N.V. Abrosimov, Vuong Huu Bao. The volume of a hyperbolic tetrahedron with symmetry group $S_4$ ... P.7-17

The problem of calculating the volume of a hyperbolic tetrahedron of general form was solved in a number of works by G. Sforza and other authors. The formulas obtained are rather cumbersome. It is known that if a polyhedron has nontrivial symmetry, then the volume formula is essentially simplified. This phenomenon was discovered by Lobachevsky, who found the volume of an ideal tetrahedron. Later, J. Milnor expressed the corresponding volume as the sum of three Lobachevsky functions. In this paper we consider compact hyperbolic tetrahedra having the symmetry group $S_4$, which is generated by a mirror-rotational symmetry of the fourth order. The latter symmetry is the composition of rotation by the angle of $\pi/2$ about an axis passing through the middles of two opposite edges and reflection with respect to a plane perpendicular to this axis and passing through the middles of the remaining four edges. We establish necessary and sufficient conditions for the existence of such tetrahedra in $\mathbb{H}^3$. Then we find relations between their dihedral angles and edge lengths in the form of a cosine law. Finally, we obtain exact integral formulas expressing the hyperbolic volume of the tetrahedra in terms of the edge lengths.

Keywords: hyperbolic tetrahedron, symmetry group, reflection followed by a rotation, hyperbolic volume.

The paper was received by the Editorial Office on June 15, 2017.

Nikolai Vladimirovich Abrosimov Cand. Sci. (Phys.-Math.), Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia, Novosibirsk State University, Novosibirsk, 630090 Russia,
e-mail: abrosimov@math.nsc.ru

Vuong Huu Bao, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia, Novosibirsk State University, Novosibirsk, 630090 Russia,
e-mail: vuonghuubao@live.com

REFERENCES

1.   Matveev S., Fomenko A. Isoenergetic surfaces of Hamiltonian systems, the enumeration of three-dimensional manifolds in order of growth of their complexity, and the calculation of the volumes of closed hyperbolic 3-manifolds. Russian Math. Surveys, 1988, vol. 43, no. 1, pp. 3–24. doi: 10.1070/RM1988v043n01ABEH001554 .

2.   Weeks J., Hyperbolic structures on 3-manifolds. Ph. D. Thesis. Princeton: Princeton University, 1985.

3.   Thurston W.P. The Geometry and topology of three-manifolds. Lecture notes. Princeton, Princeton Univ., 1980, 502 p.

4.   Gabai D., Meyerhoff R., Milley P. Minimum volume cusped hyperbolic three-manifolds. J. Amer. Math. Soc., 2009, vol. 22, no. 4, pp. 1157–1215. doi: 10.1090/S0894-0347-09-00639-0 .

5.    Milnor J. Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc., 1982, vol. 6, no. 1, pp. 9–24. doi: 10.1090/S0273-0979-1982-14958-8 .

6.   Cho Yu., Kim H. On the volume formula for hyperbolic tetrahedra. Disc. Comp. Geom., 1999, vol. 22, no. 3, pp. 347–366. doi: 10.1007/PL00009465 .

7.   Murakami J., Yano M. On the volume of a hyperbolic and spherical tetrahedron. Comm. Anal. Geom., 2005, vol. 13, no. 2, pp. 379–400. doi: 10.4310/CAG.2005.v13.n2.a5 .

8.   Ushijima A. Volume formula for generalized hyperbolic tetrahedra. In: Non-Euclidean geometries, eds. A. Prekopa, E. Molnar. 2006, ser. Mathematics and Its Applications, vol. 581, pp. 249–265.
doi: 10.1007/0-387-29555-0_13 .

9.   Derevnin D.A., Mednykh A.D. A formula for the volume of a hyperbolic tetrahedron. Russ. Math. Surv., 2005, vol. 60, no. 2, pp. 346–348. doi: 10.1070/RM2005v060n02ABEH000833 .

10.   Sforza G. Ricerche di estensionimetria differenziale negli spazi metrico-projettivi. Modena Mem. Acc., 1906. Ser. III, VIII (Appendice), pp. 21–66 (in Italian).

11.   Abrosimov N.V., Mednykh A.D. Volumes of polytopes in spaces of constant curvature. In: Rigidity and Symmetry, eds. R. Connelly, A. Ivic Weiss , W. Whiteley. N. Y.: Springer, 2014, Ser. Fields Institute Communications, vol. 70, pp. 1–26. doi: 10.1007/978-1-4939-0781-6_1 .

12.   Abrosimov N.V., Kudina E.S., Mednykh A.D. On the volume of hyperbolic octahedron with 3-symmetry. Proc. Steklov Inst. Math., 2015, vol. 288, pp. 1–9. doi: 10.1134/S0081543815010010 .

13.   Johnson N.W. Geometries and transformations. Cambridge: Cambridge University Press, 2017, 350 p. ISBN-10: 1107103401 .

14.   Ponarin Ya.P. Elementarnaya geometriya. Tom 2. Stereometriya, preobrazovaniya prostranstva. [Elementary geometry. Vol. 2: Stereometry, space transformation]. Moscow: Publ. MTsNMO, 2015, 256 p. ISBN: 978-5-4439-0369-9 .

15.   Alekseevskij D.V., Vinberg E.B., Solodovnikov A.S. Geometry of Spaces of Constant Curvature. In: Geometry II. Encyclopaedia of Mathematical Sciences, vol. 29. ed. E. B. Vinberg, Berlin, Heidelberg, Springer-Verlag, 1993, pp. 1–138. doi: 10.1007/978-3-662-02901-5_1 .