# Levi-Civita connection

On any Riemannian manifold^{} $\u27e8M,g\u27e9$, there is a unique torsion-free affine connection^{} $\nabla $ on the tangent bundle of $M$ such that the covariant derivative^{} of the metric tensor $g$ is zero, i.e. $g$ is covariantly constant. This condition can be also be expressed in terms of the inner product operation $\u27e8,\u27e9:TM\times TM\to \mathbb{R}$ induced by $g$ as follows: For all
vector fields $X,Y,Z\in TM$, one has

$$X(\u27e8Y,Z\u27e9)=\u27e8{\nabla}_{X}Y,Z\u27e9+\u27e8Y,{\nabla}_{X}Z\u27e9$$ |

and

$${\nabla}_{X}Y-{\nabla}_{Y}X=[X,Y]$$ |

This connection is called the Levi-Civita connection^{}.

In local coordinates $\{{x}_{1},\mathrm{\dots},{x}_{n}\}$, the Christoffel symbols^{} (http://planetmath.org/Connection) ${\mathrm{\Gamma}}_{jk}^{i}$ are determined by

$${g}_{i\mathrm{\ell}}{\mathrm{\Gamma}}_{jk}^{i}=\frac{1}{2}\left(\frac{\partial {g}_{j\mathrm{\ell}}}{\partial {x}_{k}}+\frac{\partial {g}_{k\mathrm{\ell}}}{\partial {x}_{j}}-\frac{\partial {g}_{jk}}{\partial {x}_{\mathrm{\ell}}}\right).$$ |

Title | Levi-Civita connection |
---|---|

Canonical name | LeviCivitaConnection |

Date of creation | 2013-03-22 13:59:25 |

Last modified on | 2013-03-22 13:59:25 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 7 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 53B05 |