Day & Time: Tuesday and Thursday 11:40pm--12:55pm Location: 203 Malott Instructor: Xiaodong Cao
Teaching assistants: Textbook:John Lee, Riemannian Manifolds: An introduction to curvature. ISBN 038798271x. It is available at the University bookstore. Prerequisites: Differential manifolds. Examinations: There will be one take home exam. |
This course will be an introduction to Riemannian geometry. We will cover the following topics: linear connections, Riemannian metric and parallel translation, covariant derivative and curvature tensors, the exponential map, the Gauss lemma and completeness of the metric, isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard, the first and second variation formulas, the index form of Morse and the theorem of Bonnet-Myers, the Rauch, Hessian, and Laplacian comparison theorems, the Morse index theorem, the conjugate and cut loci, submanifolds and the second fundamental form.
Class | Topic | Read | Exercises | Due |
Jan. 26 | Review | |||
Jan. 31 | Review | |||
Feb. 2 | Reimannian metric, linear connection | HW1 | ||
Feb. 7 | Levi-Civita connection | |||
Feb. 9 | Parallel translation | |||
Feb. 14 | Covariant derivative | |||
Feb. 16 | Riemannian curvature (I) | |||
Feb. 28 | Riemannian curvature (II) | |||
Mar. 2 | Exponential map | |||
Mar. 7 | Geodesics and minimizing curves | |||
Mar. 9 | Hopf-Rinow Theorem | HW2 | ||
Mar. 14 | Totally geodesic submanifold, second foundamental form | |||
Mar. 16 | Space forms | |||
Mar. 21 | Jacobi fields | |||
Mar. 23 | Cartan-Hadamard Theorem | |||
Mar. 28 | First and second variation formulas | |||
Mar. 30 | Application of first and second variation formulas | |||
Apr. 1 | Morse Index Form | |||
Apr. 3 | Bonnet-Myers Theorem | Textbook: P113, 6-11, P129, 7-5, P191, 10-3 | ||
Apr. 8 | Rauch comparison Theorem | |||
Apr. 10 | Other comparison Theorems | |||
Apr. 15 | Morse index Theorem | |||
Apr. 17 | From curvature to topology | Berger, Ch 12 | Hw4 | |
Apr. 29 | Symplectic geometry (by Tara Holm) | |||
May 1 | Conjugate and cut locus | |||
May 2 | Cut locus | |||
May 7 | Submanifolds and second fundamental form | |||
May 8 | Foundamental equations | |||