대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제46권6호
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  • Pages.1099-1133
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  • 2009
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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RIGONOMETRY IN EXTENDED HYPERBOLIC SPACE AND EXTENDED DE SITTER SPACE

  • Cho, Yun-Hi (Department of Mathematics University of Seoul)
  • 발행 : 2009.11.30
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초록

We study the hyperbolic cosine and sine laws in the extended hyperbolic space which contains hyperbolic space as a subset and is an analytic continuation of the hyperbolic space. And we also study the spherical cosine and sine laws in the extended de Sitter space which contains de Sitter space S$^n_1$ as a subset and is also an analytic continuation of de Sitter space. In fact, the extended hyperbolic space and extended de Sitter space are the same space only differ by -1 multiple in the metric. Hence these two extended spaces clearly show and apparently explain that why many corresponding formulas in hyperbolic and spherical space are very similar each other. From these extended trigonometry laws, we can give a coherent and geometrically simple explanation for the various relations between the lengths and angles of hyperbolic polygons, and relations on de Sitter polygons which lie on S$^2_1$, and tangent laws for various polyhedra.

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참고문헌

  1. D. V. Alekseevskij, E. B. Vinberg, and A. S. Solodovnikov, Geometry of spaces of constant curvature, Geometry, II, 1-138, Encyclopaedia Math. Sci., 29, Springer, Berlin, 1993
  2. Y. Cho and H. Kim, The analytic continuation of hyperbolic space, arXiv:math. MG/0612372
  3. Y. Cho and H. Kim, Volume of $C^{1,{\alpha}}$-boundary domain in extended hyperbolic space, J. Korean Math. Soc. 43 (2006), no. 6, 1143-1158 https://doi.org/10.4134/JKMS.2006.43.6.1143
  4. D. Derevnin and A. Mednykh, On the volume of spherical Lambert cube, arXiv:math. MG/0212301
  5. J. J. Dzan, Gauss-Bonnet formula for general Lorentzian surfaces, Geom. Dedicata 15 (1984), no. 3, 215-231 https://doi.org/10.1007/BF00147645
  6. J. J. Dzan, Sectorial measure with applications to non-Euclidean trigonometries, Mitt. Math. Ges. Hamburg 13 (1993), 179-197
  7. J. J. Dzan, Trigonometric Laws on Lorentzian sphere $S^2_1$, J. Geom. 24 (1985), no. 1, 6-13 https://doi.org/10.1007/BF01223527
  8. W. Fenchel, Elementary Geometry in Hyperbolic Space, Walter de Gruyter, Berlin New York, 1989
  9. R. Kellerhals, On the volume of hyperbolic polyhedra, Math. Ann. 285 (1989), no. 4, 541-569 https://doi.org/10.1007/BF01452047
  10. B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York, London Paris, 1983
  11. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, 149. Springer-Verlag, New York, 1994
  12. J. M. Schlenker, M´etriques sur les poly`edres hyperboliques convexes, J. Differential Geom. 48 (1998), no. 2, 323-405 https://doi.org/10.4310/jdg/1214460799
  13. W. P. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press, Princeton New Jersey, 1997
  14. E. B. Vinberg, Volumes of non-Euclidean Polyhedra, Russian Math. Surveys 48 (1993), no. 2, 15-45 https://doi.org/10.1070/RM1993v048n02ABEH001011

피인용 문헌

  1. GEOMETRIC AND ANALYTIC INTERPRETATION OF ORTHOSCHEME AND LAMBERT CUBE IN EXTENDED HYPERBOLIC SPACE vol.50, pp.6, 2013, https://doi.org/10.4134/JKMS.2013.50.6.1223
  2. On the area of a trihedral on a hyperbolic plane of positive curvature vol.25, pp.2, 2015, https://doi.org/10.3103/S1055134415020042
  3. Inequalities of trihedrals on a hyperbolic plane of positive curvature vol.58, pp.4, 2017, https://doi.org/10.1007/s13366-017-0339-5
  4. AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA vol.28, pp.4, 2013, https://doi.org/10.4134/CKMS.2013.28.4.799
  5. The analytic continuation of hyperbolic space vol.161, pp.1, 2012, https://doi.org/10.1007/s10711-012-9698-0
  6. Duality structures and discrete conformal variations of piecewise constant curvature surfaces vol.320, 2017, https://doi.org/10.1016/j.aim.2017.08.043