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# Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four

Published:2006-04-01
Printed: Apr 2006
• M. E. Fels
• A. G. Renner
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## Abstract

A method, due to \'Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with $(2,2)$ signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to $\reals^4$. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.
 Keywords: Homogeneous pseudo-Riemannian, Einstein space
 MSC Classifications: 53C30 - Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

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