Semisymmetry and Ricci-semisymmetry for hypersurfaces of semi-Riemannian space forms
type: Item
instance of: scholarly article
title: Semisymmetry and Ricci-semisymmetry for hypersurfaces of semi-Riemannian space forms
zbMATH Open document ID: 0978.53035
author: Filip Defever
author: Ryszard Deszcz
author: Dorota Kowalczyk
author: Leopold Verstraelen
published in: Arab Journal of Mathematical Sciences
publication date: 2002-01-23T00:00:00Z
review text: Let \((M, g)\) be a semi-Riemannian manifold. \(M\) is called semisymmetric if some natural product of its Riemann curvature tensor with itself is zero, and Ricci-semisymmetric if a natural product of its Riemann curvature tensor with its Ricci curvature tensor vanishes. This paper deals with a conjecture of P. J. Ryan, according to which the two notions are equivalent for a hypersurface of Euclidean \(n\)-space. The authors note that this conjecture is false, as was recently proved by one of them; it is however known to be true in a number of special cases. In this paper, they prove that it is true for hypersurfaces of semi-Riemannian space-forms for which a natural product of the Weyl curvature tensor with the Riemann tensor is zero. This result is achieved through elaborate algebraic computations on the curvature tensors of hypersurfaces.
reviewed by: Jean-Marc Schlenker
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 1513038
zbMATH Keywords: semisymmetry
zbMATH Keywords: Ricci semisymmetry
zbMATH Keywords: semi-Riemannian manifold
zbMATH Keywords: hypersurface
zbMATH Keywords: Riemann curvature tensor
zbMATH Keywords: Ricci curvature tensor
zbMATH Keywords: Weyl curvature tensor
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1513038