= Gauss–Codazzi equations for a spatial hypersurface
{c}
With signature $(-,+,+,+)$, future <unit normal> $n$ and $K_{ab}=-P^c{}_aP^d{}_b\nabla_cn_d$, the <Gauss–Codazzi equations> become $(\perp R)_{abcd}=\mathcal R_{abcd}+K_{ac}K_{db}-K_{ad}K_{cb}$ and $(\perp R)_{abcd}n^d=-D_aK_{bc}+D_bK_{ac}$, where in the second expression only the first three indices are projected. The intrinsic curvature and <spatial covariant derivative> use the spatial <induced metric>. The normal is timelike, so the normal-sign convention differs from a spacelike normal in a positive-definite ambient metric.
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