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

N. P. Mozhey

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 2, pp. 172-184

Résumé

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