Cohn-Vossen's inequality

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
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, 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
If S is the plane R², then the curvature of S is zero, and χ = 1, so the inequality is strict: 0 < 2.