Geometry of weakly symmetric spaces
type: Item
instance of: scholarly article
title: Geometry of weakly symmetric spaces
zbMATH Open document ID: 0877.53027
author: Jürgen Berndt
author: Lieven Vanhecke
published in: Journal of the Mathematical Society of Japan
publication date: 1997-08-20T00:00:00Z
review text: Weakly symmetric spaces have been introduced by \textit{A. Selberg} in [J. Indian Math. Soc., n. Ser. 20, 47-87 (1956; Zbl 0072.08201)]. An alternative definition says that a Riemannian manifold \(M\) is weakly symmetric if, for any two points \(p,q\) in \(M\), there exists an isometry of \(M\) interchanging \(p\) with \(q\). The present authors give many examples of weakly symmetric spaces which are not symmetric. First, the geodesic tubes in \(\mathbb{C} P^n\) around points or naturally embedded \(\mathbb{C} P^1, \dots, \mathbb{C} P^{n-1}\) are weakly symmetric spaces. The analogues hold for the quaternionic space \(\mathbb{H} P^n\), the Cayley plane \(\text{Cay} P^2\) and for the noncompact duals \(\mathbb{C} H^n\), \(\mathbb{H} H^n\), \(\text{Cay} H^2\) (in which the tubes are horospheres). Finally, the full classification of all simply connected weakly symmetric spaces in dimensions three and four is given. One can see that these coincide with the naturally reductive spaces. (The reviewer and \textit{R. A. Marinosci} extended this classification to dimension five using a more algebraic method [\textit{O. Kowalski} and \textit{R. A. Marinoski}, J. Geom. 58, 123-131 (1997; Zbl 0868.53038)]).
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1050414
zbMATH Keywords: tubes around submanifolds
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: weakly symmetric
reviewed by: Oldřich Kowalski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04840745
OpenAlex ID: W2071708014
DOI: 10.2969/JMSJ/04840745
description: scientific article; zbMATH DE number 1050414