# Gauss-Bonnet theorem

(Carl Friedrich Gauss and Pierre Ossian Bonnet) Given a two-dimensional compact^{} Riemannian manifold^{} $M$ with boundary,
Gaussian curvature of points $G$ and geodesic curvature of points ${g}_{x}$ on the boundary $\partial M$, it is the
case that

$${\int}_{M}G\mathit{d}A+{\int}_{\partial M}{g}_{x}\mathit{d}s=2\pi \chi (M),$$ |

where $\chi (M)$ is the Euler characteristic^{} of the manifold^{}, $dA$ denotes the measure with respect to area, and $ds$ denotes the measure with respect to arclength on the boundary. This theorem expresses a topological invariant^{} in
terms of geometrical information.

Title | Gauss-Bonnet theorem |
---|---|

Canonical name | GaussBonnetTheorem |

Date of creation | 2013-03-22 16:36:37 |

Last modified on | 2013-03-22 16:36:37 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 53A05 |