# Morse function

Let $M$ be a smooth manifold. A critical point of a map $u:M\to\mathbb{R}$ at $x\in M$ is called if the Hessian matrix $H_{u}$ (in any local coordinate system) at $x$ is non-degenerate.

A smooth function $u:M\to\mathbb{R}$ is called Morse if all its critical points are non-degenerate. Morse functions exist on any smooth manifold, and in fact form an open dense (http://planetmath.org/Dense) subset of smooth functions on $M$ (this fact is often phrased “a generic smooth function is Morse”).

Title Morse function MorseFunction 2013-03-22 13:53:15 2013-03-22 13:53:15 bwebste (988) bwebste (988) 8 bwebste (988) Definition msc 58E05 non-degenerate critical point