then the set

ℤe_{1} + ⋯ + ℤe_{n}

is called a lattice for ℝ^{n}.

(3) Let M = ℤe_{1} + ⋯ + ℤe_{n} be a lattice for ℝ^{n}. The set

[0, 1)e_{1} + ⋯ + [0, 1)e_{n}

is called the fundamental parallelepiped of the lattice M. ℝ^{n} spanned by e_{1}, …, e_{n}.

(4) A lattice of a Lie group G is a discrete subgroup Γ ⊂ G such that G/Γ has a finite volume relative to the (induced) G-invariant measure.

(5) Let X be a smooth manifold, and Γ be a discrete group. A fundamental domain of Γ is a subset D ⊂ X such that the subsets γD, where γ ∈ Γ have no common interior points and form a locally finite covering of X. For more details on fundamental domains, see for example [Rat].

The proof for the following theorem can be found in [Bo]. The proof is simple and I choose to omit it.

4.8 Theorem (Dirichlet theorem) Let X be a cone in ℝ^{n}\{0} with vertex at the origin. Let F be a positive function defined over X which satisfies

(1) F (tx) = t^{n}F (x), ∀x ∈ X, t > 0;

(2) T := {x ∈ X| F (x) < 1} is bounded and has a non-zero n-dimensional volume v(T).

Then for any lattice M in ℝ^{n} the series

f(s) = ∑_{x∈M∩X} F (x)^{−s}  (4.3)

is convergent for s > 1, and

lim_{s→1+}(s − 1)f(s) = \frac{v(T)}{Δ}  (4.4)

where Δ is the volume of the fundamental parallelepiped of the lattice M.

□

4.9 Remark Let n ∈ ℕ be greater than 1 and let X be a cone in M(n, ℝ) with a section D. We define a function F : M(n, ℝ) → ℝ by

F (x) = (det x)^{n}.

By Dirichlet theorem we have

v(T) = lim_{s→1+}(s − 1)f(s),  f(s) = ∑_{x∈M∩X}(det x)^{−ns},  (4.5)

where M := M(n, ℤ).

By theorem 4.5 the number of orbits for the left action of the group Sl(n, ℤ) on M_{m}(n, ℤ) is the same as the number of matrices which satisfies conditions (1)-(3) in theorem 4.5. Let’s count those matrices. First we see in how many ways we can choose elements y_{1,1}, …, y_{n,n} on the diagonal. Clearly that is just the number of ways the number m can be written as a product of n ordered natural numbers d_{1}, …, d_{n}. Now let’s look at the possible matrix which has y_{1,1}, …, y_{n,n} on the diagonal. Let τ_{l,k} be the number of possibilities we can fill the (l, k)-th entry for l < k. Then the number of matrices which has y_{1,1}, …, y_{n,n} on the diagonal is ∏_{l<k} τ_{l,k}. But by condition (3) in theorem 4.5 we have that τ_{l,k} = y_{k,k} for l < k so we have

∏_{l<k} b_{l,k} = y_{2,2}^{1} · y_{3,3}^{2} ⋯ y_{n,n}^{n−1}.

By letting d_{k} = y_{n−k+1,n−k+1} for k = 1, …, n we have proven the following lemma:

4.10 Lemma Let a_{m} be the number of orbits for the left action of the group Sl(n, ℤ) on M_{m}(n, ℤ). Then

a_{m} = ∑ d_{1}^{n−1}d_{2}^{n−2} ⋯ d_{n−1}  (4.6)

where the sum is taken over all the representations of the number m = d_{1} · d_{2} ⋯ d_{n}, where d_{i} ∈ ℕ.

□

4.11 Lemma The function f(s) in (4.5) can be written as

f(s) = ζ(ns)ζ(ns − 1) ⋯ ζ(ns − n + 1),  (4.7)

where ζ is the Riemann zeta function defined by

ζ(z) = ∑_{n=1}^{∞} n^{−z}  (for all z ∈ {w ∈ ℂ| Re(w) > 1}).  (4.8)

Proof: We shall expand the terms on the right side in (4.7). We have

ζ(ns)ζ(ns − 1) ⋯ ζ(ns − n + 1)
= ∑_{i_{1}=1}^{∞} i_{1}^{−ns} ∑_{i_{2}=1}^{∞} i_{2}^{−ns+1} ⋯ ∑_{i_{n}=1}^{∞} i_{n}^{−ns+n−1}

= ∑_{k_{1},k_{2},...,k_{n}=1}^{∞} i_{1_{k_{1}}}^{−ns}i_{2_{k_{2}}}^{−ns} ⋯ i_{n_{kn}}^{−ns} · i_{2_{k_{2}}}i_{3_{k_{2}}}^{2} ⋯ i_{n_{kn}}^{n−1}

which is evidently the same as f(s).

