RIGONOMETRY IN EXTENDED HYPERBOLIC SPACE AND EXTENDED DE SITTER SPACE

Title & Authors
RIGONOMETRY IN EXTENDED HYPERBOLIC SPACE AND EXTENDED DE SITTER SPACE
Cho, Yun-Hi;

Abstract
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$\small{^n_1}$ 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$\small{^2_1}$, and tangent laws for various polyhedra.
Keywords
hyperbolic space;volume;analytic continuation;
Language
English
Cited by
1.
GEOMETRIC AND ANALYTIC INTERPRETATION OF ORTHOSCHEME AND LAMBERT CUBE IN EXTENDED HYPERBOLIC SPACE,;;

대한수학회지, 2013. vol.50. 6, pp.1223-1256
2.
AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA,;

대한수학회논문집, 2013. vol.28. 4, pp.799-807
1.
GEOMETRIC AND ANALYTIC INTERPRETATION OF ORTHOSCHEME AND LAMBERT CUBE IN EXTENDED HYPERBOLIC SPACE, Journal of the Korean Mathematical Society, 2013, 50, 6, 1223
2.
On the area of a trihedral on a hyperbolic plane of positive curvature, Siberian Advances in Mathematics, 2015, 25, 2, 138
3.
Inequalities of trihedrals on a hyperbolic plane of positive curvature, Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2017, 58, 4, 723
4.
AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA, Communications of the Korean Mathematical Society, 2013, 28, 4, 799
5.
The analytic continuation of hyperbolic space, Geometriae Dedicata, 2012, 161, 1, 129
6.
Duality structures and discrete conformal variations of piecewise constant curvature surfaces, Advances in Mathematics, 2017, 320, 250
References
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

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

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

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

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

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