Properties of a scalar curvature invariant depending on two planes
type: Item
instance of: scholarly article
title: Properties of a scalar curvature invariant depending on two planes
zbMATH Open document ID: 1109.53020
author: Stefan Haesen
author: Leopold Verstraelen
published in: Manuscripta Mathematica
publication date: 2007-01-26T00:00:00Z
review text: Based on Schouten's interpretation of the Riemann-Christoffel curvature tensor \(R\) [see \textit{L. Verstraelen}, Comments on pseudo-symmetry in the sense of Ryszard Deszcz, Dillen, Franki (ed.) et al., Geometry and topology of submanifolds, VI. Singapore: World Scientific, 199--209 (1994; Zbl 0846.53034)], a geometrical meaning for the tensor \(R\cdot R\) is presented. The condition of semi-symmetry, \(R\cdot R=0\), is interpreted as an invariance of the sectional curvature of every plane after parallel transport around an infinitesimal parallelogram. A scalar curvature invariant \(L(p,\pi,\overline\pi)\) for two planes \(\pi\) and \(\overline\pi\) at the point \(p\) is constructed and interpreted in terms of the parallelogramoïds of Levi-Civita. Its isotropy with respect to \(\pi\) and \(\overline\pi\) amounts to the Riemannian manifold to be pseudo-symmetric in the sense of Deszcz.
reviewed by: Maido Rahula
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 53A55
zbMATH DE Number: 5118896
zbMATH Keywords: curvature tensor of the Riemann manifold
zbMATH Keywords: Schouten's interpretation
zbMATH Keywords: section curvature
zbMATH Keywords: semi-symmetry
zbMATH Keywords: pseudo symmetry after Deszcz
MaRDI profile type: Publication
full work available at URL: s00229-006-0056-0
OpenAlex ID: W2060128772
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DOI: 10.1007/S00229-006-0056-0
description: scientific article; zbMATH DE number 5118896