Lectures on Differential Geometry Paul Bryan June 13, 2018 1 Contents 1 Week 01 5 1.1 Week 01, Lecture 01: Introduction . 5 1.1.1 Recommended Reading . 5 1.1.2 Course Summary . 6 1.1.3 What is Riemannian Geometry? . 6 1.2 Week 01, Lecture 02: Overview of Curvature and Review of Manifolds . 12 1.2.1 What Is Curvature? . 12 1.2.2 Intrinsic Definition . 14 1.2.3 Examples . 15 1.3 Week 01, Lecture 03: Riemannian Surfaces, Tangent Vectors and Smooth Maps . 17 1.3.1 Geometry of Surfaces . 17 1.3.2 Smooth maps . 18 1.3.3 Tangent Vectors . 20 2 Week 02 22 2.1 Week 02, Lecture 01: Push Forward and The Tangent Bundle 22 2.1.1 Push Forward . 22 2.1.2 The Tangent Bundle . 25 2.2 Week 02, Lecture 02: Riemannian Metrics . 27 2.2.1 Riemannian Metrics . 27 2.3 Week 02, Lecture 03: Vector Bundles . 30 2.3.1 Vector Bundles . 30 3 Week 03 35 3.1 Week 03, Lecture 01: Multi-linear Algebra . 35 3.1.1 Hom and The Dual Space . 35 3.1.2 Remarks on Indices and the (Einstein) Summation Convention . 37 3.1.3 Tensor Products . 39 3.2 Week 03, Lecture 02: Tensor Bundles . 47 3.2.1 The Tensor Algebra . 47 3.2.2 New Bundles from Old . 49 3.3 Week 03, Lecture 03: Bundle Metrics . 55 2 3.3.1 Tensors and Change of Basis . 55 3.3.2 Inner Product on Vector Spaces . 55 3.3.3 Inner Product on Vector Bundles . 56 4 Week 04 58 4.1 Week 04, Lecture 01: Integration and the Anti-Symmetric Al- gebra . 58 4.1.1 Area of a Regular Surface . 58 4.1.2 Volume on a Riemannian Manifolds . 58 4.1.3 Anti-Symmetric Algebra . 60 4.2 Week 04, Lecture 02: Orientation, Integration and Differential Forms . 64 4.2.1 Orientation . 64 4.2.2 Differential Forms . 67 4.2.3 Exterior Derivative . 69 4.3 Week 04, Lecture 03: Connections . 72 5 Week 05 79 5.1 Week 05, Lecture 01: Dependence of Connections on Vector Fields . 79 5.1.1 Differentiation Along a Curve . 79 5.2 Week 05, Lecture 02: Parallel Transport and Holonomy . 83 5.2.1 Integral Curves . 83 5.2.2 Tensorality . 84 5.2.3 Parallel Transport . 86 5.2.4 Holonomy . 88 5.3 Week 05, Lecture 03: Levi-Civita Connection . 92 5.3.1 The Levi-Civita Connection . 92 5.3.2 Higher Derivatives . 93 6 Week 06 97 6.1 Week 06, Lecture 01: Sub-Manifolds . 97 6.1.1 Submanifolds . 97 6.2 Week 06, Lecture 02: Pull-backs Bundles . 101 6.2.1 Pull Back Bundles . 101 6.3 Week 06, Lecture 03: The Second Fundamental Form . 105 6.3.1 The Second Fundamental Form . 105 3 7 Week 07 108 7.1 Week 07, Lecture 01: The Metric Structure of a Riemannian Manifold . 108 7.1.1 Length of paths . 108 7.1.2 Riemannian Distance . 109 7.2 Week 07, Lecture 02: Geodesics . 113 7.3 Week 07, Lecture 03: The Exponential Map . 117 8 Week 08 120 8.1 Week 08, Lecture 01: Geodesic Flow and Gauss Lemma . 120 8.1.1 Geodesic Flow . 120 8.1.2 Gauss’ Lemma . 123 8.2 Week 08, Lecture 02: Length Minimisation . 125 8.3 Week 08, Lecture 03: Completeness . 129 9 Week 09 132 9.1 Week 08, Lecture 01: Higher Derivatives . 132 9.1.1 Induced Connections on Tensor Bundles . 132 9.1.2 Second Derivatives . 135 9.2 Week 09, Lecture 02: Curvature . 137 9.2.1 The Curvature Tensor . 137 9.2.2 Ricci and Scalar Curvature . 140 9.3 Week 09, Lecture 03: Curvature Operators and Sectional Cur- vature . 142 9.3.1 The Curvature Operator . 142 9.3.2 Sectional Curvature . 143 10 Week 10 146 10.1 Week 10, Lecture 01: Positive and Constant Curvature . 146 10.1.1 Positive Curvature Operator . 146 10.1.2 Constant Curvature Spaces . 148 10.2 Week 10, Lecture 02: Classical Results . 150 10.2.1 Second Variation of Length and Bonnet-Myers . 150 4 1 Week 01 1.1 Week 01, Lecture 01: Introduction 1.1.1 Recommended Reading 1. Text Books • Do Carmo: Riemannian Geometry (a classic text that is certainly relevant today but sometimes considered a little terse. Does in- clude material on differentiable manifolds.) • Lee: Riemannian Manifolds: An Introduction to Curvature (very readable, possibly a little elementary in places) • Chavel: Riemannian Geometry: A Modern Introduction (more advanced, extensive discussion of many aspects of Riemannian Geometry) • Petersen: Riemannian Geometry (more advanced, slightly non- standard approach definitely worth a look at some point) • Gallot, Hulin, Lafontaine: Riemannian Geometry (more ad- vanced, but very nice development of the formalism of Riemannian Geometry) 2. Lecture Notes • Lectures on Differential Geometry by Ben Andrews (I learned from these notes) 3. Differentiable Manifolds Some sources for