Curvature homogeneous spaces with a solvable Lie group as homogeneous model
type: Item
instance of: scholarly article
title: Curvature homogeneous spaces with a solvable Lie group as homogeneous model
zbMATH Open document ID: 0762.53031
author: Oldřich Kowalski
author: Lieven Vanhecke
author: Franco Tricerri
published in: Journal of the Mathematical Society of Japan
publication date: 1993-01-17T00:00:00Z
review text: Following I. M. Singer, a Riemannian manifold \((M,g)\) is said to be curvature homogeneous if, for every two points \(p,q\in M\), there exists a linear isometry \(F: T_ p M\to T_ q M\) such that \(F^* R_ q=R_ p\), where \(R\) denotes the Riemannian curvature tensor of \((M,g)\). Of course, any (locally) homogeneous space is curvature homogeneous. Further, a Riemannian manifold \((M,g)\) is said to have the same curvature tensor as a homogeneous space \((\widetilde M,\widetilde g)\) if, for every pair of points \(m\in M\) and \(\widetilde m\in\widetilde M\), there exists a linear isometry \(F: T_ m M\to T_{\widetilde m}\widetilde M\) such that \(F^*\widetilde R_{\widetilde m}=R_ m\). Also in this case \((M,g)\) is curvature homogeneous and \((\widetilde M,\widetilde g)\) is called a model space for \((M,g)\). The main task is to construct examples of non- homogeneous curvature homogeneous manifolds, to determine their classification and to study their geometry. An extensive study of curvature homogeneous spaces with a symmetric model has been made by several authors and, in the generic case, their classification is now known [see \textit{E. Boeckx}, \textit{O. Kowalski} and \textit{L. Vanhecke}, Nonhomogeneous relatives of symmetric spaces, to appear in Differ. Geom. Appl., for more details and further references]. We note that all these spaces are semi-symmetric manifolds. Some examples of curvature homogeneous spaces with a non-symmetric model space are already known, for examples in the class of isoparametric hypersurfaces in spheres. The main purpose of this paper is to construct new examples and to study their geometry and isometry classes. They are obtained by deforming a flat right invariant metric on a Lie group which is a semi- direct product of \(\mathbb{R}^ p\) and \(\mathbb{R}^ q\). They are all non-compact in contrast to the isoparametric examples. These last ones are the only known compact examples and the search for other ones seems to be a difficult …
reviewed by: Lieven Vanhecke
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 98459
zbMATH Keywords: model space
zbMATH Keywords: semi-symmetric manifolds
zbMATH Keywords: isoparametric hypersurfaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04430461
OpenAlex ID: W2035750855
Wikidata QID: Q115225536
DOI: 10.2969/JMSJ/04430461
description: scientific article; zbMATH DE number 98459