/ Math 110b

Math 110b: Differential Geometry

Spring 2004


Instructor: Prof. Daniel Ruberman
E-mail: ruberman@brandeis.edu
Office: Goldsmith 310
Phone: 736-3074
Class schedule: Mon Wed Thu 12-1, Goldsmith 100.
Office hours: Monday 11-12, Wednesday 1-2, Thursday 11-12, and by appointment. My schedule for spring 2004.
Textbook: Riemannian Geometry by Manfredo do Carmo.
Additional Sources:

  1. A Comprehensive Introduction to Differential Geometry, by Michael Spivak
  2. Characteristic Classes, by John Milnor and Richard Stasheff.
  3. Elliptic Operators, Topology and Asymptotic Methods, by John Roe.

Math 110b is a course in basic Riemannian geometry, with a modest amount of Lie groups. I will use the book "Riemannian Geometry," by Do Carmo; it should be available in the bookstore. Not all the material I want to cover is in the book, so I will supply some supplementary readings as necessary. The only prerequisite is Math 110a or equivalent background.

Projected outline of the course:

1. Baby Lie Groups: basic definition, Lie Algebra, left invariant vector
fields and forms. Exponential map. Basic examples. Principal bundles,
homogeneous spaces. (References: Do Carmo, Warner, Spivak vol I)

2. Riemannian metrics, connections, geodesics. (References: Do Carmo, Spivak vol II).

3. Curvature. Riemannian curvature tensor; Ricci and scalar curvature.
Examples: curvature of Lie groups. Riemannian submersions and curvature.
(Reference: Do Carmo)

4. Connections and curvature of vector bundles and (maybe) principal
bundles. (Reference: Spivak, Milnor (Characteristic Classes))

5. Assorted topics. Some possibilities: First/second variation of length
along geodesics, topology and curvature, submanifold geometry, curvature and fundamental group.... (Reference: Do Carmo).

There will be written exercises, and possibly presentation of some topics in class by students.

Homework

  1. Homework 1. Due Thursday 2/12
  2. Homework 2. Due Monday 3/8
  3. Homework 3. Due Friday 5/7