Let the double hyperbolic space @math , proposed in this paper as an extension of the hyperbolic space @math and homeomorphic to the standard unit @math -sphere @math , contain both the two sheets of the two-sheeted hyperboloid (but with the geodesics in the lower sheet @math negatively , and so is the associated volume element of @math if @math is odd), which are connected at the boundary at infinity by identifying @math projectively. We propose to extend the volume of the polytopes in @math to polytopes in @math , where the volume can possibly be complex valued. With a proper interpretation and choice of the extension in each dimension, we prove a Schl " a fli differential formula for @math , and show that the total volume of @math is equal to @math , for both even and odd dimensions. A Schl " a fli differential formula for @math is also explored when @math is odd. As a dual of @math , an extension of the de Sitter space is proposed, which is compact without boundary and homeomorphic to @math .
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.