# sectional curvature

Let $M$ be a Riemannian manifold^{}. Let $p$ be a point in $M$ and let $S$ be a two-dimensional subspace of ${T}_{p}M$. Then the *sectional curvature ^{}* of $S$ at $p$ is defined as

$$K(S)=\frac{g(R(x,y)x,y)}{g(x,x)g(y,y)-g{(x,y)}^{2}}$$ |

where $x,y$ span $S$, $g$ is the metric tensor and $R$ is the Riemann’s curvature tensor.

This is a natural generalization of the classical Gaussian curvature^{} for surfaces.

Title | sectional curvature |
---|---|

Canonical name | SectionalCurvature |

Date of creation | 2013-03-22 15:54:15 |

Last modified on | 2013-03-22 15:54:15 |

Owner | juanman (12619) |

Last modified by | juanman (12619) |

Numerical id | 5 |

Author | juanman (12619) |

Entry type | Definition |

Classification | msc 53B21 |

Classification | msc 53B20 |

Related topic | RiemannianMetric |