Wolfgang Meyer

On differentiable functions with isolated critical points

THE PURPOSE of this paper is to describe some quantitative aspects of a Morse theory for differentiable functions on a manifold which have only isolated but possibly degenerate critical points. We will show that around such points locally, a function splits into degenerate and non-degenerate parts, in terms of which the relative homology can be expressed in a certain way when passing a critical level. Our investigation originated in connection with a specific geometric problem, namely to prove the existence of infinitely many geometrically distinct periodic geodesics for a very large class of compact riemannian manifolds, see [3], but the results of this paper may be useful for other applications of Morse theory as well. We wish to thank Ralph Abraham and Alan Weinstein for helpful conversations.

Riemannsche Mannigfaltigkeiten als metrische Räume

Wir beschaftigen uns jetzt mit der naturlichen metrischen Struktur, die man jeder zusammenhangenden Riemannschen Mannigfaltigkeit zuordnen kann.

Differenzierbare Mannigfaltigkeiten und Abbildungen

Ist G offene Teilmenge von ℝn, so heist eine Abbildung f: G → ℝk differenzierbar, wenn die partiellen Ableitungen beliebiger Ordnung aller Komponentenfunktionen existieren und stetig sind. “Differenzierbar” steht also immer synonym fur “C∞”.