□

It’s easy to show that (see for example [Bo])

(s − 1)ζ(ns − n + 1) ⟶ \frac{1}{n}  as s → 1 +  (4.9)

and thus we have by Dirichlet theorem the following corollary:

4.12 Corollary The function v(T) in (4.5) is given by

v(T) = \frac{1}{n}ζ(2) ⋯ ζ(n)  (4.10)
□

4.2.2 Calculating ω_{∞}(Sl(n, ℤ)\Sl(n, ℝ))

4.13 Lemma Two left invariant n-forms on an n-dimensional Lie group G are proportional to each other.

Proof: Let ω and ω′ be two n-forms on G. We have that ω = fω′ where f is a real function. Then for every σ ∈ G we have

fω′ = l_{σ}^{∗}(fω′) = (f ∘ l_{σ})l_{σ}^{∗}(ω′) = (f ∘ l_{σ})ω′

so f must be a constant.

□

4.14 Definition Let ω be a left invariant n-forms on an n-dimensional Lie group G. Then ω is said to be a left invariant density on G.

{4.15 Remark \atop (1)} By lemma 4.13 all left invariant densities on a Lie-group G is identical up to a constant. We can thus choose a specific left invariant density by giving up its value at one point.

(2) If G is compact a left invariant density ω on G induces a left invariant Haar measure on G by requiring that

∫_{G} ω = ω_{∞}(G).

(3) If Γ is a discrete group in G such that the closure of G/Γ is compact a left invariant density ω on G induces a left invariant Haar measure on ω_{∞} on G by requiring that for a measurable fundamental domain D for Γ in G we have

∫_{D} ω = ω_{∞}(D).

I will not go through the proof of this but these results are taken from measure theory. For some discussion on Haar measure see for example [Fo].

We denote by x_{ij} the standard coordinates in M(n, ℝ). dx_{11} ∧ ⋯ ∧ dx_{nn} is the form which defines the Lebesgue measure in M(n, ℝ). Sl(n, ℝ) is a hypersurface in M(n, ℝ). We choose a connected measurable fundamental domain D for Sl(n, ℤ) in Sl(n, ℝ). It can be shown that this is indeed possible, but I’ll not show this here. We let ω be a left invariant density on Sl(n, ℝ) and denote by ω_{∞} its deduced Haar measure. We define the measure of Sl(n, ℤ)\Sl(n, ℝ) to be ω_{∞}(D) and we write ω_{∞}(Sl(n, ℤ)\Sl(n, ℝ)) to indicate the measure of Sl(n, ℤ)\Sl(n, ℝ). Since ω_{∞} is a left invariant measure on Sl(n, ℝ) our definition is independent of the choice if D. For more details and references on this subject see for example [Rat]. The mapping ϕ : I × D → T, defined by

ϕ(t, y) = ty,  (4.11)

where I = (0, 1], is clearly a homeomorphism. We extend the mapping ϕ to a homomorphism of groups ϕ : ℝ^{∗} × Sl(n, ℝ) → Gl(n, ℝ), (t, y) ↦ ty.

4.16 Lemma The form

β := (det(·))^{−n}dx_{11} ∧ ⋯ ∧ dx_{nn}

is a left invariant density on Gl(n, ℝ).

