Levi-Civita connection

On any Riemannian manifold $⟨M,g⟩$, 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 $⟨,⟩:TM\times TM\to ℝ$ induced by $g$ as follows: For allvector fields $X,Y,Z\in TM$, one has

$$X(⟨Y,Z⟩)=⟨{\nabla }_{X}Y,Z⟩+⟨Y,{\nabla }_{X}Z⟩$$

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

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