Preprint Article Version 1 This version is not peer-reviewed

# Infinitesimal Transformations of Locally Conformal Kähler Manifolds

Version 1 : Received: 21 June 2019 / Approved: 24 June 2019 / Online: 24 June 2019 (08:45:40 CEST)

How to cite: Cherevko, Y.; Berezovski, V.; Hinterleitner, I.; Smetanová, D. Infinitesimal Transformations of Locally Conformal Kähler Manifolds. Preprints 2019, 2019060232 (doi: 10.20944/preprints201906.0232.v1). Cherevko, Y.; Berezovski, V.; Hinterleitner, I.; Smetanová, D. Infinitesimal Transformations of Locally Conformal K&auml;hler Manifolds. Preprints 2019, 2019060232 (doi: 10.20944/preprints201906.0232.v1).

## Abstract

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-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. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. Also we have calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding K\"{a}hlerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to a some subgroup of homothetic group of the coresponding local K\"{a}hlerian metric.

## Subject Areas

Hermitian manifold; locally conformal Kähler manifold; Lee form; diffeomorphism; conformal transformation; Lie derivative