In any smooth coordinate chart for \(M\), the coefficients of a Levi-Civita connection are given by
\begin{equation*} \Gamma_{ij}^k=\frac{1}{2}g^{kl}(\partial_ig_{jl}+\partial_jg_{il}-\partial_lg_{ij}). \end{equation*}The connection coefficients are called Christoffel symbols.