Morse function

Summary

A Morse function is a smooth real-valued function on a smooth manifold all of whose critical points are nondegenerate. This means that the Hessian at each critical point is invertible. Near each critical point it has a simple quadratic normal form, given by the Morse lemma. Morse functions exist on every smooth manifold and are generic among smooth functions. They are the basic tool of Morse theory, which reads the topology of a manifold off the critical points of such a function.