On the volume of a non-Euclidean tetrahedron
数学专题报告
报告题目(Title)：On the volume of a non-Euclidean tetrahedron
报告人(Speaker)：Nikolay Abrosimov (Sobolev Institute of Mathematics)
地点(Place)：后主楼1124
时间(Time)：2025年4月16日 10:00-11:00
邀请人(Inviter)：程志云
报告摘要
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], where a version of the Sforza formula for the volume of a spherical tetrahedron is also given. 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 second part 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 hyperbolic trirectangular 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.
At the end of the talk, we will present a new formula for the volume of a spherical trirectangular tetrahedron [5]. The list of spherical Coxeter tetrahedra was constructed by H.S.M. Coxeter [6]. He shown that there are 11 types of Coxeter tetrahedra in S3. We will show that exactly 5 of these types belong to the family of trirectangular tetrahedra. We will calculate their volumes in order to check our formula.
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.
[4] N. Abrosimov, S. Stepanishchev, The volume of a trirectangular hyperbolic tetrahedron. Siberian Electronic Mathematicsl Reports, 2023, V. 20, No. 1, P. 275–284.
[5] N. Abrosimov, B. Bayzakova, The volume of a spherical trirectangular tetrahedron. Siberian Electronic Mathematical Reports (in print).
[6] H.S.M. Coxeter, Discrete groups generated by reflections. Ann. Math., 1934, V. 35, P. 588-621.
