Lectures on Differential Geometry Math 240BC John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: [email protected] June 5, 2009 Preface This is a set of lecture notes for the course Math 240BC given during the Winter and Spring of 2009. The notes evolved as the course progressed and are still somewhat rough, but we hope they are helpful. Starred sections represent digressions are less central to the core subject matter of the course and can be omitted on a first reading. Our goal was to present the key ideas of Riemannian geometry up to the generalized Gauss-Bonnet Theorem. The first chapter provides the foundational results for Riemannian geometry. The second chapter provides an introduction to de Rham cohomology, which provides prehaps the simplest introduction to the notion of homology and cohomology that is so pervasive in modern geometry and topology. In the third chapter we provide some of the basic theorem relating the curvature to the topology of a Riemannian manifold|the idea here is to develop some intuition for curvature. Finally in the fourth chapter we describe Cartan's method of moving frames and focus on its application to one of the key theorems in Riemannian geometry, the generalized Gauss-Bonnet Theorem. The last chapter is more advanced in nature and not usually treated in the first-year differential geometry course. It provides an introduction to the theory of characteristic classes, explaining how these could be generated by looking for extensions of the generalized Gauss-Bonnet Theorem, and describes applications of characteristic classes to the Atiyah-Singer Index Theorem and to the existence of exotic differentiable structures on seven-spheres. i Contents 1 Riemannian geometry 2 1.1 Review of tangent and cotangent spaces . 2 1.2 Riemannian metrics . 5 1.3 Geodesics . 9 1.3.1 Smooth paths . 9 1.3.2 Piecewise smooth paths . 13 1.4 Hamilton's principle . 14 1.5 The Levi-Civita connection . 20 1.6 First variation of J .......................... 24 1.7 Lorentz manifolds . 27 1.8 The Riemann-Christoffel curvature tensor . 30 1.9 Curvature symmetries; sectional curvature . 36 1.10 Gaussian curvature of surfaces . 39 1.11 Review of Lie groups . 42 1.12 Lie groups with biinvariant metrics . 45 1.13 Grassmann manifolds . 49 1.14 The exponential map . 53 1.15 The Gauss Lemma . 56 1.16 Curvature in normal coordinates . 57 1.17 Riemannian manifolds as metric spaces . 60 1.18 Completeness . 62 1.19 Smooth closed geodesics . 64 2 Differential forms 68 2.1 Tensor algebra . 68 2.2 The exterior derivative . 71 2.3 Integration of differential forms . 75 2.4 Theorem of Stokes . 80 2.5 de Rham Cohomology . 83 2.6 Poincar´eLemma . 85 2.7 Mayer-Vietoris Sequence . 90 2.8 Singular homology* . 94 2.8.1 Definition of singular homology* . 94 2.8.2 Singular cohomology* . 98 ii 2.8.3 Proof of the de Rham Theorem* . 102 2.9 The Hodge star . 106 2.10 The Hodge Laplacian . 112 2.11 The Hodge Theorem . 116 2.12 d and δ in terms of moving frames . 119 2.13 The rough Laplacian . 122 2.14 The Weitzenb¨ock formula . 123 2.15 Ricci curvature and Hodge theory . 125 2.16 The curvature operator and Hodge theory . 126 2.17 Proof of Gallot-Meyer Theorem* . 128 3 Curvature and topology 131 3.1 The Hadamard-Cartan Theorem . 131 3.2 Parallel transport along curves . 133 3.3 Geodesics and curvature . 134 3.4 Proof of the Hadamard-Cartan Theorem . 138 3.5 The fundamental group . 140 3.5.1 Definition of the fundamental group . 140 3.5.2 Homotopy lifting . 142 3.5.3 Universal covers . 145 3.6 Uniqueness of simply connected space forms . 147 3.7 Non simply connected space forms . 149 3.8 Second variation of action . 151 3.9 Myers' Theorem . 153 3.10 Synge's Theorem . 155 4 Cartan's method of moving frames 159 4.1 An easy method for calculating curvature . 159 4.2 The curvature of a surface . 163 4.3 The Gauss-Bonnet formula for surfaces . 167 4.4 Application to hyperbolic geometry . 171 4.5 Vector bundles . 177 4.6 Connections on vector bundles . 179 4.7 Metric connections . 183 4.8 Curvature of connections . 185 4.9 The pullback construction . 187 4.10 Classification of connections in complex line bundles . 189 4.11 Classification of U(1)-bundles* . 195 4.12 The Pfaffian . 196 4.13 The generalized Gauss-Bonnet Theorem . 197 4.14 Proof of the generalized Gauss-Bonnet Theorem . 200 iii 5 Characteristic classes 208 5.1 The Chern character . 208 5.2 Chern classes . 213 5.3 Examples of Chern classes . 215 5.4 Invariant polynomials . 217 5.5 The universal bundle* . 220 5.6 The Clifford algebra . 225 5.7 The spin group . 232 5.8 Spin structures and spin connections . 235 5.9 The Dirac operator . 240 5.10 The Atiyah-Singer Index Theorem . 243 5.10.1 Index of the Dirac operator . 243 5.10.2 Spinc structures* . 246 5.10.3 Dirac operators on general manifolds* . 248 5.10.4 Topological invariants of four-manifolds* . 252 5.11 Exotic spheres* . 253 Bibliography 257 1 Chapter 1 Riemannian geometry 1.1 Review of tangent and cotangent spaces We will assume some familiarity with the theory of smooth manifolds, as pre- sented, for example, in the first four chapters of [5]. Suppose that M is a smooth manifold and p 2 M, and that F(p) denotes the space of pairs (U; f) where U is an open subset of M containing p and f : U ! R is a smooth function. If φ = (x1; : : : ; xn): U ! Rn is a smooth coordinate system on M with p 2 U, and (U; f) 2 F(p), we define @ −1 i (f) = Di(f ◦ φ )(φ(p)) 2 R; @x p where Di denotes differentiation with respect to the i-th component. We thereby obtain an -linear map R @ i : F(p) −! R; @x p called a directional derivative operator, which satisfies the Leibniz rule, ! ! @ @ @ i (fg) = i (f) g(p) + f(p) i (g) ; @x p @x p @x p and in addition depends only on the \germ" of f at p, @ @ f ≡ g on some neighborhood of p ) (f) = (g): @xi p @xi p The set of all linear combinations n X i @ a @xi i=1 p 2 of these basis vectors comprises the tangent space to M at p and is denoted by 1 n TpM. Thus for any given smooth coordinate system (x ; : : : ; x ) on M, we have a corresponding basis ! @ @ 1 ;:::; n @x p @x p for the tangent space TpM. The notation we have adopted makes it easy to see how the components (ai) of a tangent vector transform under change of coordinates. If = (y1; : : : ; yn) is a second smooth coordinate system on M, the new basis vectors are related to the old by the chain rule, n j j @ X @x @ @x j −1 = (p) ; where (p) = Di(x ◦ )( (p)): @yi @yi @xj @yi p j=1 p The disjoint union of all of the tangent spaces forms the tangent bundle [ TM = fTpM : p 2 Mg; 1 n which has a projection π : TM ! M defined by π(TpM) = p. If φ = (x ; : : : ; x ) is a coordinate system on U ⊂ M, we can define a corresponding coordinate system φ~ = (x1; : : : ; xn; x_ 1;:::; x_ n) on π−1(U) ⊂ TM by letting 0 1 0 1 n n i X j @ i i X j @ i x @ a A = x (p); x_ @ a A = a : (1.1) @xj @xj j=1 p j=1 p For the various choices of charts (U; φ), the corresponding charts (π−1(U); φ~) form an atlas making TM into a smooth manifold of dimension 2n, as you saw in Math 240A. ∗ The cotangent space to M at p is simply the dual space Tp M to TpM. Thus ∗ an element of Tp M is simply a linear map α : TpM −! R: Corresponding to the basis ! @ @ 1 ;:::; n @x p @x p of TpM is the dual basis ! ( 1 n i @ i 1; if i = j, dx jp; : : : ; dx jp ; defined by dx jp j = δj = @x p 0; if i 6= j. 3 ∗ The elements of Tp M, called cotangent vectors, are just the linear combinations of these basis vectors n X i aidx jp i=1 Once again, under change of coordinates the basis elements transform by the chain rule, n X @yi dyij = (p)dxjj : p @xj p j=1 An important example of cotangent vector is the differential of a function at a point. If p 2 U and f : U ! R is a smooth function, then the differential of ∗ 1 n f at p is the element dfjp 2 Tp M defined by dfjp(v) = v(f). If (x ; : : : ; x ) is a smooth coordinate system defined on U, then n X @f dfj = (p)dxij : p @xi p i=1 Just as we did for tangent spaces, we can take the disjoint union of all of the cotangent spaces forms the cotangent bundle ∗ [ ∗ T M = fTp M : p 2 Mg; 1 n which has a projection π : TM ! M defined by π(TpM) = p. If φ = (x ; : : : ; x ) is a coordinate system on U ⊂ M, we can define a corresponding coordinate system ~ 1 n −1 φ = (x ; : : : ; x ; p1; : : : ; pn) on π (U) ⊂ TM by letting 0 1 0 1 n n i X j @ i X j x @ a A = x (p); pi @ ajdx jpA = ai: @xj j=1 p j=1 For the various choices of charts (U; φ), the corresponding charts (π−1(U); φ~) form an atlas making T ∗M into a smooth manifold of dimension 2n.