Morse function

Let $M$ be a smooth manifold. A critical point of a map $u:M\to ℝ$ 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 $u:M\to ℝ$ 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”).