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Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four
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- Journal:
- Canadian Journal of Mathematics / Volume 58 / Issue 2 / 01 April 2006
- Published online by Cambridge University Press:
- 20 November 2018, pp. 282-311
- Print publication:
- 01 April 2006
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- Article
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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.
