= Solution
Take the orientation induced by the future-directed <unit normal> and positive <lapse function> $\alpha$. In coordinates adapted to the <3+1 decomposition of spacetime>, any component $\epsilon_{\ell ijk}$ with all four indices spatial vanishes: only three distinct spatial indices are available. The <shift vector> therefore contributes nothing, and
$$
\epsilon_{\mu ijk}n^\mu=\alpha^{-1}\epsilon_{0ijk}
=\alpha^{-1}\sqrt{|g|}\,[ijk]
=\sqrt\gamma\,[ijk].
$$
Here $[ijk]$ is the <Levi-Civita symbol>, and the last equality uses the <metric determinant in a 3+1 decomposition>, $|g|=\alpha^2\gamma$. The right side is precisely the spatial <metric volume tensor>, with component $\sqrt\gamma$ at $ijk=123$, the appropriate permutation sign, and zero for repeated indices. Hence
$$
\boxed{\tilde\epsilon_{ijk}=n^\mu\epsilon_{\mu ijk}.}
$$
More invariantly, the spatial <volume form> is the restriction to the <hypersurface> of the contraction of the spacetime <volume form> with $n$.
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