On canonscal quasi-geodesic mappings of recurrent-parabolic spaces

Abstract

The article is devoted to the problem of holomorphically projective transformations of locally conformal Kaehler manifolds. it's worth to be noted, that J. Mikes and Z. Radulovich have proved that a locally conformal Kaehler manifold does not admit finite nontrivial holomorphically projective mappings for a Levi-Civita connection. Earlier we had proved that a locally conformal Kaehler manifold also does not admit nontrivial infinitesimal holomorphically projective transformations for a Levi-Civita connection. But since the Weyl connection defined by Lee form on a locally conformal Kaehler manifold is F-connection, hence for the connection nontrivial infinitesimal holomorphically projective transformations are admitted. Then we rewrote the system of partial differential equations for the Levi-Civita connection. So we introduced so called infinitesimal conformal holomorphically projective transformations. We have got the necessary and sufficient conditions in order that the a locally conformal Kaehler manifold admits a group of infinitesimal conformal holomorphically projective transformations. Also we have calculated the number of parameters which the group depend on. We have got invariants, i. e. a tensor and a non-tensor which are preserved by the transformations. And finally, we have proved that a vector field which generates infinitesimal conformal holomorphically projective transformations of a compact locally conformal Kaehler manifold is contravariant almost analytic.

Keywords

• Ермітові многовиди
• конформно келерові мнговиди
• форма Лі
• конформно голоморфно-проективні перетворення