Title：On the volume of hyperbolic tetrahedron
Reporter：Nikolay Abrosimov
Work Unit：Sobolev Institute of Mathematics
Time：2023/08/20 10:30-11:30
Address：Room 313, Zhengxin Building
Summary of the report: The talk will give an overview of the latest results on finding exact formulas for calculating the volumes of hyperbolic tetrahedra. The classical formula of G. Sforza [1] expresses the volume of a hyperbolic tetrahedron of a general form in terms of dihedral angles. Its modern proof is proposed in [2]. The formula in terms of edge lengths is obtained in the recent joint work of the author with B. Vuong [3]. The known formulas for the volume of a hyperbolic tetrahedron of a general form are very complicated and cannot always be applied to calculate the volumes of more complex polyhedra. So, natural question arises to find more convenient and simple formulas for sufficiently wide families of hyperbolic tetrahedra. At the end of the talk we will consider hyperbolic tetrahedra of special types: ideal, biorthogonal, 3-orthogonal and their generalizations. The volume of the ideal and biorthogonal tetrahedron was known to N.I. Lobachevsky. We will present new formulas for calculating volumes and normalized volumes of a trirectangular hyperbolic tetrahedron [4] as well as its generalization for 4-parameter family of tetrahedra with one edge orthogonal to the face. The latter formulas can be used to calculate the volumes of more complex polyhedra in the Lobachevsky space.
References:
[1] G. Sforza, Spazi metrico-proiettivi // Ricerche di Estensionimetria Integrale, Ser. III, VIII (Appendice), 1907, P. 41–66.
[2] N.V. Abrosimov, A.D. Mednykh, Volumes of polytopes in constant curvature spaces // Fields Inst. Commun., 2014, V. 70, P. 1–26. arXiv:1302.4919.
[3] N. Abrosimov, B. Vuong, Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths // Journal of Knot Theory and Its Ramifications, 2021, V. 30, No. 10, 2140007. arXiv:2107.03004.
[4] N. Abrosimov, S. Stepanishchev, The volume of a trirectangular hyperbolic tetrahedron // Siberian Electronic Mathematicsl Reports, 2023, V. 20, No. 1, P. 275–284.
Introduction of the Reporter: Professor Nikolay Abrosimov is a professor of the department of Mechanic-Mathmatics of Novosivirsk State University. He received a Candidate of Sciences in physics and mathematics in 2009 from Sobolev Institute of Mathematics. His Research interests are hyperbolic geometry, volumes of non-Euclidean polytopes, geometry and topology of three-dimensional manifolds, orbifolds, knots and links. He has more than 25 research publications. He is a chairman of the Council of Scientific Youth of the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
