# Variationsrechnung im Grossen

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*H. Seifert; W. Threlfall*

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

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.