= Scalar curvature of an umbilic vacuum hypersurface
{title2=${}^{(3)}R=6\varepsilon\lambda^2$}
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
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