Isolated singularities of hypersurfaces

TL;DR

The paper investigates the existence of cylindrical neighborhoods around isolated critical points of smooth functions, proving the conjecture for certain classes of critical points including cone-like and Rothe H hypothesis satisfying points.

Contribution

It proves the conjecture that all isolated critical points have cylindrical ball neighborhoods for specific classes of critical points, advancing understanding in singularity theory.

Findings

Conjecture holds for cone-like critical points.

Conjecture holds for Cornea reasonable critical points.

Conjecture holds for critical points satisfying Rothe H hypothesis.

Abstract

Introduced by Seifert and Threlfall, cylindrical neighborhoods is an essential tool in the Lusternik-Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate.