Homogeneous Riemannian spaces and Lie algebras of Killing fields

Details Homogeneous Riemannian spaces and Lie algebras of Killing fields Homogeneous Riemannian spaces and Lie algebras of Killing fields

Jun 1971

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1971gregr...2..105s&link_type=abstract

General Relativity and Gravitation, Volume 2, Issue 2, pp.105-120

Physics

1

Scientific paper

The following new results are proved: (1) Every Riemannian space admitting a multiply transitive Lie algebra of Killing fields is locally isometric to a homogeneous Riemannian space. (2) For every closed connceted subgroupH 0 of the invariance group of a non degenerate quadratic form a homogeneous Riemannian space exists whose isotropy group containsH 0. (3) Necessary and sufficient conditions are derived for a Lie algebra\underset{raise0.3emsmashriptscriptstyle-}{G} to have a realization as multiply transitive Killing fields. These conditions are constructire in the sense that, for a given linear connected isotropy group, Lie algebras\underset{raise0.3emsmashriptscriptstyle-}{G} can be calculated algebraically, (4) It is shown how the Riemann tensor of a bomogencous space and its covariant derivatives can be expressed in terms of the metric at one point and the structure constants of the Lie algebra of Killing fields.

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