 Totally umbilic hypersurface

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.