Let \(\nabla\) be a connection on a manifold. Then the Christoffel symbols of \(\nabla \) for \((\partial_i)\) are smooth functions defined by
\begin{equation*} \nabla_{\partial_i}\partial_j=\Gamma_{ij}^k\partial_k. \end{equation*}Let \(\nabla\) be a connection on a manifold. Then the Christoffel symbols of \(\nabla \) for \((\partial_i)\) are smooth functions defined by
\begin{equation*} \nabla_{\partial_i}\partial_j=\Gamma_{ij}^k\partial_k. \end{equation*}