Cornell Math - MATH 662, Spring 2000

MATH 662 — Spring 2000
Riemannian Geometry

I will cover the following topics: Linear connections, Riemannian metrics and parallel translation. Covariant differentiation and curvature tensors. The exponential map, the Gauss Lemma and completeness of the metric. Isometries and space forms. Jacobi fields and the theorem of Cartan-Hadamard. The first and second variation formulas. The index form of Morse and the theorem of Bonnet-Myers. The Rauch, Hessian, and Laplacian comparison theorems. The Morse index theorem. The conjugate and cut loci. Submanifolds and the Second Fundamental form.