Scalar Gauss equation
= Scalar Gauss equation
{c}
For the convention $K_{\mu\nu}=-\perp^\rho{}_\mu\nabla_\rho n_\nu$, contraction of the <Gauss equation> gives
$$
{}^{(4)}R+2R_{\mu\nu}n^\mu n^\nu
={}^{(3)}R+K^2-K_{\mu\nu}K^{\mu\nu}.
$$
= Scalar Gauss equation
{c}
For the convention $K_{\mu\nu}=-\perp^\rho{}_\mu\nabla_\rho n_\nu$, contraction of the <Gauss equation> gives
$$
{}^{(4)}R+2R_{\mu\nu}n^\mu n^\nu
={}^{(3)}R+K^2-K_{\mu\nu}K^{\mu\nu}.
$$