Pseudo-Riemannian Spaces With a Special Riemann Tensor

Abstract

The paper considers pseudo-Riemannian spaces, the Riemann tensor of which has a special structure. The structure of the Riemann tensor is given as a combination of special symmetric and obliquely symmetric tensors. Tensors are selected so that the results can be applied in the theory of geodetic mappings, the theory of holomorphic-projective mappings of Kähler spaces, as well as other problems arising in differential geometry and its application in general relativity, mechanics and other fields. Through the internal objects of pseudo-Riemannian space, others are determined, which are studied depending on what problems are solved in the study of pseudo-Riemannian spaces. By imposing algebraic or differential constraints on internal objects, we obtain special spaces. In particular, if constraints are imposed on the metric we will have equidistant spaces. If on the Ricci tensor, we obtain spaces that allow φ (Ric)-vector fields, and if on the Einstein tensor, we have almost Einstein spaces. The paper studies pseudo-Riemannian spaces with a special structure of the curvature tensor, which were introduced into consideration in I. Mulin paper. Note that in his work these spaces were studied only with the requirement of positive definiteness of the metric. The proposed approach to the specialization of pseudo-Riemannian spaces is interesting by combining algebraic requirements for the Riemann tensor with differential requirements for its components. In this paper, the research is conducted in tensor form, without restrictions on the sign of the metric. Depending on the structure of the Riemann tensor, there are three special types of pseudo-Riemannian spaces. The properties which, if necessary, satisfy the Richie tensors of pseudoriman space and the tensors which determine the structure of the curvature tensor are studied. In all cases, it is proved that special tensors satisfy the commutation conditions together with the Ricci tensor. The importance and usefulness of such conditions for the study of pseudo-Riemannian spaces is widely known. Obviously, the results can be extended to Einstein tensors. Proven theorems allow us to effectively investigate spaces with constraints on the Ricci tensor.