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
∬ S K d A ≤ 2 π χ ( S ) , {\displaystyle \iint _{S}K\,dA\leq 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
∬ S K d A = 2 π χ ( S ) − ∫ ∂ S k g d s {\displaystyle \iint _{S}K\,dA=2\pi \chi (S)-\int _{\partial S}k_{g}\,ds}
where k g {\displaystyle k_{g}} 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 R2, then the curvature of S is zero, and χ(S) = 1, so the inequality is strict: 0 < 2π.
Notes and references
Cohn-Vossen, Stefan (1935). "Kürzeste Wege und Totalkrümmung auf Flächen". Compositio Mathematica. 2: 69–13. JFM 61.0789.01. MR 1556908. Zbl 0011.22501. Huber, Alfred (1957). "On subharmonic functions and differential geometry in the large". Commentarii Mathematici Helvetici. 32 (1): 13–72. doi:10.1007/BF02564570. hdl:2027/mdp.39015095254580. MR 0094452. Zbl 0080.15001. Li, Peter (2000). "Curvature and function theory on Riemannian manifolds". In Yau, S.-T. (ed.). Surveys in Differential Geometry. Vol. 7. Somerville, MA: International Press. pp. 375–432. doi:10.4310/SDG.2002.v7.n1.a13. ISBN 1-57146-069-1. MR 1919432. Zbl 1066.53084. Shiohama, Katsuhiro; Shioya, Takashi; Tanaka, Minoru (2003). The geometry of total curvature on complete open surfaces. Cambridge Tracts in Mathematics. Vol. 159. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511543159. ISBN 0-521-45054-3. MR 2028047. Zbl 1086.53056.
External links Gauss–Bonnet theorem, in the Encyclopedia of Mathematics, including a brief account of Cohn-Vossen's inequality