Gauss–Codazzi equations for a non-null hypersurface (${\mathcal{R}}_{abcd}=(PR{)}_{abcd}+\epsilon (K_{ac}{K}_{bd}-K_{ad}{K}_{bc})$)

For a hypersurface with unit normal vector $n$, put $\epsilon =g(n,n)=\pm 1$, $P=1-\epsilon n\otimes n^{♭}$, and $K(X,Y)=g({\nabla }_{X}n,Y)$ for tangent vectors. The induced metric is nondegenerate precisely when the normal is non-null. With $R(X,Y)Z=[{\nabla }_X,{\nabla }_Y]Z-{\nabla }_{[X,Y]}Z$ and $R_{abcd}=g(e_a,R(e_c,e_d)e_b)$, the Gauss formula is ${\nabla }_{X}Y=D_{X}Y-\epsilon K(X,Y)n$. Substituting twice into the ambient Riemann curvature tensor and projecting tangentially gives the displayed Gauss equation. Its normal projection gives $g(n,R(X,Y)Z)=-(D_{X}K)(Y,Z)+(D_{Y}K)(X,Z)$, the curved-ambient Codazzi equation. A null hypersurface needs additional geometric data because its induced metric is degenerate.