A characteristic property of the sphere

Abstract

On an ovaloid S with Gaussian curvature K > 0 K > 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}}}} . It is shown that S is a sphere if K II = c H s K r {K_{{\text {II}}}} = c{H^s}{K^r} , where c, s and r are constants, H is the mean curvature of S and 0 ⩽ s ⩽ 1 0 \leqslant s \leqslant 1 .