Given a Riemannian manifold \((M,g)\). A connection \(\nabla\) on \(TM\) is called compatible with \(g\) or simply metric connection if it satisfies for \(X,Y,Z\in \mathfrak{X}(M)\)
\begin{equation*} \nabla_X\langle Y, Z\rangle=\langle \nabla_XY, Z\rangle+\langle Y, \nabla_XZ\rangle. \end{equation*}.
Remarks
- Consider a metric connection. Then \(\lvert \gamma'(t)\rvert\) is constant if and only if \(D_t\gamma'(t)\) is orthogonal to \(\gamma'(t)\) for all \(t\in I\) [@lee2013smooth_manifolds, Corollary 5.6].