On the three-dimensional spaces which admit a continuous group of motions
type: Item
instance of: scholarly article
title: On the three-dimensional spaces which admit a continuous group of motions
zbMATH Open document ID: 1027.53001
DOI: 10.1023/A:1015357132699
author: Luigi Bianchi
published in: General Relativity and Gravitation
publication date: 2002-12-12T00:00:00Z
review text: This is a translated reprint of Luigi Bianchi's classical paper ``Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti'' which originally appeared in [Mem. Mat. Fis. Soc. Ital. Sci. (3) 11, 267-352 (1898; JFM 28.0586.01 and JFM 28.0642.01] and was also reprinted in: Opere [The Collected Works of Luigi Bianchi)], Rome, Edizione Cremonese Vol. 9, 17-109 (1958; Zbl 0100.00109)]. The author systematically derives all possible Lie algebras and local Lie groups of isometries of a three-dimensional Riemannian manifold together with a normal form of the metric of the latter. There are isometry groups of dimension \(1,2,3,4,6\) but none of dimension 5. Dimension 6 leads to the three-dimensional spaces of constant curvature. Bianchi's paper greatly influenced the later development in Riemannian geometry, Lie theory, and cosmology.
Mathematics Subject Classification ID: 53-03
Mathematics Subject Classification ID: 22-03
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 01A75
zbMATH DE Number: 1844821
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: isometry
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: cosmology
reviewed by: Rainer Schimming
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1844821