Normal forms of functions with degenerate singularities on surfaces equipped with semi-free circle actions

TL;DR

This paper develops a normal form classification for a class of smooth functions with degenerate singularities on surfaces with semi-free circle actions, generalizing Morse-Bott functions without saddles.

Contribution

It establishes a normal form theorem for these functions, showing they can be represented via a composition involving a simple Morse function, a diffeomorphism, and a smooth reparameterization.

Findings

Normal form representation for functions with degenerate singularities.

Extension of Morse-Bott function theory to new classes of functions.

Applicable to surfaces like cylinders, tori, disks, and spheres.

Abstract

This article is devoted to the study of a certain class of smooth circle-valued functions on a cylinder $S^1\times [0,1]$, a torus $T^2$, a disk $D^2$ and a sphere $S^2$ which is a generalization of Morse-Bott functions without saddles. We established a "normal form" for functions from this class, namely, we proved that any such function $f$ can be presented in the form $f = \varkappa\circ f_0\circ h^{-1}$, where $f_0$ is the ``simplest'' Morse function on the given surface for some diffeomorphism $h$ and a smooth function $\varkappa$ satisfying some natural conditions.