does any one know a way to prove or dispove (a counter example) that for a moving hypersurface in a Riemannian manifold, with velocity $v = v_{n} n$ where $n$ is a unit normal field, then the covariant derivative $\overline{\nabla}_{n}n$ is given by the $\overline{\nabla}_{n}n = - \overline{\nabla} v_{n}$ being $\overline{\nabla}$ the Riemannian connection on the ambient manifold?