Christoffel symbol

Definition

Since affine connections are local, then it makes sense to talk about them in relation to local coordinates. Let $x:W\to {\mathbb{R}}^n$ be local coordinates on an open set $W\subset M$. The Christoffel symbols are defined by

$${\nabla }_{\frac{\partial }{\partial x^i}}\frac{\partial }{\partial x^j}={\Gamma }_{ij}^k\frac{\partial }{\partial x^k}$$