## Math 110b Spring 2003

**Instructor: **Daniel Ruberman

**Hours:** Monday, Wednesday, Thursday 9-10 (subject to change).

Math 110b is a course in basic Riemannian geometry, with a modest amount of
Lie groups. I will use the book "Riemannian Geometry," by Gallot, Hulin, and
Lafontaine, referred to as GLH in the description below; 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:

- Baby Lie Groups: basic definition, Lie Algebra, left invariant vector
fields and forms. Exponential map. Basic examples. Principal bundles,
homogeneous spaces. (References: GLH, Warner, Spivak vol I)
- Riemannian geometry: Riemannian metrics, connections, geodesics. (References: GLH, Spivak vol II).
- Riemannian curvature tensor; Ricci and scalar curvature.
Examples: curvature of Lie groups. Riemannian submersions and curvature.(Reference: GLH)
- Connections and curvature of vector bundles and (maybe) principal
bundles. (Reference: Spivak, Milnor (Characteristic Classes))
- Assorted topics. Some possibilities: First/second variation of length
along geodesics, Topology and curvature, submanifold geometry, curvature and
fundamental group....

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