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

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.