On the total volume of the double hyperbolic space

Abstract

Let the \emph{double hyperbolic space} $\mathbb{DH}^n$, proposed in this paper as an extension of the hyperbolic space $\mathbb{H}^n$, contain a two-sheeted hyperboloid with the two sheets connected to each other along the boundary at infinity. We propose to extend the volume of convex polytopes in $\mathbb{H}^n$ to polytopes in $\mathbb{DH}^n$, where the volume is invariant under isometry but can possibly be complex valued. We show that the total volume of $\mathbb{DH}^n$ is equal to $i^n V_n(\mathbb{S}^n)$ for both even and odd dimensions, and prove a Schl\"{a}fli differential formula (\SDF{}) for $\mathbb{DH}^n$. For $n$ odd, the volume of a polytope in $\mathbb{DH}^n$ is shown to be completely determined by its intersection with $\partial\mathbb{H}^n$ and induces a new intrinsic \emph{volume} on $\partial\mathbb{H}^n$ that is invariant under M\"{o}bius transformations.