Introduction to Geometry and geometric analysis

Introduction to Geometry and geometric analysis

Oliver Knill

This is an introduction into Geometry and geometric analysis, taught in the fall term 1995 at Caltech. It introduces geometry on manifolds, tensor analysis, pseudo Riemannian geometry. General relativity is used as a

guiding example in the last part. Exercises, midterm and final with solutions as well as 4 appendices listing some results and definitions in real analysis,

geometry, measure theory and differential equations are located at the end of the text. The material contains hardly anything which can not be found in

the union of the textbooks listed in the bibliography. In retrospect the material appears a bit too condensed for a 9 week undergraduate course.

Contents

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.2 Some problems in geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 Manifolds

7

1.1 Definition of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Examples of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 A lemma for constructing manifolds . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 The theorem of Sard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7 Whitney's embedding theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 Brower's fixed point theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.9 Classification of one dimensional manifolds . . . . . . . . . . . . . . . . . . . 19

2 Tensor analysis

21

2.1 General tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Antisymmetric tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Tangent space and tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Integration on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Chains and boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Theorem of Stokes for chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8 Theorem of Stokes for oriented manifolds . . . . . . . . . . . . . . . . . . . . 33

3 Riemannian geometry

37

3.1 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Hodge star operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Theorem of Stokes for Riemannian manifolds . . . . . . . . . . . . . . . . . . 44

3.5 Interior product and Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.7 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.8 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.9 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.10 Two dimensional hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . 54

3.11 Riemannian Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.12 Ricci tensor and scalar curvature . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.13 The second Bianchi identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.14 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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CONTENTS

4 Exercises with solutions

69

4.1 Week 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Week 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Week 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4 Week 4: Midterm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Week 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.6 Week 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Week 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.8 Week 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 Appendices

107

5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4 Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Preliminaries

0.1 Introduction

What is geometry? The partition of mathematics into topics is a matter of fashion and depends on the time period. It is therefore not so easy to define what part of mathematics is geometry.

? The original meaning of geometry origins in the pre-Greek antiquity, where measurement of the earth had priority. However, in ancient Greek, most mathematics was considered geometry.

? Felix Klein's Erlanger program proposed to classify mathematics and especially geometry algebraically by group of isomorphisms. For example, conformal geometry is left invariant under conformal transformations.

? Today, geometry is mainly used in the sense of differential topology, the study of differentiable manifolds. Since differentiable manifolds can also be discrete, this does not exclude finite geometries.

Geometry and other parts of Mathematics. It is maybe useless to put boundaries in a specific mathematical field. Any major field of mathematics influences the other major fields. An indication of this fact is that one could take any ordered pair of the set of topics {"algebra", "geometry", "analysis", "probability theory","number theory"} and form pairs like {geometric, analysis} which defines itself a branch of mathematics like in the example "geometric analysis" or analytic geometry. There are other branches of geometry like Differential geometry, the study of Riemannian manifolds, Algebraic geometry, the study of varieties=algebraic manifolds, Symplectic geometry, the study of symplectic manifolds, Geometry of Gauge fields, differential geometry on fiber bundles, Spectral Geometry, the spectral theory of differential operators on a manifold. Non-commutative Geometry, geometry on spaces with fractal dimensions, foliations or discrete manifolds.

Why is geometry important? Many classical physical theories can be described in a purely geometrical way. Examples are classical mechanics, electromagnetism and other gauge field theories, general relativity or the standard model in particle physics.

About this course This course is the first part of an introduction into geometry and geometric analysis. It does not rely on a specific book. The material treated in this term is covered quite well by the books [16] and [1]. But careful, these two sources together add up to almost 2000 pages.

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CONTENTS

We will build up differential calculus on manifolds, do some differential geometry and use an introduction into general relativity as an application.

0.2 Some problems in geometry

We first look at a selected list of problems which belong to the realm of geometric analysis. The list should give a glimpse onto the subject. It is not so important to understand the problems now. You will encounter one or the other during the course or in the second or third part of the course.

1) The Ham-Sandwich-Salad problem: 1 Given a sandwich made of bread, salad and ham. Can you cut the sandwich into two pieces in such a way that both parts have the same amount of ham and the same amount of bread and the same amount of salad? A mathematical reformulation is: given three sets B, H, S R3. Is there a plane in R3 such that the volumes of H and B and S are the same on each side of the plane?

Remark: the solution of this problem is called the Stone-Tukey theorem and is a pleasant application of the Borsuk-Ulam theorem, which says that for any continuous mapping f : S R2 from a sphere S to the plane, there is a point p in S such that f (p) = f (p), where p is the antipode of p.

Example: The Borsuk-Ulam's theorem implies for example that there exists always two antipodal points on the earth which have both the same temperature and the same pressure.

Explanation. Let us explain, how the more abstract theorem of Borsuk-Ulam gives the solution of the Ham-Sandwich-Salad problem: for any set X and any line l, there exists a point P (l, X) which lies on l and such that the plane perpendicular to l through P (l, X) cuts X into two pieces of the same volume. Define a map fX from the sphere S to R by defining fX (x) = |x - P (l(x), X)|, where l(x) is the line through x and the origin. Clearly fX (x) = 2 - fX (x). Define now the map g : S R2 by g(x) = (fB(x) - fS(x), fH (x) - fS(x)). By Borsuk-Ulam, there exists a point p on the sphere, such that g(p) = g(p). Since g(x) = -g(x) for all x, we have g(p) = 0. This however assures that fB(p) = fH (p) = fH (p).

A proof of the Borsuk-Ulam theorem can be obtained using the notion of the degree of a continuous map f from one manifold to an other manifold.

2) The annulus problem and the Scho?nflies problem: given two centered standard balls B1, B2 in Rd of radius 1 and 2. The region between B1 and B2 is called an annulus. The problem is to show: for a smooth embedding of a d-dimensional ball into a d-dimensional ball, the region between these two regions is homeomorphic to the annulus. This has been shown in dimensions d = 4. The requirement of smoothness was important and we want to illustrate, why it can't be dropped: consider the Alexander horned sphere inside the ball in R3. Every closed curve in a three dimensional annulus can be deformed to any other closed curve. This is not true for the region between the horned sphere and the sphere: put a loop around one horn. We can not take it out without breaking it. But the horned sphere is homeomorphic to the three dimensional ball. However, this embedding is not smooth at accumulation points of

1I learned this problem as a high school student during an "open door event" at the ETH Zu?rich. The inspiring lecture for high school students who were shopping for a study discipline was given by Peter Henrici. The lecture convinced me: "math is it".

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