Homogeneous 4-dimensional Kaehler--Weyl Structures

TL;DR

This paper demonstrates that all algebraic configurations of the alternating Ricci tensor in 4-dimensional Kaehler-Weyl structures can be realized geometrically on Lie groups, enriching the understanding of these structures.

Contribution

It shows that every algebraic possibility for the alternating Ricci tensor corresponds to a left-invariant Kaehler-Weyl structure on a 4D Lie group, in both Hermitian and para-Hermitian cases.

Findings

All algebraic types of the alternating Ricci tensor are realizable geometrically.

Left-invariant Kaehler-Weyl structures can be constructed on 4D Lie groups.

The results apply to both Hermitian and para-Hermitian settings.

Abstract

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebraic possibility in this representation space can in fact be geometrically realized by a left-invariant Kaehler-Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting.