# level sets of smooth functions on manifolds

Let $f:{\mathbb{R}}^{n}\to \mathbb{R}$ be smooth. Further suppose that the gradient^{} of $f$ differs from zero at every point of a level set. Then it follows from the implicit function theorem^{} that that level set is a smooth hypersurface. Furthermore, at any point of the level set, the gradient of the function^{} at that point is orthogonal^{} to the level set.

One can generalize this observation to manifolds. Suppose that $M$ is a smooth manifold and that $f:M\to \mathbb{R}$ is smooth. Further suppose that the gradient of $f$ differs from zero at every point of a level set. Then it follows from the implicit function theorem that that level set is a smooth hypersurface. If one chooses a Riemannian metric on the manifold, the gradient of the function at that point will be orthogonal to the level set.

Title | level sets of smooth functions on manifolds |
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Canonical name | LevelSetsOfSmoothFunctionsOnManifolds |

Date of creation | 2013-03-22 15:20:02 |

Last modified on | 2013-03-22 15:20:02 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 4 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 03E20 |