Mathematics G4403. Modern Geometry II, Spring 2012
Tentative Syllabus
Riemannian Geometry (continued).
- Complete manifolds, Hopf-Rinow and Cartan-Hadamard theorems
- Manifolds of constant sectional curvature
- First and second variations of energy
- Bonnet-Myers theorem and Synge-Wienstein theorem
- The Rauch comparison theorem
References: [dC] Chapter 7-10, [GHL] Chapter 3, [CE] Chapter 1
Principal Bundles.
- Principal bundles and associate bundles
- Connections and curvatures on a principal bundle
- Induced connections and curvatures on associated vector bundles
- Parallel transport and holonomy
- Characteristic classes
References: [KN] Chapter I Section 5, Chapter II, Chapter XII, [M] Chapter 5, 6
Witten's Proof of the Positive Mass Theorem. Reference: [PT], [W]