The analytic continuation of hyperbolic space

We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space. Such a construction inspires a new concrete insight for the study of the hyperbolic geometry and Lorentzian geometry as a unified object. We also discuss the advantages of this new geometric model as well as some of its applications.