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 $\inn{,} \colon TM \times TM \to \mathbb{R}$ induced by $g$ as follows: For all vector fields $X, Y, Z \in TM$ , one has$$ X(\inn{Y,Z})=\inn{\nabla_X Y, Z}+\inn{Y, \nabla_X Z}$$ and$$ \nabla_X Y-\nabla_Y X=[X,Y]$$

This connection is called the Levi-Civita connection.

In local coordinates $\{x_1,\ldots,x_n\}$ , the Christoffel symbols $\Gamma^i_{jk}$ are determined by $$g_{i\ell}\Gamma^i_{jk}=\frac 12\left(\frac{\partial g_{j\ell}}{\partial x_k}+ \frac{\partial g_{k\ell}}{\partial x_j}-\frac{\partial g_{jk}}{\partial x_\ell}\right).$$