Cherevko Y. Geometry of special diffeomorphisms of locally conformal Kähler manifolds

The work is devoted to exploring of locally Conformal Kähler manifolds and their diffeomorphisms. We find tensors and non-tensor which are preserved by conformal mappings. Regarding infinitesimal conformal transformations of 1.c.K. manifolds we have found the expression for the Lie derivative of a Lee form. Also we have obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Finally we have got the necessary and sufficient conditions in order that an 1.c.K. manifold admit a group of conformal motions. Also we have calculated the number of parameters which the group depend on. We have proved that a group of conformal motions admitted by an 1.c.K. manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. Also we introduced so called infinitesimal conformal holomorphically projective transformations. We have got the necessary and sufficient conditions in order that the an 1.c.K. manifold admit 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. tensor and non-tensor which are preserved by the transformations. Additionally, we have found the necessarily and sufficient conditions for a LCK-manifold to admit immersion of complex hypersurface so that the Lee field and the anti-Lee field to be normal to the hypersurface. We propose call such LCK-manifolds as the Pseudo-Vaisman manifolds. Also we obtain the necessarily and sufficient conditions for Riemannian manifolds(not necessarily 1.c.K. ones) admitting conformal mappings preserving the Generalized Einstein tensor.