On convex hypersurfaces in Euclidean space
type: Item
instance of: scholarly article
title: On convex hypersurfaces in Euclidean space
zbMATH Open document ID: 0633.53085
DOI: 10.1007/BF01210720
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: 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].
Mathematics Subject Classification ID: 53C45
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 4030458
zbMATH Keywords: characterization of spheres
zbMATH Keywords: hyperovaloid
zbMATH Keywords: second fundamental form
author: Christos Baikoussis
author: T. Koufogiorgos
reviewed by: Udo Simon
MaRDI profile type: Publication
cites work: Q5678153
cites work: Two Characteristic Properties of the Sphere
cites work: Anwendungen der De Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie
cites work: Characterizations of the Sphere by the Curvature of the Second Fundamental Form
cites work: Smoothness theorems for the principal curvatures and principal vectors of a hypersurface
cites work: Closed Convex Hypersurfaces with Second Fundamental Form of Constant Curvature
description: scientific article; zbMATH DE number 4030458