Hilbert's lemma

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

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

References
  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.
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