The connected groups acting by isometries on either the real or the complex hyperbolic spaces are determined. A Lie-theoretic description of the homogeneous Riemannian, respectively Kähler, structures of linear type on these spaces is then found. On both spaces, examples that are not of linear type are given.
"Homogeneous structures on real and complex hyperbolic spaces." Illinois J. Math. 53 (2) 561 - 574, Summer 2009. https://doi.org/10.1215/ijm/1266934792