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Open AccessArticle

Infinitesimal Transformations of Locally Conformal Kähler Manifolds

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Department of Economic Cybernetics and Information Technologies, Odesa National Economic University, 65082 Odesa, Ukraine
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Department of Mathematics and Physics, Uman National University of Horticulture, 20300 Uman, Ukraine
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Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, 60190 Brno, Czech Republic
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Department of Informatics and Natural Sciences, Faculty of Technology, Institute of Technology and Business in České Budějovice, 37001 Czech Budejovice, Czech Republic
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(8), 658; https://doi.org/10.3390/math7080658
Received: 21 June 2019 / Revised: 16 July 2019 / Accepted: 20 July 2019 / Published: 24 July 2019
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
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. We have also 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. We have also 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 Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric. View Full-Text
Keywords: Hermitian manifold; locally conformal Kähler manifold; Lee form; diffeomorphism; conformal transformation; Lie derivative Hermitian manifold; locally conformal Kähler manifold; Lee form; diffeomorphism; conformal transformation; Lie derivative
MDPI and ACS Style

Cherevko, Y.; Berezovski, V.; Hinterleitner, I.; Smetanová, D. Infinitesimal Transformations of Locally Conformal Kähler Manifolds. Mathematics 2019, 7, 658.

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