differential manifolds. There are many resources available, and some of the resources listed above treat this topic before moving on to Riemannian Geometry. The fol- lowing should be sufficient background reading. • Lee: Introduction to Smooth Manifolds • Hitchin: Differentiable Manifolds 5 1.1.2 Course Summary This course is about Riemannian geometry, that is the extension of geometry to spaces where differential/integral calculus is possible, namely to manifolds. We will study how to define the notions of length, angle and area on a smooth manifold, which leads to the definition of a Riemannian Manifold. Important concepts are • Riemannian metrics • Connections (differentiation of vector fields) • Length, angle • Geodesics (shortest paths) • Area • Curvature At the end of the course, we will touch on some global aspects of Rieman- nian geometry, and discuss a little about the interaction between curvature, geometry and topology. Such interaction was studied heavily in the mid to late 20’th century and is currently still an active area of research. A famous example is the Hamilton-Perelman resolution of the Poincaré conjecture, one of the Clay Foundation’s seven Millennium Prizes, was resolved only this century. 1.1.3 What is Riemannian Geometry? What follows is an imprecise overview of the basic ideas behind Riemannian Geometry. No proofs, or references are given. That will happen throughout the course! For now, we just want to a basic feel for the topics to be studied in this course. 1. Curves and Surfaces in Euclidean Space (The Genesis of Riemannian Geometry) (a) Curves γ :[a; b] ! R2 (or R3) that are differentiable and regular (gamma0 6= 0). Why regular? We really want to study the image 6 C = γ([a; b]); γ is only a parmetrisation of this set that imbues it with a structure allowing for differential and integral calculus on C. Without regularity, the calculus on [a; b] does not trans- fer to C via γ: if f :[a; b] ! R is differentiable, then we can define f¯ : C ! R by f¯(γ(t)) = f(t), or (slightly informally, as f¯(x) = f(γ−1(x)) for x 2 C). But notice that (formally) by the chain-rule, ¯ ¯0 0 @tf(γ(t)) = f (γ(t)) · γ (t) 0 0 will be zero at irregular points t0 (γ (t0) = 0) even if f 6= 0 at such a point. Thus the function f¯◦ γ(t) may have zero derivative at t0 even if this is not a critical point of the function f, in which case the first derivative test for extremal points would fail! The regularity assumption rules out this "bad" behaviour. (b) Surfaces 3 2 A set S ⊂ R covered by local parametrisations fφα : Uα ⊂ R ! 3 1 S ⊂ R g where each Uα is open and each φα is C as a map R2 ! R3 and is a homeomorphism with it’s image (the image is equipped with the subspace topology). In general, we cannot expect a single, global parametrisation to cover S (e.g. the sphere requires at least two local parametrisations to cover it). Calculus on S is achieved by doing calculus on Uα and transferring it to S by means of Uα. Again we require regularity: Dφα is a rank 2 matrix everywhere on Uα so that calculus on S can be identified with calculus on Uα. A major advance was the realisation in the early 20th century that the parametrisations φα should also be compatible. For con- venience, let us write Vα = φα[Uα] = fφα(u): u 2 Uαg. Then compatibility means that for each α; β, −1 −1 2 −1 2 φα ◦ φβ : φβ [Vα] ⊂ R ! φα [Vβ] ⊂ R is a diffeomorphism (differentiable with differentiable inverse) be- tween open sets in R2. This requirement ensures that our calculus on S is independent of the choice of local parametrisation. Such luminaries as Gauss, Riemann and even Einstein did have this compatability condition at their disposal and considerable effort was required in order to show that calculations were independent 7 of the choice of parametrisation. It was Herman Weyl (1912) who gave the first rigourous definition of a smooth manifold, and which wasn’t widely accepted until the work of Hassler Whitney in the 1930’s. 2. Differentiable manifolds Forget S is a subset of R3: Simply require the compatibility condition to obtain the definition of a 2-dimensional dif- ferentiable manifold! An n-dimensional manifold is defined analogously. This is the intrinsic definition of a manifold. What about geometry? The length of a curve γ :[a; b] ! S is defined as Z b L[γ] = jγ0(t)j dt a where γ0 2 R3 is the tangent vector to the curve thought of as a curve in R3.