Weakly symmetric groups of Heisenberg type
type: Item
instance of: scholarly article
title: Weakly symmetric groups of Heisenberg type
zbMATH Open document ID: 0921.53018
DOI: 10.1016/S0926-2245(98)00014-X
author: Jürgen Berndt
author: Fulvio Ricci
author: Lieven Vanhecke
published in: Differential Geometry and its Applications
publication date: 1998-09-08T00:00:00Z
review text: A weakly symmetric space is a Riemannian manifold on which any two points can be interchanged by an isometry. This class of Riemannian homogeneous spaces was introduced by \textit{A. Selberg} [J. Indian Math. Soc., New Ser. 20, 47-87 (1956; Zbl 0072.08201)]. In the present paper, the authors classify all generalized Heisenberg groups which are weakly symmetric. It turns out that these are precisely those generalized Heisenberg groups which are commutative spaces. A commutative space is a Riemannian manifold on which the algebra of all isometry-invariant differential operators is commutative. The commutative generalized Heisenberg groups have been classified previously by the second author in [\textit{F. Ricci}, J. London Math. Soc. 32, 265-271 (1985; Zbl 0592.22009)]. The authors also classify all generalized Heisenberg groups on which any two points can be interchanged by an isometry belonging to the identity component of the full isometry group. Some weakly symmetric generalized Heisenberg groups turn out not to have this property. The authors further provide a brief survey about weakly symmetric spaces.
reviewed by: Jürgen Berndt
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1186364
zbMATH Keywords: \(H\)-type groups
zbMATH Keywords: generalized Heisenberg groups
zbMATH Keywords: weakly symmetric spaces
zbMATH Keywords: commutative spaces
Wikidata QID: Q115337494
MaRDI profile type: Publication
cites work: Geodesics in weakly symmetric spaces
cites work: Symmetric-like Riemannian manifolds and geodesic symmetries
cites work: Q4344575
cites work: Generalized Heisenberg groups and Damek-Ricci harmonic spaces
cites work: Geometry of weakly symmetric spaces
cites work: Q3331943
cites work: Q5521308
cites work: Fundamental Solutions for a Class of Hypoelliptic PDE Generated by Composition of Quadratic Forms
cites work: Riemannian nilmanifolds attached to Clifford modules
cites work: Q3671655
cites work: Q5593040
cites work: Weakly symmetric spaces in dimension five
cites work: Q4887322
cites work: Weakly symmetric spaces in complex and quaternionic space forms
cites work: Commutative Algebras of Invariant Functions on Groups of Heisenberg Type
cites work: Explicit Spin Representations and Lie Algebras of Heisenberg Type
cites work: Q3237356
cites work: Q3138495
cites work: Q4747189
cites work: Q5682519
cites work: Isometry groups on homogeneous nilmanifolds
cites work: Q4887328
description: scientific article; zbMATH DE number 1186364