On convex hypersurfaces in Euclidean space
For an hyperovaloid M in Euclidean space \(E^{n+1}\) (n\(\geq 2)\) the second fundamental form II defines a Riemannian manifold (M,II) with sectional curvature K(II). R. Schneider proved that \(K(II)=const\). iff M is a round sphere. Several authors generalized this result, and this paper contains further interesting results of this type, e.g.: Let M be a hyperovaloid and assume that K(II)\(\leq CK(I)\), where \(C^ n=A(II)^ 2 \omega_ n^{-2}\), then M is a round sphere \([A(II)=area\) of (M,II); \(\omega_ n=area\) of \(S^ n(1)\); K(I) sectional curvature of the first fundamental form].
scientific article; zbMATH DE number 4030458