Hilbert's lemma was proposed at the end of the 19th century by mathematician
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
. The lemma describes a property of the
principal curvatures
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by d ...
of surfaces. It may be used to prove
Liebmann's theorem that a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
surface with constant
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
must be a sphere.
Statement of the lemma
Given a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
in three dimensions that is
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
and
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
over a patch containing the point ''p'', where ''k'' and ''m'' are defined as the principal curvatures and ''K''(''x'') is the
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
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.
[.]
See also
*
Hilbert's theorem (differential geometry)
In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative gaussian curvature K immersed in \mathbb^. This theorem answers the question for the negative case of which surfaces in ...
References
Lemmas
Differential geometry
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