Natural almost Hermitian structures on conformally foliated 4-dimensional Lie groups with minimal leaves

TL;DR

This paper classifies almost Hermitian structures on 4-dimensional Riemannian Lie groups with conformal foliations, identifying conditions for almost Kähler, integrable, and Kähler structures, and provides new examples in each class.

Contribution

It offers a classification of almost Hermitian structures on specific Lie groups with conformal foliations, including new explicit examples for each structure type.

Findings

Classification of almost Hermitian structures into almost Kähler, integrable, and Kähler.

Construction of new multi-dimensional examples in each class.

Identification of conditions under which these structures exist.

Abstract

Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\F$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\F$. We classify such structures $J$ which are almost K\"ahler $(\A\K)$, integrable $(\I)$ or K\"ahler $(\K)$. Hereby we construct several new multi-dimensional examples in each class.