Proof: Let A ∈ Gl(n, ℝ). For x = (x_{1,1}, …, x_{n,n}) we define

C = [ \begin{matrix}c_{1,1} & ⋯ & c_{1,n} \\ ⋮ & ... & ⋮ \\ c_{n,1} & ⋯ & c_{n,n} \end{matrix} ] := [ \begin{matrix}A_{1,1} & ⋯ & A_{1,n} \\ ⋮ & ... & ⋮ \\ A_{n,1} & ⋯ & A_{n,n} \end{matrix} ] · [ \begin{matrix}x_{1,1} & ⋯ & x_{1,n} \\ ⋮ & ... & ⋮ \\ x_{n,1} & ⋯ & x_{n,n} \end{matrix} ] .

and c_{i,j} is given by

c_{i,j} = ∑_{k=1}^{n} A_{i,k} · x_{k,j}.  (4.12)

We then have by (2.16) that

(l_{A}^{∗}β)_{x}

=(det l_{A}x)^{−n} det(J_{i,j}(l_{A}))_{x} dx_{11} ∧ ⋯ ∧ dx_{nn}

=(det l_{A}x)^{−n} | \begin{matrix}\frac{∂C_{1,1}}{∂x_{1,1}} & ⋯ & \frac{∂C_{1,n}}{∂x_{1,1}} & ⋯ & ⋯ & \frac{∂C_{n,1}}{∂x_{1,1}} & ⋯ & \frac{∂C_{n,n}}{∂x_{1,1}} \\ ⋮ & ... & ⋮ & & & ⋮ & ... & ⋮ \\ \frac{∂C_{1,1}}{∂x_{1,n}} & ⋯ & \frac{∂C_{1,n}}{∂x_{1,n}} & ⋯ & ⋯ & \frac{∂C_{n,1}}{∂x_{1,n}} & ⋯ & \frac{∂C_{n,n}}{∂x_{1,n}} \\ ⋮ & & ⋮ & ... & & ⋮ & & ⋮ \\ ⋮ & & ⋮ & & ... & ⋮ & & ⋮ \\ \frac{∂C_{1,1}}{∂x_{n,1}} & ⋯ & \frac{∂C_{1,n}}{∂x_{n,1}} & ⋯ & ⋯ & \frac{∂C_{n,1}}{∂x_{n,1}} & ⋯ & \frac{∂C_{n,n}}{∂x_{n,1}} \\ ⋮ & ... & ⋮ & & & ⋮ & ... & ⋮ \\ \frac{∂C_{1,1}}{∂x_{n,n}} & ⋯ & \frac{∂C_{1,n}}{∂x_{n,n}} & ⋯ & ⋯ & \frac{∂C_{n,1}}{∂x_{n,n}} & ⋯ & \frac{∂C_{n,n}}{∂x_{n,n}} \end{matrix} | dx_{11} ∧ ⋯ ∧ dx_{nn}

=(det A)^{−n}(det x)^{−n} | \begin{matrix}A & & & & & 0 \\ & A \\ & & . & . & . \\ 0 & & & & & A \end{matrix} | dx_{11} ∧ ⋯ ∧ dx_{nn}

=(det A)^{−n}(det x)^{−n}(det A)^{n}dx_{11} ∧ ⋯ ∧ dx_{nn} = (det x)^{−n}dx_{11} ∧ ⋯ ∧ dx_{nn} = β_{x}.

□

If we let n = 1 in the above lemma we get the following corollary.

4.17 Corollary (·)^{−1}dt is left invariant form on ℝ^{∗}.

□

4.18 Lemma If β on Ω(H) is a left invariant form and ϕ : G → H is a homomorphism of Lie groups then ϕ^{∗}(β) is a left invariant form on Ω(G).

Proof: Since ϕ is a homomorphism it’s easy to show that ϕ ∘ l_{g} = l_{ϕ(g)} ∘ ϕ. Therefore for all g ∈ G we have

\begin{aligned}l_{g}^{∗}ϕ^{∗}(β) & = (ϕ ∘ l_{g})^{∗}β = (l_{ϕ(g)} ∘ ϕ)^{∗}β \\ & = ϕ^{∗}l_{ϕ(g)}^{∗}β = ϕ^{∗}(β), \end{aligned}

which proves the lemma.

□

4.19 Corollary (1) ϕ^{∗}((det(·))^{−n}dx_{11} ∧ ⋯ ∧ dx_{nn}) = α(·)^{−1}dt ∧ ω, where α ∈ ℝ^{∗}.

(2) ϕ^{∗}(dx_{11} ∧ ⋯ ∧ dx_{nn}) = α(·)^{n^{2}−1}dt ∧ ω.

where α ∈ ℝ. Proof:

(1) In previous lemmas we have shown that ϕ^{∗}((det(·))^{−n}dx_{11} ∧ ⋯ ∧ dx_{nn}) and (·)^{−1}dt ∧ ω are both left invariant forms on R^{∗} × Sl(n, ℝ), so the result follows from lemma 4.13.

