Equal sized hyperbolic fish from M. C. Escher

Math 537B Spring 2006

Room/Time: Tuesday, Thursday 12:30-1:45 in Math East 246
Instructor: David Glickenstein
Office: Mathematics 213
Office Phone: 621-2463
Email:glickenstein@math.arizona.edu

Presentations on May 4

Presentations will be given May 4 (Dead Day) in Math 320 1pm to 5pm (if it takes that long).

Syllabus summary

We will begin by finishing Lee's book, Chapters 7-11. After that, we will explore one or more of the following topics: Geometrization of three-manifolds, Laplacian comparison, Bishop comparison theorem, the sphere theorem, Gromov's compactness theorem, Cheeger's finiteness theorem.

Printed resources

The books we are currently using are:

Riemannian Manifolds: An Introduction to Curvature by John Lee. He has errata here.
Ricci Flow: An Introduction by Ben Chow and Dan Knopf

Other books of note:

Elementary Differential Geometry Lecture Notes by Gilbert Weinstein.
Riemannian Geometry by M. P. Do Carmo
A Comprehensive Introduction to Differential Geometry by M. Spivak
Riemannian Geometry: A Modern Introduction by I. Chavel
Lectures on Differential Geometry by R. Schoen and S. T. Yau
Comparison Theorems in Riemannian Geometry by J. Cheeger and D. Ebin
Riemannian Geometry by Peter Petersen

Homework assignments

Lecture notes

These lecture notes are not particularly well edited and not for general distribution, which is why the files are password protected.

Possible project topics

The following are possible project topics. Please talk with me about the scope of the work. You are also welcome to suggest other topics.

Simplectic geometry
Spherical and hyperbolic trigonometry
Hodge theorem
Finsler geometry
Nonlinear averaging and center of mass
Moduli space of flat metrics
Frobenius theorem
Grassmannians
Bochner technique
Riemannian submersions
Chern-Gauss-Bonnet theorem
Morse Index Theorem

Last updated May 2, 2006