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AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA
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 Title & Authors
AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA
Cho, Yunhi;
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 Abstract
We defined and studied a naturally extended hyperbolic space (see [1] and [2]). In this study, we describe Sforza`s formula [7] and Santal`s formula [6], which were rediscovered and later discussed by many mathematicians (Milnor [4], Surez-Peir [8], J. Murakami and Ushijima [5], and Mednykh [3]) in the spherical space in an elementary way. Thereafter, using the extended hyperbolic space, we apply the same method to prove their results in the hyperbolic space.
 Keywords
hyperbolic space;spherical space;polyhedron;volume;
 Language
English
 Cited by
 References
1.
Y. Cho, Trigonometry in extended hyperbolic space and extended de Sitter space, Bull. Korean Math. Soc. 46 (2009), no. 6, 1099-1133. crossref(new window)

2.
Y. Cho and H. Kim, The analytic continuation of hyperbolic space, Geom. Dedicata 161 (2012), no. 1, 129-155. crossref(new window)

3.
A. D. Mednykh, Hyperbolic and spherical volume for knots, links and polyhedra, Summer school and conference on Geometry and Topology of 3-manifolds, Trieste-Italy, 6-24 June 2005.

4.
J. Milnor, The Schlafli Differential Equality, Collected papers, Vol. 1, Publish or Perish, Houston, Texas, 1994.

5.
J. Murakami and A. Ushijima, A volume formula for hyperbolic tetrahedra in terms of edge lengths, J. Geom. 83 (2005), no. 1-2, 153-163. crossref(new window)

6.
L. Santalo, Integral Geometry and Geometric Probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley, 1976.

7.
G. Sforza, Spazi metrico-proiettivi, Ricerche di Estensionimetria differenziale, Serie III, VIII (1906), 3-66.

8.
E. Suarez-Peiro, A Schlafli differential formula for simplices in semi-Riemannian hyperquadrics, Gauss-Bonnet formulas for simplices in the de Sitter sphere and the dual volume of a hyperbolic simplex, Pacific J. Math. 194 (2000), no. 1, 229-255. crossref(new window)