Locally conformally Kaehler structures on homogeneous spaces

TL;DR

This paper surveys and classifies 4-dimensional homogeneous and locally homogeneous locally conformally Kähler manifolds using Lie algebra techniques, providing new insights and a comprehensive overview of the field.

Contribution

It offers a complete classification of 4D homogeneous and locally homogeneous l.c.K. manifolds based on Lie algebra analysis, including new results and observations.

Findings

Complete classification of 4D homogeneous l.c.K. manifolds

Survey of known results in l.c.K. geometry

Introduction of new observations in the field

Abstract

We will discuss in this paper homogeneous locally conformally Keahler (or shortly homogeneous l.c.K.) manifolds and locally homogeneous l.c.K. manifolds from various aspects of study in the field of l.c.K. geometry. We will provide a survey of known results along with some new results and observations; in particular we make a complete classification of 4-dimensional homogeneous and locally homogeneous l.c.K. manifolds in terms of Lie algebras.