The inverse spacetime metric decomposes as
$$
g^{\mu\nu}=\perp^{\mu\nu}-n^\mu n^\nu.
$$
Contracting the <covariant derivative> of $Z_\nu$ and recognizing the fully projected contraction as its <spatial covariant derivative> gives
$$
\boxed{
\nabla^\mu Z_\mu
=-n^\mu n^\nu\nabla_\mu Z_\nu+D^\mu Z_\mu.
}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 1 / a / iii / Solution

{title2=$D_i$}

The spatial covariant derivative projects the spacetime <covariant derivative> and every free tensor index onto a spatial hypersurface. It is the <Levi-Civita connection> of the induced spatial metric.

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