On the total volume of the double hyperbolic space

TL;DR

This paper introduces the double hyperbolic space ^n, extends volume concepts to it, and finds that its total volume relates to the sphere volume, with implications for Mf6bius invariance and boundary geometry.

Contribution

The paper defines the double hyperbolic space ^n, extends volume measures to it, and establishes a Schla4fli formula, revealing new invariants and boundary volume structures.

Findings

Total volume of ^n equals i^n times the sphere volume.

Volume in ^n can be complex-valued and is invariant under isometries.

For odd dimensions, volume is determined by boundary intersection and induces a Mf6bius-invariant boundary volume.

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.