Marston Morse

Bimeasures and their integral extensions

SummaryThe object of this paper is to motivate three papers byM. Morse andW. Transue, which will presently appear, where the Authors will lay the foundations of a general theory of C-bimeasures and their integral extensions.

THE ELIMINATION OF CRITICAL POINTS OF A NON-DEGENERATE FUNCTION ON A DIFFERENTIABLE MANIFOLD,

Abstract : M is a compact, connected, orientable, differentiable n-manifold of class C, and f is a non-degenerate function of class C. This note reveals certain fundamental topological characteristics of M by combining the study of the critical points of f with the study of certain differentiable submanifolds of M associated with the respective critical points of f and termed bowls of f. An attempt is made to modify f so as to eliminate as many critical points as possible, replacing f by another nondegenerate function. This is accomplished mainly through the development of a theorem concerning bowls and the existence of a non-degenerate function without critical points in a given neighborhood. (Author)