MATH 567-RIEMANNIAN GEOMETRY I-FALL 2015
Syllabus
Course outline
8/20 Manifolds: definition, examples, partition of unity
8/25 tangent vector at a point, directional derivative, smooth maps and differentials, immersions and embeddings
8/27 submanifolds, proper maps and embeddings, regular surface in R^n, preimage of reg. value (Sard's tneorem)
9/1 Examples: O(n), properly discontinuous actions, orientability.
9/3 Tangent bundle, vector fields and their flows, Lie derivative
HW 1: (due 9/10) CH.0, Problems 3, 5, 8, 9
9/8 Cotangent bundle, differential forms, tensors on manifolds
9/10 class canceled
9/15 Riemannian metrics- examples, left-invariant metrics on Lie groups (taught by Ken Knox)
9/17 Volume form, connections, Levi-Civita connection (taught by Ken Knox)
9/22 Covariant derivative of tensors, covariant derivative along a curve, parallel transport
9/24 pullback connection, geodesics, geodesics as critical points of length.
9/29 continuation/discussion of HW problems
10/1 L-C connection on a submanifold/ parallel transport on the sphere/discusion of problems
10/6 continuation
10/8 Exponential map, Gauss Lemma (taught by Ken Knox)
10/13 Class canceled
10/15 FALL BREAK
10/20 Gauss lemma/minimizing properties of geodesics
10/22 length-minimizing implies geodesic/ geodesics of hyperbolic plane
10/27 discussion of problems (Ch. 3)
10/29 gradient, divergence, Hessian, Laplacian
11/3 Riemann curvature tensor/ example (hyperbolic plane)
11/5 sectional curvature/ symmetries of the Riemann tensor/ curvature operator
11/10 irreducible decomp. of curvature tensors/Weyl tensor/ Differential Bianchi identity
11/12 isometric imersions and submanifolds: second fundamental form
11/17 curvature equations for submaniflds/ fundamental theorem
11/19 Jacobi fields, conjugate points
11/24 totally geodesic submanifolds/ Cartan's theorem/ complete manifolds
11/26 Thanksgiving
12/ 1 Hopf-Rionow theorem/ Cartan-Hadamard theorem