Math 865 Advanced Topics in Geometry (Fall 2011)

Lectures

Recommended Textbooks

1. Riemannian Geometry by Peter Petersen.
2. Differential Geometric Structures by Poor.
3. Einstein Manifolds by Besse.
4. Lecture notes in Geometric Analysis (Old notes, see new ones below).

Mailing List

Selected Topics

• First few lectures will be a quick review of tensor calculus and Riemannian geometry: metrics, connections, curvature tensor, Bianchi identities, commuting covariant derivatives, etc.
• Decomposition of curvature tensor into irreducible summands.
• Bochner-Weitzenbock formulas: various curvature conditions yield topological restrictions on a manifold.
• Review of elliptic theory in Holder and Sobolev spaces. Theory of elliptic operators on Riemannian manifolds with basic Fredholm Theory, with applications to Hodge Theory.
• On non-compact manifolds we will consider Fredholm operators on weighted spaces, such as weighted Sobolev and Holder spaces. This has applications to the study of asymptotically locally Euclidean spaces (ALE) spaces, such as the Eguchi-Hanson metric.

Lectures

• Course lecture notes (will be frequently updated).