AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA

- Journal title : Communications of the Korean Mathematical Society
- Volume 28, Issue 4, 2013, pp.799-807
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2013.28.4.799

Title & Authors

AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA

Cho, Yunhi;

Cho, Yunhi;

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

References

1.

Y. Cho, Trigonometry in extended hyperbolic space and extended de Sitter space, Bull. Korean Math. Soc. 46 (2009), no. 6, 1099-1133.

2.

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

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.

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.