Totally umbilic hypersurface
= Totally umbilic hypersurface
{title2=$K_{ab}=\lambda h_{ab}$}
At every point its second fundamental form is a scalar multiple of its induced metric. Equivalently its shape operator is scalar, so all principal normal curvatures agree. For a three-dimensional hypersurface, $K=3\lambda$ and $K_{ab}K^{ab}=3\lambda^2$. The sign of its intrinsic scalar in a vacuum Lorentzian ambient space then follows from the <Gauss equation for a nonnull hypersurface>.