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Hilbert's lemma was proposed at the end of the 19th century by mathematician David Hilbert. The lemma describes a property of the principal curvatures of surfaces. It is a result of the proof of Liebmann's lemma.
Statement of the lemma
Given a manifold in three dimensions that is smooth and differentiable over a patch containing the point p, where k and m are defined as the principal curvatures and K(x) is the Gaussian curvature at a point x, if k has a max at p, m has a min at p, and k is strictly greater than m at p, then K(p) is a non-positive real number.