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GEOMETRIC AND ANALYTIC INTERPRETATION OF ORTHOSCHEME AND LAMBERT CUBE IN EXTENDED HYPERBOLIC SPACE

  • Cho, Yunhi ;
  • Kim, Hyuk
  • Received : 2012.10.03
  • Published : 2013.11.01

Abstract

We give a geometric proof of the analyticity of the volume of a tetrahedron in extended hyperbolic space, when vertices of the tetrahedron move continuously from inside to outside of a hyperbolic space keeping every face of the tetrahedron intersecting the hyperbolic space. Then we find a geometric and analytic interpretation of a truncated orthoscheme and Lambert cube in the hyperbolic space from the viewpoint of a tetrahedron in the extended hyperbolic space.

Keywords

hyperbolic space;volume;analytic continuation

References

  1. D. V. Alekseevskij, E. B. Vinberg, and A. S. Solodovnikov, Geometry of Space of Constant Curvature, Geometry, II, 1-138, Encyclopaedia Math. Sci., 29, Springer, Berlin, 1993.
  2. B. B¨ohm and H. C. Im Hof, Flacheninhalt Verallgemeinerter Hyperbolischer Dreiecke, Geom. Dedicata 42 (1992), no. 2 223-233.
  3. Y. Cho, Trigonometry in extended hyperbolic space and extended de Sitter space, Bull. Korean Math. Soc. 46 (2009), no. 6, 1099-1133. https://doi.org/10.4134/BKMS.2009.46.6.1099
  4. Y. Cho and H. Kim, On the volume formula for hyperbolic tetrahedra, Discrete Comput. Geom. 22 (1999), no. 3, 347-366. https://doi.org/10.1007/PL00009465
  5. Y. Cho and H. Kim, Volume of $C^{1,{\alpha}}$-boundary domain in extended hyperbolic space, J. KoreanMath. Soc. 43 (2006), no. 6, 1143-1158. https://doi.org/10.4134/JKMS.2006.43.6.1143
  6. Y. Cho and H. Kim, The analytic continuation of hyperbolic space, Geom. Dedicata 161 (2012), 129-155. https://doi.org/10.1007/s10711-012-9698-0
  7. P. Doyle and G. Leibon, 23040 symmetries of hyperbolic tetrahedra, arXiv:math. GT/0309187.
  8. R. Kellerhals, On the volume of hyperbolic polyhedra, Math. Ann. 285 (1989), 541-569. https://doi.org/10.1007/BF01452047
  9. A. Kolpakov and J. Murakami, Volume of a doubly truncated hyperbolic tetrahedron, arXiv:math.MG/1203.1061v4.
  10. J. Milnor, Hyperbolic geometry: the first 150 years, Bull. Am. Math. Soc. (N.S.) 6 (1982), no. 1, 9-24. https://doi.org/10.1090/S0273-0979-1982-14958-8
  11. J. Murakami and M. Yano, On the volume of a hyperbolic and spherical tetrahedron, Comm. Anal. Geom. 13 (2005), no. 2, 379-400. https://doi.org/10.4310/CAG.2005.v13.n2.a5
  12. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, 149. Springer-Verlag, New York, 1994.
  13. A. Ushijima, A volume formula for generalised hyperbolic tetrahedra, Non-Euclidean geometries, 249-265, Math. Appl. (N. Y.), 581, Springer, New York, 2006. https://doi.org/10.1007/0-387-29555-0_13
  14. E. B. Vinberg, Volumes of non-Euclidean Polyhedra, Russian Math. Surveys 48 (1993), no. 2, 15-45. https://doi.org/10.1070/RM1993v048n02ABEH001011

Cited by

  1. Combinatorial Decompositions, Kirillov–Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra 2016, https://doi.org/10.1080/10586458.2016.1242441

Acknowledgement

Supported by : Korea Research Foundation(KRF)