Math 6620: Riemannian Geometry

This is an introduction to Riemannian Geometry. Topics will include Riemannian metrics, the curvature tensor, geodesics, the Cartan-Hadamard theorem, and various comparison theorems.

Time and Place: TR 11:40-12:55, Malott 205.
Instructor: Jason Manning, Email: jfmanning@math.cornell.edu
Office hours: TR 2:45-3:45 in Malott 573.
Web page: http://pi.math.cornell.edu/~jfmanning/teaching/2019S6620.html

Prerequisites

You should be comfortable with differential topology (as covered in Math 6520) and with covering spaces and fundamental group (as covered in Math 6510). Please talk with me if you haven't taken or tested out of the 6510/6520 sequence.

References

(The links should work on campus, or off-campus, using PassKey. They should give access to the full text of the references.)

[L] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature. This is the main text for the course. The other books below may be used for supplemental material or alternate perspectives.
[B] M. Berger, A Panoramic View of Riemannian Geometry.
[CE] J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry. (No link, sorry.)
[dC] M. do Carmo, Riemannian Geometry. (No link.)

Problem sets

Here are the problem sets posted so far:

Homework 1, due February 7.
Homework 2, due February 19.
Homework 3, due March 19.
Homework 4, due April 24.

Schedule

The following schedule is incomplete and subject to change. Days marked with asterisks will either have a guest lecturer or be rescheduled.

Date	Topic	Reading
22 Jan	Introduction to the course. Review.	[L], Ch. 1-2.
24 Jan	Riemannian metric, some examples.	[L], Chapter 3
29 Jan	Connections	[L], Chapter 4
31 Jan	Parallel translation
5 Feb	Riemannian geodesics	[L], Chapter 5
7 Feb	Exponential map
12 Feb	Local minimization	[L], Chapter 6
14 Feb	Gauss Lemma
19 Feb	Hopf-Rinow
21 Feb	Curvature	[L], Chapter 7
26 Feb	NO CLASS (February Break)
* 28 Feb	NO CLASS
5 Mar	Curvature identities
7 Mar	Riemannian submanifolds	[L], Chapter 8
12 Mar	Sectional curvature
14 Mar	Gauss-Bonnet	[L], Chapter 9
19 Mar	Gauss-Bonnet ctd.
21 Mar	Gauss-Bonnet
26 Mar	Jacobi Fields	[L], Chapter 10
28 Mar	Jacobi Fields
2/4 Apr	NO CLASS (Spring break)
9 Apr	Gauss-Bonnet
11 Apr	Applications of Jacobi fields
16 Apr	Comparison theorems	[L], Chapter 11
18 Apr
23 Apr
25 Apr
30 Apr
2 May
7 May

Jason Manning's home page.

Last Updated 2019-04-15