(2) We have

ϕ^{∗}((det(·))^{−n}dx_{1,1} ∧ ⋯ ∧ dx_{n,n})(t, y)
=ϕ^{∗}((det(·))^{−n})(t, y)ϕ^{∗}(dx_{1,1} ∧ ⋯ ∧ dx_{n,n})
=((det(·))^{−n} ∘ ϕ)(t, y)ϕ^{∗}(dx_{1,1} ∧ ⋯ ∧ dx_{n,n})
=(det ty)^{−n}ϕ^{∗}(dx_{1,1} ∧ ⋯ ∧ dx_{n,n})
=(·)^{−n^{2}}ϕ^{∗}(dx_{1,1} ∧ ⋯ ∧ dx_{n,n}),

and the results follows.

□

4.20 Remark T_{E}Gl(n, ℝ) is naturally isomorphic to ℝ^{n^{2}}. Since T_{1}ℝ^{∗} is naturally isomorphic to ℝ we have that T_{(1,E)}(ℝ^{∗} × Sl(n, ℝ)) is isomorphic to ℝ × 𝔰𝔩(n, ℝ) by remark 2.55. Thus via this isomorphism a vector in T_{(1,E)}(ℝ^{∗} × Sl(n, ℝ)) can be written on the form (s, A) where s ∈ ℝ and A ∈ 𝔰𝔩(n, ℝ).

4.21 Lemma Let s ∈ ℝ and A ∈ 𝔰𝔩(n, ℝ). Then

(1) ϕ_{∗}(s, 0) = sE;

(2) ϕ_{∗}(0, A) = A.

Proof:

(1) Let τ :]a, b[→ ℝ^{∗} × Sl(n, ℝ) be defined by

τ(t) = (1, E) + (st, 0)

τ is thus a representation curve for the tangent vector (s, 0). So we have

\begin{aligned}ϕ_{∗}(s, 0) & = \frac{d}{dt}|_{t=0}ϕ ∘ τ(t) = \frac{d}{dt}|_{t=0}((1 + st)E) \\ & = sE ∈ T_{ϕ(1,E)}Gl(n, ℝ). \end{aligned}

(2) Let τ :]a, b[→ ℝ^{∗} × Sl(n, ℝ) be defined by

τ(t) = (1, E) + (0, tA).

τ is thus a representation curve for the tangent vector (0, A). Thus we get

\begin{aligned}ϕ_{∗}(0, A) & = \frac{d}{dt}|_{t=0}ϕ ∘ τ(t) = \frac{d}{dt}|_{t=0}(E + tA) \\ & = A ∈ T_{ϕ(1,E)}Gl(n, ℝ). \end{aligned}

□

In the next two lemmas we let A_{n} be the (n × n)-matrix defined by

A_{n} := [ \begin{matrix}1 & 1 & 1 & ⋯ & 1 & 1 \\ 1 & −1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & −1 & ⋯ & 0 & 0 \\ ⋮ & ... & ... & ... & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & −1 \end{matrix} ] .

4.22 Lemma |A_{n}| = (−1)^{n+1}n

Proof: I’ll use induction. The postulate is obvious for n = 1, 2. Let’s assume it’s right for n < p. We’ll expand A_{p} on the first column, i.e.

|A_{p}| = 1 | \begin{matrix}−1 & & & 0 \\ 1 & −1 \\ & ... & ... \\ 0 & & & −1 \end{matrix} | − |A_{p−1}|.

If p is even the first term on the right hand side equals −1 and by our induction hypothesis we have that |A_{p−1}| = p − 1. Hence |A_{p}| = −p. On the other hand if p is odd the first term on the right hand side equals 1 and by our induction hypothesis we have that |A_{p−1}| = −(p − 1). Thus |A_{p}| = p. We have thus proved the lemma.

