# 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 .

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

 $g_{i\ell}\Gamma^{i}_{jk}=\frac{1}{2}\left(\frac{\partial g_{j\ell}}{\partial x% _{k}}+\frac{\partial g_{k\ell}}{\partial x_{j}}-\frac{\partial g_{jk}}{% \partial x_{\ell}}\right).$
Title Levi-Civita connection LeviCivitaConnection 2013-03-22 13:59:25 2013-03-22 13:59:25 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Definition msc 53B05