Riemannian Geometry (MATH 6620, Spring 2017)

[ Schedule of Lectures ]

Day & Time: Tuesday and Thursday 11:40pm--12:55pm

Location: 203 Malott

Instructor: Xiaodong Cao

Office Hours: Tuesday & Thursday 2:00-2:50pm or by appointment, in 521 Malott

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.

 [textbook image]

Introduction

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.

Schedule of Lectures (Tentative!)

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