Classification of left invariant Hermitian structures on 4-dimensional non-compact rank one symmetric spaces

Srdjan Vukmirović,Andrijana Dekić,Marijana Babić

doi:10.33044/revuma.v60n2a04

Abstract

The only 4-dimensional non-compact rank one symmetric spaces are CH2 and RH4. By the classical results of Heintze, one can model these spaces by real solvable Lie groups with left invariant metrics. In this paper we classify all possible left invariant Hermitian structures on these Lie groups, i.e., left invariant Riemannian metrics and the corresponding Hermitian complex structures. We show that each metric from the classification on CH2 admits at least four Hermitian complex structures. One class of metrics on CH2 and all the metrics on RH4 admit 2-spheres of Hermitian complex structures. The standard metric of CH2 is the only Einstein metric from the classification, and also the only metric that admits K¨ahler structure, while on RH4 all the metrics are Einstein. Finally, we examine the geometry of these Lie groups: curvature properties, self-duality, and holonomy.

Highlights

Thanks to Heintze [9], it is known that all non-compact rank one symmetric spaces are exactly the hyperbolic spaces
From Heintze’s papers [9, 10], we know that a connected homogeneous manifold of non-positive curvature can be represented as a connected solvable Lie
The classification of non-isometric left invariant Riemannian metrics on the Lie group RH4 is presented in Theorem 3.1

Summary

Introduction

Thanks to Heintze [9], it is known that all non-compact rank one symmetric spaces are exactly the hyperbolic spaces. We give the classification of non-isometric left invariant Riemannian metrics on the Lie group CH2 in Theorem 2.1. The classification of non-isometric left invariant Riemannian metrics on the Lie group RH4 is presented in Theorem 3.1. These results are partially known (see [6, Table 2, A14,9 for ch, A14,,15 for rh4]). The main result of this paper is a classification of left invariant Hermitian complex structures on CH2 with respect to all non-isometric left invariant Riemannian metrics (Theorem 2.2). Snow [18] investigated 4-dimensional solvable -connected real Lie groups with commutator subalgebra of dimension less than three He classified left invariant complex structures with respect to certain subalgebras of the complexification of real Lie algebras. We are very grateful to our colleagues Neda Bokan and Tijana Sukilovic for many valuable comments and suggestions

Preliminaries

Classification of left invariant Hermitian structures on CH2

Classification of left invariant Hermitian structures on RH4

Findings

Left invariant geometry of metric Lie groups CH2 and RH4