Cohn-Vossen's inequality
Relates the integral of Gaussian curvature of surfaces to the Euler characteristic From Wikipedia, the free encyclopedia
Relates the integral of Gaussian curvature of surfaces to the Euler characteristic From Wikipedia, the free encyclopedia
In differential geometry, Cohn-Vossen's inequality, named after Stefan Cohn-Vossen, relates the integral of Gaussian curvature of a non-compact surface to the Euler characteristic. It is akin to the Gauss–Bonnet theorem for a compact surface.
A divergent path within a Riemannian manifold is a smooth curve in the manifold that is not contained within any compact subset of the manifold. A complete manifold is one in which every divergent path has infinite length 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 and finite Euler characteristic, we have[1]
where K is the Gaussian curvature, dA is the element of area, and χ is the Euler characteristic.
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