Math G4403. Modern Geometry II, Spring 2010
Tentative Syllabus
Riemannian Geometry (continued):
- Hopf-Rinow and Cartan-Hadamard theorems
- Manifolds of constant sectional curvature
- First and second variations of energy
- Bonnet-Myers theorem
- The Rauch comparison theorem
- Curvature and the fundamental group
- Curvature tensors and representations of the orthogonal group
Vector Bundles and Principal Bundles:
- Real and complex vector bundles
- Metrics, connections, and curvature on vector bundles
- Preliminaries in differential topology
- Chern, Pontryagin, and Euler classes
- Principal bundles
- Connections and curvature on principal bundles
- Parallel transport and holonomy
- Elliptic complexes
If time permits, we will discuss some other topics
(to be determined later).