A classification of simple closed geodesics on regular tetrahedra in the Lobachevsky space

Authors

  • A.A. Borisenko B. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv
  • D.D. Sukhorebska B. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv

DOI:

https://doi.org/10.15407/dopovidi2019.04.003

Keywords:

closed geodesics, Lobachevsky space, regular tetrahedra

Abstract

The full classification of simple closed geodesics on regular tetrahedra in the hyperbolic space is described. The asymptotics of the number of simple closed geodesics of length not more than L, with L tending to infinity, is found.

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References

Lusternik, L. A. & Schnirelmann, L. G. (1927). On the problem of three closed geodesic on surfaces of genus zero. C. R. Acad. Sci., 189, pp. 269-271.

Fet, A. I. (1965). On a periodicity problem in the calculus of variations. Dokl. AN SSSR, 160, No. 2, pp. 287-289 (in Russian).

Huber, H. (1961). On the analytic theory hyperbolic spatial forms and motion groups II. Math. Ann., 143, pp. 463-464 (in German). doi: https://doi.org/10.1007/BF01470758

Rivin, I. (2001). Simple curves on surfaces. Geometriae Dedicata, 87, pp. 345-360. doi: https://doi.org/10.1023/A:1012010721583

Mirzakhani, M. (2008). Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. Math., 168, pp. 97-125. doi: https://doi.org/10.4007/annals.2008.168.97

Fuchs, D. & Fuchs, E. (2007). Closed geodesics on regular polyhedra. Mosc. Math. J., 7, No. 2, pp. 265-279. doi: https://doi.org/10.17323/1609-4514-2007-7-2-265-279

Fuchs, D. (2009). Geodesics on a regular dodecahedron. Preprints of Max Planck Institute for Mathematik, No. 91. Bonn. Retrieved from http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2010/2009_91.pdf

Protasov, V. Yu. (2007). Closed geodesics on the surface of a simplex. Sb. Math., 198, No. 2, pp. 243-260. doi: https://doi.org/10.1070/SM2007v198n02ABEH003836

Hardy, G. H. & Wright, E. M. (1975). An introduction to the theory of numbers. Oxford: Oxford University Press.

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Published

21.04.2024

How to Cite

Borisenko, A., & Sukhorebska, D. (2024). A classification of simple closed geodesics on regular tetrahedra in the Lobachevsky space . Reports of the National Academy of Sciences of Ukraine, (4), 3–9. https://doi.org/10.15407/dopovidi2019.04.003

Issue

No. 4 (2019): Reports of the National Academy of Sciences of Ukraine

Section

Mathematics

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