Toponogov's Theorem and Applications

These notes have been prepared for a series of lectures given at the College on Diierential Geometry at Trieste in the Fall of 1989. The lectures center around To-ponogov's triangle comparison theorem, critical point theory and applications. In the short amount of time available not all the aspects can becovered. We focus on those applications which seem to be most important and at the same time most suitable for an exposition. Some basic knowledge in geometry will be assumed. It has been provided by K. Grove in the rst series of these lectures. Nevertheless we try to keep the lectures selfcontained and independent as much as possible. For the result about the sum of Betti numbers in section 3.5 a lemma from algebraic topology is needed. A proof for this result has been provided in the appendix. I am indebted to U. Abresch for many helpful conversations and also for writing and typing the appendix.