□

4.23 Lemma (dx_{11} ∧ ⋯ ∧ dx_{nn})_{E}(E, E_{11} − E_{22}, …) = ±n

Proof: By making the suitable change of the permutation of dx_{11} ∧ ⋯ ∧ dx_{nn} we have that

(dx_{11} ∧ ⋯ ∧ dx_{nn})_{E}(E, E_{11} − E_{22}, …) = ± | \begin{matrix}A_{n} & 0 \\ 0 & I, \end{matrix} |

and the results follows from the previous lemma.

4.3. VOLUME OF SO(N, ℝ) AND FURTHER STUDY 87

□

4.24 Theorem

ϕ^{∗}(dx_{11} ∧ ⋯ ∧ dx_{nn})_{(1,E)}(1, E_{11} − E_{22}, …) = ±n.

Proof:

ϕ^{∗}(dx_{11} ∧ ⋯ ∧ dx_{nn})_{(1,E)}((1, 0), (0, E_{11} − E_{22}), ⋯)
= (dx_{11} ∧ ⋯ ∧ dx_{nn})_{ϕ(1,E)}(ϕ_{∗}(1, 0), ϕ_{∗}(0, E_{11} − E_{22}), ⋯)
= (dx_{11} ∧ ⋯ ∧ dx_{nn})_{E}(E, E_{11} − E_{22}, ⋯)  (by lemma 4.21)
= ±n  (by lemma 4.23).

□

4.25 Remark A natural selection of our invariant density ω is to require that at the identity its value on the zero trace basis is equal to ±1. By previous theorem we deduce that |α| = n in corollary 4.19.

Thus

\begin{aligned}v(T) & = ∫_{T} dx_{11} ∧ ⋯ ∧ dx_{nn} = ∫_{D} ∫_{0}^{1} ϕ^{∗}(dx_{11} ∧ ⋯ ∧ dx_{nn}) \\ & = ∫_{D} ∫_{0}^{1} nt^{n^{2}−1}dt ∧ ω = \frac{1}{n}ω_{∞}(D). \end{aligned}  (4.13)

By combining (4.10) and the above equation we have

4.26 Theorem

ω_{∞}(Sl(n, ℤ)\Sl(n, ℝ)) = ζ(2) ⋯ ζ(n)  (4.14)
□

4.3 Volume of SO(n, ℝ) and further study

SO(n, ℝ) is a Lie group of dimension n(n − 1)/2. In [Fl, He] it is shown how a basis of linearly independent left invariant 1-forms ω_{i,j} where 1 ⩽ i < l ⩽ n can be constructed. The resulting basis is known as the Maurer-Cartan basis. We define an invariant density on SO(n, ℝ) by

α_{n} = ⋀_{i>j} ω_{i,j},  (4.15)

where we require that the induced measure on SO(n, ℝ),

α_{n}(SO(n, ℝ)) := ∫_{SO(n,ℝ)} α_{n},

is equal to the volume of SO(n, ℝ) with respect to the standard density. We look at SO(n − 1, ℝ) as a subgroup of SO(n, ℝ) by

SO(n − 1, ℝ) = {A ∈ SO(n, ℝ)| a_{1,1} = 1, a_{i,1} = a_{1,i} = 0 for all i = 2, …, n}. (4.16)

Let’s write α_{n} = u_{n−1} ∧ α̃_{n−1} where u_{n−1} = ⋀_{i>1} ω_{i,1} and α̃_{n−1} = ⋀_{i>j>1} ω_{i,j}. Then it can be shown that

α_{n}(SO(n, ℝ)) = α_{n−1}(SO(n − 1, ℝ)) · u(S^{n−1})  (4.17)

where u is the volume form on S^{n−1}. From this we can calculate α_{n}(SO(n, ℝ)) for all n, and I’ll do that in the following theorem.

4.27 Theorem For n ∈ ℕ we have

α_{n}(SO(n, ℝ)) = 2^{n−1}π^{n(n+1)/4} ∏_{a=1}^{n} Γ (\frac{a}{2})^{−1} .  (4.18)

Proof: We’ll use induction. From chapter 1 we have that

u(S^{m}) = \frac{2π^{(m+1)/2}}{Γ (\frac{m+1}{2})} .

