On the geometry of three-dimensional pseudo-Riemannian homogeneous spaces. I

The problem of establishing links between the curvature and the topological structure of a manifold is one of the important problems of geometry. In general, the purpose of the research of manifolds of various types is rather complicated. Therefore, it is natural to consider this problem for a narrower class of pseudo-Riemannian manifolds, for example, for the class of homogeneous pseudo-Riemannian manifolds. The basic notions — such as an isotropically-faithful pair, a pseudo-Riemannian homogeneous space, an affine connection, curvature and torsion tensors, Levi–Cevita connection, Ricci tensor, Ricci-flat space, Einstein space, Ricci-parallel space, locally-symmetric space, conformally-flat space — are defined. In this paper, for all three-dimensional Riemannian homogeneous spaces, it is determined under what conditions the space is Ricci-flat, Einstein, Ricci-parallel, locally-symmetric or conformally-flat. In addition, Levi–Cevita connections, curvature and torsion tensors, holonomy algebras, scalar curvatures, Ricci tensors are written out in explicit form for all these spaces. The results can be applied in mathematics and physics, since many fundamental problems in these fields are reduced to the study of invariant objects on homogeneous spaces.