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

Abstract:

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