The homogeneous geometries of complex hyperbolic space

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.