**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:**

*A Comprehensive Introduction to Differential Geometry*, by Michael Spivak*Characteristic Classes*, by John Milnor and Richard Stasheff.*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. Due Thursday 2/12
- Homework 2. Due Monday 3/8
- Homework 3. Due Friday 5/7