Affine hypersurfaces with the Ricci semi-symmetry
type: Item
instance of: scholarly article
title: Affine hypersurfaces with the Ricci semi-symmetry
zbMATH Open document ID: 1075.53049
DOI: 10.1007/BF03323016
author: Yoshio Matsuyama
published in: Results in Mathematics
publication date: 2005-06-09T00:00:00Z
review text: A differentiable manifold \(M\) with the curvature tensor field \(R\), defined by a torsion-free affine connection \(\nabla\) on \(M\), is called locally symmetric resp. semi-symmetric if \(\nabla R= 0\) resp. \(RR:= R(X, Y)R= \nabla_X\nabla_Y R-\nabla_Y\nabla_X R-\nabla_{[X,Y]}R= 0\) for all smooth tangent vector fields \(X\), \(Y\) on \(M\). In the same way one speaks of local Ricci symmetry, resp., Ricci semi-symmetry of \(M\) if \(\nabla\,\text{Ric}= 0\), resp., \(R\text{\,Ric}= 0\) for the Ricci-tensor field Ric of \(M\). The author applies these definitions to the induced connnection \(\nabla\) of an affine hypersurface \(f: M^n\to \mathbb R^{n+1}\) which is induced by an affine normal vector field of \(f\). His main result is: If \(f\) with \(n\geq 3\) is a (nondegenerate) Blaschke immersion, then \(M^n\) is Ricci semi-symmetric with respect to the induced connection \(\nabla\) of \(f\) iff \(f\) is either a proper affine hypersphere or an affine cylinder where the rank of the shape operator \(S\) of \(f\) is \(\leq 1\). In the first case semi-symmetry is equivalent to Ricci semi-symmetry.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53B05
zbMATH DE Number: 2174565
zbMATH Keywords: induced connection of a Blaschke immersion
zbMATH Keywords: affine cylinder
zbMATH Keywords: Ricci symmetry
zbMATH Keywords: Ricci semi-symmetry
reviewed by: Kurt Leichtweiß
MaRDI profile type: Publication
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Wikidata QID: Q125932521
description: scientific article; zbMATH DE number 2174565