DE RHAM DECOMPOSITION AND COHOMOLOGY THREE-DIMENSIONAL HOMOGENEOUS SPACE PSEUDO-RIEMANNIAN MANIFOLDS
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Abstract
In this paper we study three-dimensional homogeneous Riemannian (pseudo-Riemannian) manifolds. We describe de Rham decomposition and cohomology this manifolds. The cohomology allows us to answer the question of when closed forms on a manifold are exact. The local classification of homogeneous spaces is equivalent to the description of effective pairs of Lie algebras. We describe all invariant symmetric nondegenerate bilinear forms on those homogeneous spaces. A complete locally homogeneous Riemannian (pseudo-Riemannian) manifold is locally isometric to a globally homogeneous Riemannian (pseudo-Riemannian) space. Also we describe curvature Riemannian (pseudo-Riemannian) homogeneous spaces. We restrict ourselves to the case of a nontrivial stationary subgroup, all other pseudo-Riemannian homogeneous spaces in this dimension are just Lie groups with a left-invariant metric. We use the algebraic approach for description of manifolds, methods of the theory of Lie groups and Lie algebras and homogeneous spaces.
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