On convex hypersurfaces in a space form
type: Item
instance of: scholarly article
title: On convex hypersurfaces in a space form
zbMATH Open document ID: 0672.53052
DOI: 10.1007/BF01295286
published in: Monatshefte für Mathematik
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178404
review text: The aim of this paper is to generalize some known results which are valid for an ovaloid of \(E^ n\). The main result is the following: Let f: \(M\to \bar M\) be an isometric immersion of a smooth, compact, connected and orientable n-dimensional Riemannian manifold M into an \((n+1)\)- dimensional space form \(\bar M(c)\). If the sectional curvature K of M is nonnegative (when \(c<0)\) or \(K>c\) (when \(c\geq 0)\) then \[ \int_{M}\nabla_{II}(Tr L,\log Tr L)dM\geq 0\quad and\quad \int_{M}\nabla_{II}(Tr L^{-1},\det L)dM\leq 0 \] where L is the Weingarten map of f and \(\nabla_{II}\) is the first Beltrami operator with respect to the second fundamental form. Equality holds iff f is totally umbilical (when \(c\geq 0)\) or, either f is totally umbilical or L has two distinct constant eigenvalues at every point (when \(c<0)\).
Mathematics Subject Classification ID: 53C45
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 4100117
zbMATH Keywords: isometric immersion
zbMATH Keywords: space form
zbMATH Keywords: Weingarten map
zbMATH Keywords: second fundamental form
zbMATH Keywords: totally umbilical
author: Christos Baikoussis
author: T. Koufogiorgos
reviewed by: Christos Baikoussis
MaRDI profile type: Publication
cites work: On convex hypersurfaces in $E^{n+1}$
cites work: A formula of Simons' type and hypersurfaces with constant mean curvature
OpenAlex ID: W2058841845
description: scientific article; zbMATH DE number 4100117