Gauss equation for a nonnull hypersurface (${}^{\Sigma }{R}_{abcd}=h_a{}^e{h}_b{}^f{h}_c{}^g{h}_d{}^k{R}_{efgk}+\epsilon (K_{ac}{K}_{bd}-K_{ad}{K}_{bc})$)

Let the unit normal satisfy $n^a{n}_a=\epsilon =\pm 1$, let $h_{ab}=g_{ab}-\epsilon n_a{n}_b$, and set $K_{ab}=h_a{}^c{h}_b{}^d{\nabla }_c{n}_d$. The normal component of an ambient derivative of tangent fields is $-\epsilon K(X,Y)n$. Substitute that decomposition into the curvature commutator and project tangentially to obtain the displayed equation. Its scalar contraction is ${}^{\Sigma }R=R-2\epsilon R_{ab}{n}^a{n}^b+\epsilon (K^2-K_{ab}{K}^{ab})$. Both metric signature and curvature-slot convention must be specified when assigning a sign.