Given a Riemannian manifold \((M,g)\). A connection \(\nabla\) on \(TM\) is said to be symmetric if
\begin{equation*} \nabla_XY-\nabla_YX=[X,Y] \end{equation*}for all \(X,Y\in \mathfrak{X}(M)\). Note, \([X,Y]\) denotes the Lie-bracket of \(X\) and \(Y\).
Remark
- It is called symmetric because the connection coefficients in every coordinate frame satisfy \begin{equation*} \Gamma^k_{ij}=\Gamma^k_{ji}, \end{equation*} for all \(k,i,j\).