$P$-categories and functions with degenerate singular submanifolds

Abstract

We study smooth functions on manifolds whose sets of critical points is a disjoint union of submanifolds each diffeomorphic to the same manifold P. These submanifolds are not necessarily non-degenerate. We obtain sufficent conditions for existence of functions with minimal number of such submanifolds. This a translation from Ukrainian of PhD defended in 1993 at the Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine. Scientific advisor: Volodymyr Sharko.