Hilbert's lemma

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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 may be used to prove Liebmann's theorem that a compact surface with constant Gaussian curvature must be a sphere.[1]

Statement of the lemma[edit]

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.[2]

See also[edit]


  1. ^ Gray, Mary (1997), "28.4 Hilbert's Lemma and Liebmann's Theorem", Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.), CRC Press, pp. 652–654, ISBN 9780849371646 .
  2. ^ O'Neill, Barrett (2006), Elementary Differential Geometry (2nd ed.), Academic Press, p. 278, ISBN 9780080505428 .