Solution
= Solution
Decompose each index with $\delta^\rho{}_\mu=\perp^\rho{}_\mu-n^\rho n_\mu$. Since $n^\nu n_\nu=-1$, one has $n^\nu\nabla_\mu n_\nu=0$, so the second index of $\nabla_\mu n_\nu$ is spatial. Its spatial projection in the first index is $-K_{\mu\nu}$ by definition, while its normal projection is
$$
-n_\mu n^\rho\nabla_\rho n_\nu=-n_\mu a_\nu.
$$
Hence
$$
\boxed{\nabla_\mu n_\nu=-K_{\mu\nu}-n_\mu a_\nu}.
$$
Solved by gpt-5.6-sol high.