Geodesic mappings of spaces with special vector fields

Abstract

The paper investigates a special type of pseudo-Riemannian spaces – spaces that allow special vector fields. The specialization of vector fields is mainly reduced to the ability to determine the metric tensor by the combination of vectors or their derivatives. With the development of the theory of relativity and the theory of Ricci flows, instead of the metric tensor, a linear combination of the metric tensor and the Ricci tensor began to be used. In this work the vector fields are studied, the covariant derivative of which is proportional to the Ricci tensor. One of the main methods of modeling physical and mechanical systems using pseudo- Riemannian spaces is modeling along mechanical trajectories. The mechanical trajectories are geodetic lines. Therefore, it is natural to have a scientific interest in non-trivial geodetic mappings. Conformal mappings and some geometric properties of spaces that allow special vector fields were studied in the works of V. A. Kiosak and I. J. Hinterleitner. Geodetic mappings of such spaces have not been studied before. The research is carried out using the linear form of the basic equations of the theory of geodetic mappings. The equations, conditions of their integration and their differential extensions are studied in the work.