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Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four

Published online by Cambridge University Press:  20 November 2018

M. E. Fels  and
A. G. Renner
M. E. Fels
Affiliation:
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, U.S.A. e-mail: fels@math.usu.eduarenner@cc.usu.edu
A. G. Renner
Affiliation:
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, U.S.A. e-mail: fels@math.usu.eduarenner@cc.usu.edu
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Abstract

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A method, due to Élie 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 ${{\mathbb{R}}^{4}}$. 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

53C30Homogeneous pseudo-RiemannianEinstein space
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 2 , 01 April 2006 , pp. 282 - 311
Copyright
Copyright © Canadian Mathematical Society 2006

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