ON GEODESIC MAPPINGS OF PSEUDO-RIEMANNIAN SPACES WITH SPECIAL VECTOR FIELDS

Abstract

The paper investigates geodesic mappings of pseudo-Riemannian spaces, which constitute one of the fundamental objects of modern differential geometry. The theory of geodesic mappings originated in the works of E. Beltrami, U. Dini, and T. Levi-Civita and has remained an important field of geometric research for more than a century and a half. Significant contributions to its development were made by H. Weyl, L. P. Eisenhart, T. Y. Thomas, V. F. Kagan, M. S. Sinyukov, and their followers, who established an invariant tensor apparatus for the study of mappings and developed effective methods for the analysis of the corresponding differential equations. The relevance of the present study is determined by the need for a deeper understanding of the geometric properties of pseudo-Riemannian spaces and the conditions ensuring the existence of their geodesic mappings. Interest in such investigations is motivated not only by intrinsic problems of differential geometry but also by numerous applications in mathematical physics, mechanics, gravitation theory, and the modeling of dynamical processes, where trajectories of motion may be interpreted as geodesic lines of the corresponding spaces. The aim of the paper is to investigate new characteristics of special pseudo-Riemannian spaces and their geodesic mappings. The objects of study are special classes of pseudo-Riemannian spaces, whereas the subject of the investigation comprises systems of differential equations, their integrability conditions, and differential prolongations characterizing the possibility of the existence of geodesic mappings. To achieve this goal, classical methods of Riemannian geometry, local methods in the class of sufficiently smooth functions, and tensor analysis are employed. The obtained results contribute to the further development of the theory of geodesic mappings and provide a deeper insight into the geometric structure of pseudo-Riemannian spaces.