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Mathematics > Metric Geometry

arXiv:1409.3355 (math)
[Submitted on 11 Sep 2014 (v1), last revised 3 Jun 2016 (this version, v5)]

Title:The dual Jacobian of a generalised tetrahedron, and volumes of prisms

Authors:Alexander Kolpakov, Jun Murakami
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Abstract:We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the corresponding volume formulae.
Also, we obtain a volume formula for a hyperbolic $n$-gonal prism: the proof requires the above mentioned Jacobian, employed in the analysis of the edge lengths behaviour of such a prism, needed later for the Schläfli formula.
Comments: 26 pages, 11 figures, 1 table; to appear in Tokyo Journal of Mathematics
Subjects: Metric Geometry (math.MG); Complex Variables (math.CV); Differential Geometry (math.DG)
MSC classes: 51M25 (Primary), 51M09 (Secondary)
Cite as: arXiv:1409.3355 [math.MG]
  (or arXiv:1409.3355v5 [math.MG] for this version)
  https://doi.org/10.48550/arXiv.1409.3355

Submission history

From: Alexander Kolpakov [view email]
[v1] Thu, 11 Sep 2014 08:41:41 UTC (223 KB)
[v2] Wed, 24 Sep 2014 17:49:22 UTC (223 KB)
[v3] Sat, 25 Jul 2015 15:48:06 UTC (263 KB)
[v4] Wed, 16 Dec 2015 14:48:19 UTC (263 KB)
[v5] Fri, 3 Jun 2016 08:19:27 UTC (263 KB)
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