Abstract
Let the double hyperbolic space \({\mathbb {D}}{\mathbb {H}}^n\), 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 \({\mathbb {D}}{\mathbb {H}}^n\), where the volume is invariant under isometry but can possibly be complex valued. We show that the total volume of \({\mathbb {D}}{\mathbb {H}}^n\) is equal to \(i^n V_n({\mathbb {S}}^n)\) for both even and odd dimensions, and prove a Schläfli differential formula (SDF) for \({\mathbb {D}}{\mathbb {H}}^n\). For n odd, the volume of a polytope in \({\mathbb {D}}{\mathbb {H}}^n\) is shown to be completely determined by its intersection with \(\partial {\mathbb {H}}^n\) and induces a new intrinsic volume on \(\partial {\mathbb {H}}^n\) that is invariant under Möbius transformations.
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Notes
It should not be confused with changing the metric from \(ds^2\) to \(-ds^2\), which does change the sign of the constant curvature \(\kappa \). In this paper, to clear up any confusion, for a sign change of metric we always specify whether we are referring to \(-ds\) or \(-ds^2\).
In fact, this is expected even without using complex analysis. Let x and \(-x\) be the vertices of a half-space in \({\mathbb {D}}{\mathbb {H}}^1\), and y and \(-y\) be the vertices of a “smaller” half-space sitting completely inside the first one. If we expect all half-spaces to have a fixed length c, then the distance from \(-x\) to \(-y\) inside \({\mathbb {H}}^1_{-}\) has to be negative.
In this paper the term “double covering” is used loosely to only refer to the open part of a space, usually with the boundary points ignored, and the metrics on the two covers need not agree.
Let U be a unit ball centered at the origin. For \(U'\), let \(U'_{+}=U_{+}\) and \(U'_{-}\) be a translation of \(U_{-}\) such that \(U'_{-}\) is still centered on \(x_0=0\) but \(U'_{-}\cap U_{-}=\varnothing \). By Proposition 4.1, U is \(\mu \)-measurable; by (4.2) we have \(\mu _{\epsilon }(U')=\mu _{\epsilon }(U)\), so \(U'\) is also \(\mu \)-measurable. But \(U\cap U'=U_{+}\) is a half-space in \({\mathbb {H}}^n\), not \(\mu \)-measurable.
For \(m=1\), for any closed region U in \(\partial {\mathbb {H}}^3\) with piecewise smooth boundary, a potential definition of the volume \(V_{\infty ,2}(U)\) (that is still finitely additive) is as follows. First partition U into some simplicial regions with piecewise smooth boundaries, for each region define the volume as \(\alpha +\beta +\gamma -\pi \) where \(\alpha \), \(\beta \) and \(\gamma \) are the dihedral angles, then sum them up to define \(V_{\infty ,2}(U)\). It can be shown that \(V_{\infty ,2}(U)\) is well defined and independent of the partition (see also Corollary 12.7). By the definition, \(V_{\infty ,2}(U)\) is not only invariant under Möbius transformations, but also invariant under any conformal mappings on the closed region U. It should not be confused with the Riemman mapping theorem (for mapping to an open disk) whose subjects are simply connected open regions where the mapping may not be necessarily conformal on the boundary.
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I would like to thank Wei Luo and a referee for carefully reading the manuscript and making many helpful suggestions.
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Zhang, L. On the total volume of the double hyperbolic space. Beitr Algebra Geom 64, 403–443 (2023). https://doi.org/10.1007/s13366-022-00640-4
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DOI: https://doi.org/10.1007/s13366-022-00640-4