Comparison Theorems in Riemannian Geometry: How Curvature Affects Topology

The main subject of this work are results of global Riemannian geometry. In all of these theorems we assume some local property of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including the topological type of the manifold. In the first part of the work, we study the Bonnet-Myers theorem, the Rauch comparison theorem and the Cartan-Hadamard theorem. The second part is devoted to the sphere theorem, which was proven by Berger and Klingenberg. After extensive preparations, we give a complete proof of the theorem. Finally, we briefly discuss some recent developments that build on the sphere theorem.

The author was born in 1986 in Vienna. After finishing school in June 2004, he studied Mathematics, Physics and Geography at the University of Vienna. After receiving his Diploma in Mathematics in September 2010, he moved to Potsdam for his doctorate. He finshed his doctorate studies in December 2013.