Natural Almost Hermitian Structures on Conformally Foliated 4-Dimensional Lie Groups with Minimal Leaves

TL;DR

This paper classifies almost Hermitian structures on 4D Lie groups with conformal foliations, identifying conditions for almost Kähler, integrable, and Kähler structures, and constructs numerous families of such structures.

Contribution

It provides a comprehensive classification of almost Hermitian structures on specific 4D Lie groups, including explicit families for each type, advancing understanding of geometric structures on these groups.

Findings

16 almost Kähler families constructed

18 integrable families identified

11 Kähler families classified

Abstract

Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\mathcal{F}$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\mathcal{F}$. The corresponding Lie algebra $\mathfrak{g}$ must then belong to one of 20 families $\mathfrak{g}_1,\dots,\mathfrak{g}_{20}$ according to S. Gudmundsson and M. Svensson. We classify such structures $J$ which are almost K\"{a}hler $(\mathcal{A}\mathcal{K})$, integrable $(\mathcal{I})$ or K\"{a}hler $(\mathcal{K})$. Hereby, we construct 16 multi-dimensional almost K\"{a}hler families, 18 integrable families and 11 K\"{a}hler families.