Variationsrechnung Im Grossen

Presents some of the significant developments in mathematics in the first half of the 20th Century. This book covers basic topological aspects and applications of Morse Theory. It focuses mainly on the variational problem of geodesics in a Riemannian manifold joining two given points.

This is an excellent account of what has now become known as ''Morse Theory'', written not long after the appearance of the seminal work by Morse. In the interest of simplicity and readability, the authors have not attempted to give the most general versions of the theorems. In one hundred pages, the reader is engagingly introduced to one of the most significant developments in mathematics in the first half of the 20th Century. The basic topological aspects and applications of Morse Theory are covered in the first chapter. The introduction includes an explanation of the familiar special case of the torus. The later two chapters cover the analysis that is used to establish the general results. In particular, the last chapter focuses mainly on the variational problem of geodesics in a Riemannian manifold joining two given points. This analysis then leads to results such as the Morse inequality and conditions for the equality on manifolds with a Riemannian metric.