Lectures on Differential Geometry Math 240C John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail:
[email protected] June 6, 2011 Preface This is a set of lecture notes for the course Math 240C given during the Spring of 2011. The notes will evolve as the course progresses. The starred sections are less central to the course, and may be omitted by some readers. i Contents 1 Riemannian geometry 1 1.1 Review of tangent and cotangent spaces . 1 1.2 Riemannian metrics . 4 1.3 Geodesics . 8 1.3.1 Smooth paths . 8 1.3.2 Piecewise smooth paths . 12 1.4 Hamilton's principle* . 13 1.5 The Levi-Civita connection . 19 1.6 First variation of J: intrinsic version . 25 1.7 Lorentz manifolds . 28 1.8 The Riemann-Christoffel curvature tensor . 31 1.9 Curvature symmetries; sectional curvature . 39 1.10 Gaussian curvature of surfaces . 42 1.11 Matrix Lie groups . 48 1.12 Lie groups with biinvariant metrics . 52 1.13 Projective spaces; Grassmann manifolds . 57 2 Normal coordinates 64 2.1 Definition of normal coordinates . 64 2.2 The Gauss Lemma . 68 2.3 Curvature in normal coordinates . 70 2.4 Tensor analysis . 75 2.5 Riemannian manifolds as metric spaces . 84 2.6 Completeness . 86 2.7 Smooth closed geodesics . 88 3 Curvature and topology 94 3.1 Overview . 94 3.2 Parallel transport along curves . 96 3.3 Geodesics and curvature . 97 3.4 The Hadamard-Cartan Theorem . 101 3.5 The fundamental group* .