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:

  1. 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)
  2. Riemannian geometry: Riemannian metrics, connections, geodesics. (References: GLH, Spivak vol II).
  3. Riemannian curvature tensor; Ricci and scalar curvature. Examples: curvature of Lie groups. Riemannian submersions and curvature.(Reference: GLH)
  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....

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