A Study of Riemannian Geometry

Abstract

In this reading project I studied some interesting results in Riemannian geometry. Starting from the definition of Riemannian metric, geodesics and curvature this thesis covers deep results such as Gauss-Bonnet theorem, Cartan-Hadamard theorem, Hopf-Rinow theorem and the Morse index theorem. Along the way it introduces useful tools such as Jacobi fields, variation formulae, cut locus etc. It finally builds up to the proof of the celebrated Sphere theorem using some basic Morse theory.