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