differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, Cohn-Vossen's inequality, named after

Stefan Cohn-Vossen Stefan Cohn-Vossen (28 May 1902 – 25 June 1936) was a mathematician, who was responsible for Cohn-Vossen's inequality and the Cohn-Vossen transformation is also named for him. He proved the first version of the splitting theorem. He was also ...

, relates the integral of

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 ...

of a non-compact

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

to the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

. It is akin to the

Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a ...

for 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. A divergent path within a

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

is a smooth curve in the manifold that is not contained within any

subset of the manifold. A complete manifold is one in which every divergent path has infinite

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

with respect to the Riemannian metric on the manifold. Cohn-Vossen's inequality states that in every complete Riemannian 2-manifold ''S'' with finite

total curvature In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: :\int_a^b k(s)\,ds. The total curvature of a closed curve i ...

and finite Euler characteristic, we haveRobert Osserman, ''A Survey of Minimal Surfaces'', Courier Dover Publications, 2002, page 86. :

\iint_S K \, dA \le 2\pi\chi(S),

where ''K'' is the Gaussian curvature, ''dA'' is the element of area, and ''χ'' is the Euler characteristic.

Examples

* If ''S'' is a compact surface (without boundary), then the inequality is an equality by the usual Gauss–Bonnet theorem for compact manifolds. * If ''S'' has a boundary, then the Gauss–Bonnet theorem gives ::

\iint_S K\, dA = 2\pi\chi(S) - \int_k_g\,ds

:where

k_g

is the

geodesic curvature In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's ...

of the boundary, and its integral the

which is necessarily positive for a boundary curve, and the inequality is strict. (A similar result holds when the boundary of ''S'' is piecewise smooth.) * If ''S'' is the plane R², then the curvature of ''S'' is zero, and ''χ''(''S'') = 1, so the inequality is strict: 0 < 2.

Notes and references

* * * *{{cite book, mr=2028047, last1=Shiohama, first1=Katsuhiro, last2=Shioya, first2=Takashi, last3=Tanaka, first3=Minoru, title=The geometry of total curvature on complete open surfaces, series=Cambridge Tracts in Mathematics, volume=159, publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, location=Cambridge, year=2003, isbn=0-521-45054-3, doi=10.1017/CBO9780511543159, zbl=1086.53056

External links

Gauss–Bonnet theorem, in the ''Encyclopedia of Mathematics''
including a brief account of Cohn-Vossen's inequality Theorems in differential geometry Inequalities