The only 4-dimensional non-compact rank one symmetric spaces are $mathbb{C}H^2$ and $mathbb{R}H^4$. 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 $mathbb{C}H^2$ admits at least four Hermitian complex structures. One class of metrics on $mathbb{C}H^2$ and all the metrics on $mathbb{R}H^4$ admit 2-spheres of Hermitian complex structures. The standard metric of $mathbb{C}H^2$ is the only Einstein metric from the classification, and also the only metric that admits K?hler structure, while on $mathbb{R}H^4$ all the metrics are Einstein. Finally, we examine the geometry of these Lie groups: curvature properties, self-duality, and holonomy.
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机译：唯一的4维非紧致秩为1的对称空间是$ mathbb {C} H ^ 2 $和$ mathbb {R} H ^ 4 $。根据Heintze的经典结果，可以通过具有左不变度量的实际可解Lie组对这些空间建模。在本文中，我们对这些Lie组上所有可能的左不变Hermitian结构进行分类，即左不变黎曼度量和相应的Hermitian复杂结构。我们显示，来自$ mathbb {C} H ^ 2 $的分类中的每个度量均接受至少四个Hermitian复杂结构。 $ mathbb {C} H ^ 2 $上的一类度量和$ mathbb {R} H ^ 4 $上的所有度量都承认Hermitian复杂结构的2个球体。 $ mathbb {C} H ^ 2 $的标准度量标准是分类中唯一的爱因斯坦度量标准，也是唯一允许采用K？hler结构的度量标准，而$ mathbb {R} H ^ 4 $的所有度量标准是爱因斯坦。最后，我们检查了这些李群的几何形状：曲率性质，自对偶性和完整性。
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