= Gauss–Codazzi equations for a non-null hypersurface
{c}
{title2=$\mathcal R_{abcd}=(PR)_{abcd}+\varepsilon(K_{ac}K_{bd}-K_{ad}K_{bc})$}
For a <hypersurface> with unit <normal vector> $n$, put $\varepsilon=g(n,n)=\pm1$, $P=1-\varepsilon n\otimes n^\flat$, and $K(X,Y)=g(\nabla_Xn,Y)$ 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 $\nabla_XY=D_XY-\varepsilon K(X,Y)n$. Substituting twice into the ambient <Riemann curvature tensor> and projecting tangentially gives the displayed <Gauss equation>. Its normal projection gives $g(n,R(X,Y)Z)=-(D_XK)(Y,Z)+(D_YK)(X,Z)$, the curved-ambient <Codazzi equation>. A null hypersurface needs additional geometric data because its induced metric is degenerate.
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