The homogeneous geometries of real hyperbolic space

TL;DR

This paper classifies the holonomy algebras of homogeneous structures on real hyperbolic spaces, revealing the types of homogeneous tensors and showing the moduli space has two connected components.

Contribution

It provides a complete description of holonomy algebras and homogeneous tensor types for all dimensions of real hyperbolic spaces, and identifies the moduli space's connected components.

Findings

Holonomy algebras of all homogeneous structures are classified.

Homogeneous tensors are categorized into specific types.

The moduli space of structures has exactly two connected components.

Abstract

We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use our analysis to show that the moduli space of homogeneous structures on real hyperbolic space has two connected components.