Homogeneous locally conformally Kaehler and Sasaki manifolds

TL;DR

This paper classifies homogeneous locally conformally Kaehler and Sasaki manifolds, establishing conditions under which they are of Vaisman type and identifying all such structures on reductive Lie groups.

Contribution

It provides new classification results for homogeneous locally conformally Kaehler manifolds, especially relating to Vaisman type and structures on reductive Lie groups.

Findings

Homogeneous locally conformally Kaehler manifolds of reductive groups are Vaisman if the normalizer of the isotropy group is compact.

The classification does not hold for noncompact normalizers.

All left-invariant locally conformally Kaehler structures on reductive Lie groups are determined.

Abstract

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kaehler manifold of a reductive group is of Vaisman type, if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of noncompact normalizer and determine all left-invariant locally conformally Kaehler structures on reductive Lie groups.