НЕКОТОРЫЕ ВОПРОСЫ ГЕОМЕТРИИ ЛОКАЛЬНО КОНФОРМНО-КЕЛЕРОВЫХ МНОГООБРАЗИЙ

Abstract

Torso-forming vector fields, as well as their special cases (concircular, special concircular and recurrent) are used in many areas of differential geometry, for example, in conformal, geodesic, almost geodesic, holomorphically projective and other mappings and transformations. The presence of torso-forming vector fields on the space under consideration makes the geometry of this space more meaningful. It is of interest to study the geometry of spaces that admit recurrent vector fields. In this paper, the authors consider locally conformally Kahler manifolds with a recurrent Lie vector. Such manifolds are called recurrent locally conformally Kahler manifolds. The Lie form and the Lie vector are calculated explicitly. It is proved that the Lie vector of a locally conformally Kahler manifold of constant curvature is a concircular field. A recurrence criterion for a conformally flat locally conformally Kahler manifold is obtained. Some properties of conformally flat locally conformally Kahler manifolds are proved. It is proved that a compact manifold of constant curvature does not admit its own recurrent locally conformal Kahler structure.