= Solution
The definition of the <extrinsic curvature of a spatial hypersurface> and $\perp^\rho{}_\mu=\delta^\rho{}_\mu+n^\rho n_\mu$ give
$$
K_{\mu\nu}
=-\left(\delta^\rho{}_\mu+n^\rho n_\mu\right)
\nabla_\rho n_\nu
=-\nabla_\mu n_\nu-n_\mu a_\nu,
$$
where $a_\nu=n^\rho\nabla_\rho n_\nu$ is the <normal acceleration>. Hence
$$
\boxed{-\nabla_\mu n_\nu=K_{\mu\nu}+n_\mu a_\nu.}
$$
Differentiating $n^\nu n_\nu=-1$ shows $n^\nu a_\nu=0$, so the acceleration is spatial as required.
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