A New Proof of Liebmann Classical Rigidity Theorem for Surfaces in Space Forms
Juan A. Aledo, Luis J. Alı́as, Alfonso Romero · Rocky Mountain Journal of Mathematics · 2005
In this paper we provide a new direct proof of the Liebmann classical rigidity theorem for surfaces in space forms, showing that the only compact surfaces with constant Gaussian curvature which are immersed into the Euclidean space E 3 , into the hyperbolic space H 3 , or into an open hemisphere S 3 + are the totally umbilical round spheres.Our proof is an application of the Gauss-Bonnet theorem along with a formula involving the Gaussian curvatures of the first and second fundamental forms of the surface, which is interesting per se.