The purpose of this talk is twofold. First, we will describe a
coordinate-free approach to several classic geometries. The approach is
applicable, say, to hyperbolic (real, complex, quaternionic), elliptic
(spherical, Fubini-Study), and Lorentzian (de Sitter, anti de Sitter)
spaces. We will then illustrate how the introduced methods and tools lead
to new examples of complex hyperbolic manifolds. The construction of
manifolds that admit a geometric structure modelled on the complex
hyperbolic plane is an important part of the study of uniformization
problems in real dimension 4.