On the total volume of the double hyperbolic space

Abstract

Let the double hyperbolic space DH n , proposed in this paper as an extension of the hyperbolic space H n and homeomorphic to the standard unit n-sphere S n , contain both the two sheets of the two-sheeted hyperboloid (but with the geodesics in the lower sheet H n − negatively valued, and so is the associated volume element of H n − if n is odd), which are connected at the boundary at infinity by identifying ∂H n = ∂H n − projectively. We propose to extend the volume of the polytopes in H n to polytopes in DH n , where the volume can possibly be complex valued. With some mild assumption, we prove a Schläfli differential formula for DH n , and show that the total volume of DH n is equal to i n V n (S n ) for both even and odd dimensions. When n = 2m + 1, on top of the conformal structure of ∂H 2m+1 it also induces an intrinsic volume on regions in ∂H 2m+1 that we call polytopes, and we explore a Schläfli differential formula for ∂H 2m+1 as well. As a dual of DH n , an extension of the de Sitter space is proposed, which is compact without boundary and homeomorphic to S n−1 × S 1 ; and it also induces an extension of R n−1,1 , reminiscent of the conformal compactification of spacetime.