Characterizations of the sphere by the curvature of the second fundamental form

Abstract

On an ovaloid S S with Gaussian curvature K ( I) > 0 K({\text {I)}} > 0 in Euclidean three-space E 3 {E^3} the second fundamental form defines a nondegenerate Riemannian metric with curvature K ( II ) K({\text {II}}) . R. Schneider [7] proved that the spheres in Euclidean space E n + 1 {E^{n + 1}} are the only closed hypersurfaces on which the second fundamental form defines a nondegenerate Riemannian metric of constant curvature. For surfaces in E 3 {E^3} we give a common generalization of Schneider’s theorem and the classical theorem of Liebmann [6] (which states that any ovaloid in E 3 {E^3} with constant Gaussian curvature is a sphere).