# Morse theory without nondegeneracy

## TL;DR

This paper extends Morse theory to smooth functions on compact Riemannian manifolds without requiring nondegeneracy, only assuming finitely many connected components in the critical locus.

## Contribution

It introduces a generalized Morse theory framework that relaxes the classical nondegeneracy condition, broadening its applicability.

## Key findings

- Morse theory applies to functions with degenerate critical points.
- The critical locus can have finitely many connected components.
- The theory provides new tools for analyzing smooth functions on manifolds.

## Abstract

We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any nondegeneracy assumptions except that the critical locus must have only finitely many connected components.