## You are here

HomeLevi-Civita connection

## Primary tabs

# Levi-Civita connection

On any Riemannian manifold $\langle M,g\rangle$, 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 $\langle,\rangle\colon TM\times TM\to\mathbb{R}$ induced by $g$ as follows: For all vector fields $X,Y,Z\in TM$, one has

$X(\langle Y,Z\rangle)=\langle\nabla_{X}Y,Z\rangle+\langle Y,\nabla_{X}Z\rangle$ |

and

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

This connection is called the Levi-Civita connection.

In local coordinates $\{x_{1},\ldots,x_{n}\}$, the Christoffel symbols $\Gamma^{i}_{{jk}}$ are determined by

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

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

53B05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections