On the three-dimensional spaces which admit a continuous group of motions
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.
scientific article; zbMATH DE number 1844821