“Hilbert's lemma”的意思、由来-开放百科全书

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Hilbert's lemma

释义

Statement of the lemma
See also
References

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]

References

1. ^{{citation|title=Modern Differential Geometry of Curves and Surfaces with Mathematica|first=Mary|last=Gray|edition=2nd|publisher=CRC Press|year=1997|isbn=9780849371646|contribution=28.4 Hilbert's Lemma and Liebmann's Theorem|pages=652–654|url=https://books.google.com/books?id=-LRumtTimYgC&pg=PA652}}.
2. ^{{citation|title=Elementary Differential Geometry|first=Barrett|last=O'Neill|edition=2nd|publisher=Academic Press|year=2006|isbn=9780080505428|page=278|url=https://books.google.com/books?id=OtbNXAIve_AC&pg=PA278}}.

2 : Lemmas|Differential geometry

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Statement of the lemma

See also

References