(4.18) is obvious for n = 1. Let’s assume it’s true for n = p − 1, where p ⩾ 2. By (4.17) and our induction hypothesis we have

α_{p}(SO(p, ℝ)) = α_{p−1}(SO(p − 1, ℝ)) · u_{p−1}(S^{p−1})

= 2^{p−2}π^{(p−1)p/4} ∏_{a=1}^{p−1} Γ (\frac{a}{2})^{−1} · \frac{2π^{p/2}}{Γ (\frac{p}{2})}

= 2^{p−1}π^{(p^{2}−p)/4+2p/4} ∏_{a=1}^{p} Γ (\frac{a}{2})^{−1}

= 2^{p−1}π^{p(p+1)/4} ∏_{a=1}^{p} Γ (\frac{a}{2})^{−1}

so (4.18) is also true for n = p and thus (4.18) is true for all n so the proof is complete.

□

In [Gilm] the volume of many classical Lie groups and quotients of Lie groups is calculated. We have seen that for the fibre bundle

SO(n − 1) → SO(n) → S^{n−1}

we got a very nice relation between the volume of these spaces. One might ask if this was true in general. Let’s look at the following question:

Question:

Let’s assume that F and B are compact Riemannian manifolds with invariant densities μ_{F} and μ_{B} respectively. If E is a manifold such that F → E → B is a fibre bundle, can we find a Riemannian metric on E such that

μ_{E}(E) = μ_{F} (F) · μ_{B}(B)

where μ_{E}(E) is the volume of E with respect to some invariant density μ_{E} ?

Part of answer:

We saw that this was true, when we calculated the volume of SO(n, ℝ). According to [Mac] the answer to the above question is “yes” if F, E and B are compact Lie groups. MacDonald’s proof is above my understanding and so is a simpler proof given by [Ha]. Those who are familiar with characteristic classes and algebraic topology could however find the proofs interesting. For some more discussion of this

see: http://www.lehigh.edu/~dmd1/p2000.html. Many classical fibre bundles are

given in [Br]. [Hus] is also a great book about fibre bundles. I think that studying this further could be very interesting, but it’s beyond the scope of this thesis.

Index

(T_{p}P)^{⊥}, 41 ; (∂_{i}), 31 ; (dx_{i}), 31 ; A^{t}, 29 ; B^{n}, 9 ; B_{p}^{n}(R), 9 ; C^{∞}(M), 15 ; D_{γ}, 17 ; E_{i,j}, 75 ; Gl(n, R̃), 29 ; I_{s}, 60 ; K^{M}, 36 ; K_{hol}, 69 ; M, 15 ; M_{m}(n, R̃), 29 ; N, 44 ; O(n, R̃), 29 ; P_{h}(W_{k}), 59 ; P_{t}, 47 ; Q_{k}(A), 76 ; R(t), 48, 71 ; R_{N}, 47 ; R_{u}, 71 ; S, 47 ; S′(t), 48 ; S(t), 48 ; SO(n, R̃), 29 ; S^{n}, 9 ; S_{p}^{n}(R), 9 ; Sl(n, R̃), 30 ; T, 49 ; T (P, r), 46 ; T M, 20 ; T^{(k,l)}(V), 19 ; T^{(k,l)}M, 20 ; T^{∗}M, 20 ; T_{p}M, 17 T_{u}, 49

T_{abu}, 49 T_{ab}, 49 W_{k}, 58 [X, Y], 31 [[i, j]], 13 Λ^{k}M, 20 Ω^{k}(M), 21 Ξ_{m}, 46 ∐, 13 δ_{i,j}, 13 ⟨F ⟩, 61 ⟨·, ·⟩, 30 𝕂_{hol}^{n}(λ), 70 K^{n}(λ), 56 E(M), 21 E(M, p), 16 E_{ν}, 42 T^{(k,l)}(M), 21 X (M), 21 X (P, p), 44 X (P, p)^{⊥}, 44 X (P, p)^{⊤}, 44 𝔖_{s}, 14 𝔤, 28 dist, 34 grad, 45 ν, 41 ω(M), 22 ϕ^{∗}, 24 π_{1}, 20 ρ, 36 σ, 44 τ, 36 ξ_{v}, 34 d, 23 k-form, 19 k-form bundle, 20 l_{σ}, 28 q(Q_{k}(A)), 76 q), 60 fix, 76 floor, 76 inv, 27 m, 27 2-cycle, 14

action, 27 almost complex structure, 67 almost complex manifold, 67 almost Hermitian manifold, 68 almost Hermitian metric, 68 alternating covariant tensor, 19 atlas, 15

