# Cohn-Vossen's inequality

In differential geometry, **Cohn-Vossen's inequality**, named after Stephan 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.

## 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

- where is the geodesic curvature of the boundary, and its integral the total curvature 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**R**^{2}, then the curvature of*S*is zero, and*χ*(*S*) = 1, so the inequality is strict: 0 < 2π.

## Notes and references

- Robert Osserman,
*A Survey of Minimal Surfaces*, Courier Dover Publications, 2002, page 86.

- S. E. Cohn-Vossen,
*Some problems of differential geometry in the large*, Moscow (1959) (in Russian)

## External links

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