The homogeneous geometries of complex hyperbolic space

TL;DR

This paper classifies the holonomy algebras and homogeneous structures of complex hyperbolic spaces, providing explicit formulas and linking geometric structures to algebraic submodules.

Contribution

It offers a comprehensive description of holonomy algebras and homogeneous structures on complex hyperbolic spaces, including explicit formulas and classifications.

Findings

Holonomy algebras of all canonical connections are described.

Formulas for all Kahler homogeneous structures are provided.

Connections between homogeneous structures and specific algebraic submodules are established.

Abstract

We describe the holonomy algebras of all canonical connections and their action on complex hyperbolic spaces $\mathbb{C}\mathrm{H}(n)$ in all dimensions ($n\in\mathbb{N}$). This thorough investigation yields a formula for all Kahler homogeneous structures on complex hyperbolic spaces. Finally, we have related the belonging of the homogeneous structures to the different Tricerri and Vanhecke's (or Abbena and Garbiero's) orthogonal and irreducible $\mathrm{U}(n)$-submodules with concrete and determined expressions of the holonomy.