An elementary proof of Sforza-Santalo relation for spherical and hyperbolic polyhedra
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We defined and studied a naturally extended hyperbolic space(see [1] and [2]). In this study, we describe Sforza’s formula [7] and San-tal´o’s formula [6], which were rediscovered and later discussed by manymathematicians (Milnor [4], Su´arez-Peir´o [8], J. Murakami and Ushijima[5], and Mednykh [3]) in the spherical space in an elementary way. There-after, using the extended hyperbolic space, we apply the same method toprove their results in the hyperbolic space.
Abstract
We defined and studied a naturally extended hyperbolic space(see [1] and [2]). In this study, we describe Sforza’s formula [7] and San-tal´o’s formula [6], which were rediscovered and later discussed by manymathematicians (Milnor [4], Su´arez-Peir´o [8], J. Murakami and Ushijima[5], and Mednykh [3]) in the spherical space in an elementary way. There-after, using the extended hyperbolic space, we apply the same method toprove their results in the hyperbolic space.
참고문헌 (8)
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- G. Sforza, "Spazi metrico-proiettivi", Ricerche di Estensionimetria differenziale, Serie III, VIII, 3~66 (1906)
- E. Su´arez-Peir´o, "A Schl¨afli 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(1), pp.229~255 (2000)