A volume formula for generalized hyperbolic tetrahedra

Akira Ushijima

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A volume formula for generalized hyperbolic tetrahedra

Akira Ushijima

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Abstract

A generalized hyperbolic tetrahedra is a polyhedron (possibly non-compact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that a volume formula for ordinary hyperbolic tetrahedra devised by J. Murakami and M. Yano can be applied to such ones. There are two key tools for the proof; one is so-called Schlafli's differential formula for hyperbolic polyhedra, and the other is a necessary and sufficient condition for given numbers to be the dihedral angles of a generalized hyperbolic simplex with respect to their dihedral angles.

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Akira Ushijima. 2003-10-30. A volume formula for generalized hyperbolic tetrahedra. https://arxiv.org/abs/math/0309216

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