Some basic theorems on Killing vector fields are reviewed. In particular, the topic of a constant curvature space is examined. A detailed proof is given for a theorem describing the most general form of the metric of a homogeneous isotropic space-time. Although this theorem can be considered commonly known, its complete proof is difficult to find in the literature. An example metric is presented which, while all its spatial cross sections correspond to constant curvature spaces, still is not homogeneous and isotropic as a whole. An equivalent definition of a homogeneous isotropic space-time in terms of embedded manifolds is also given.

Keywords: Killing vector field, homogeneous universe, isotropic universe, Friedmann metric PACS: 04.20.−q DOI: 10.3367/UFNe.2016.05.037808 URL: https://ufn.ru/en/articles/2016/7/d/ 000386357600004 2-s2.0-84991721633 2016PhyU...59..689K Citation: Katanaev M O "Killing vector fields and a homogeneous isotropic universe" Phys. Usp. 59 689–700 (2016) BibTexBibNote ® (generic)BibNote ® (RIS)Medline RefWorks RT Journal T1 Killing vector fields and a homogeneous isotropic universe A1 Katanaev,M.O. PB Physics-Uspekhi PY 2016 FD 10 Jul, 2016 JF Physics-Uspekhi JO Phys. Usp. VO 59 IS 7 SP 689-700 DO 10.3367/UFNe.2016.05.037808 LK https://ufn.ru/en/articles/2016/7/d/ Received: 4th, December 2015, accepted: 16th, May 2016 Оригинал: Катанаев М О «Векторные поля Киллинга и однородная и изотропная вселенная» УФН 186 763–775 (2016); DOI: 10.3367/UFNr.2016.05.037808