Homogeneous Riemannian spaces and Lie algebras of Killing fields

Abstract

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 $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{G} $$ 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 $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{G} $$ 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.