Totally umbilic hypersurface ($K_{ab}=\lambda h_{ab}$)

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.

For a timelike unit normal with squared norm epsilon and vanishing ambient Ricci tensor, contraction of the Gauss equation for a nonnull hypersurface gives the displayed formula. In signature $(+---)$ the timelike normal has epsilon plus one, so the scalar of the induced negative-definite metric is nonnegative. In signature $(-+++)$ the positive-definite spatial metric instead has scalar $-6{\lambda }^2$. Replacing the induced metric by its negative reverses the scalar, so the inequality must not be transferred between these conventions without adjustment.

// Target: general-relativity.bigb