maximal, 15 average, 61

ball, 9 base, 20 boundary, 15

chain rule, 18 chart, 15 compatible with, 71 complete Riemannian manifold, 34 cone, 80 connection, 31

compatible with, 31 Levi-Civita, 32 linear, 31 Riemannian, 32 symmetric, 31

constant sectional curvature, 70 contraction operators, 58 contravariant, 19 coordinate frame, 31 cotangent bundle, 20 covariant, 19 covariant derivative, 47 curvature transformation, 36 curvature tensor, 36 curve, 32

admissible, 32 length, 32 locally minimizing, 33 minimizing, 32 regular, 32 segment, 32

cut locus, 55

cut point along a geodesic, 55

diagonalizes, 50 differential, 18 differential form, 21 differential operator, 16 directional derivative, 17 disjoint union, 13 double form, 57 dual coframe, 31

Einstein summation convention, 37 elementary invariant, 59 embedded submanifold, 18 embedding, 18 exponential map

domain for, 34 exponential map, 35 domain for ν, 42

exponential map of the normal bundle,

42 exterior derivative, 23 exterior product, 22, 57

Fermi coordinates, 42

Fermi field normal, 44 tangential, 44

fiber, 20

first Bianchi identity, 36 Fubini-Study metric, 73 fundamental domain, 80 fundamental parallelepiped, 80

Gauss equation, 63 Gauss map, 45 Gaussian curvature, 36 geodesic, 33 geodesic ball, 35 gradient vector field, 45 group

Lie, 28 orthogonal, 29 permutation, 14 special linear, 30 special orthogonal group, 29 topological, 27

harmonic space, 73 Hodge star operator, 38 holomorphic orthonormal frame, 68 holomorphic section, 69 holomorphic sectional curvature, 69 hypersurface, 18

immersion, 18 infinitesimal change of volume function,

51 integral, 25, 26 integrated mean curvature, 58 invariant, 59

Kähler form, 68 Kähler manifold, 68 Kronecker symbol, 13

lattice, 80 left invariant density, 82 left invariant form, 28 left invariant vector field, 28 left translation, 28 Lie algebra, 28 Lie algebra of the Lie group, 29 Lie brackets, 31 Lie group, 28 lie in the orientation, 24 local frame, 30 local trivilization, 20 locally finite, 14 locally symmetric, 71

manifold, 15 complex, 15 with boundary, 15

Maurer-Cartan basis, 87 maximal atlas, 16 natural isomorphism of T_{p}V, 17 normal bundle, 41

orbit, 27 orient preserving, 26 orientable, 16 orientation, 24 oriented manifold, 16 oriented vector space, 24 orthogonal group, 29

paracompact, 15 parallel, 68 partition of unity, 16 permutation, 13 permutation group, 14 polynomial of degree h, 59 positive definite, 30 principal curvatures

of a hypersurface, 50 principal curvature vector fields, 50 principal curvature functions, 50 product in 𝔖_{s}, 14 projection, 20

radial distance function, 45 radial vector, 41 refinement, 14 representation curve, 17 Riccati differential equation, 48 Ricci curvature, 36 Riemann zeta function, 81 Riemannian geodesic, 33 Riemannian metric, 30 row equivalent, 75 row operations, 75

scalar curvature, 36 second fundamental form, 49 section, 21, 80 sectional curvature, 36 shape operator, 47 sign, 14 smooth function, 15 smooth map, 15 space of constant sectional curvature, space of constant holomorphic sectional

curvature, 70 special linear group, 30 special orthogonal group, 29 sphere, 9 spherical vector, 41 Stoke’s Theorem, 27 subatlas, 16 subdivision, 32 submanifold, 18 submersion, 18 support

of a function, 25 symmetric, 30 symmetric double form, 57

tangent bundle, 20 tangent space, 17 tangent vector, 17 tensor, 19 tensor bundle, 20 Tensor Characterization Lemma, 22 tensor field, 21, 22 topological group, 27 total space, 20 transpose, 29 tube, 46 tubular hypersurface, 47

unit speed curve, 32

vector bundle, 20 vector field, 21

along a curve, 33 radial, 45

volume form, 38

wedge product, 22 Weingarten map, 49

zero section, 21 zero trace basis, 75 94 INDEX

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