# 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 |
---|---|

Canonical name | MorseFunction |

Date of creation | 2013-03-22 13:53:15 |

Last modified on | 2013-03-22 13:53:15 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 8 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 58E05 |

Defines | non-degenerate critical point |