= Gauss equation for a nonnull hypersurface
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{title2=${}^\Sigma R_{abcd}=h_a{}^eh_b{}^fh_c{}^gh_d{}^kR_{efgk}+\varepsilon(K_{ac}K_{bd}-K_{ad}K_{bc})$}
Let the unit normal satisfy $n^an_a=\varepsilon=\pm1$, let $h_{ab}=g_{ab}-\varepsilon n_an_b$, and set $K_{ab}=h_a{}^ch_b{}^d\nabla_cn_d$. The normal component of an ambient derivative of tangent fields is $-\varepsilon K(X,Y)n$. Substitute that decomposition into the curvature commutator and project tangentially to obtain the displayed equation. Its scalar contraction is ${}^\Sigma R=R-2\varepsilon R_{ab}n^an^b+\varepsilon(K^2-K_{ab}K^{ab})$. Both metric signature and curvature-slot convention must be specified when assigning a sign.
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