|
|||
Current volumePast volumes
1952-1968
1944-1951
1936-1944 |
Classification of left invariant Hermitian structures on 4-dimensional
non-compact rank one symmetric spaces
Volume 60, no. 2
(2019),
pp. 343–358
https://doi.org/10.33044/revuma.v60n2a04
Abstract
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 Kähler 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.
|
||
|
Published by the Unión Matemática Argentina |
|||