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Статья
2021

Asymptotic Lines on Pseudospheres and the Angle of Parallelism


A. V. KostinA. V. Kostin
Русская математика
https://doi.org/10.3103/S1066369X21060037
Abstract / Full Text

The angle between the asymptotic lines―and generally between the lines of the Chebyshev network―on surfaces of constant curvature is usually analytically interpreted as a solution of the second-order partial differential equation. For surfaces of constant negative curvature in Euclidean space, this is the sine-Gordon equation. Conversely, surfaces of constant negative curvature are used to construct and interpret solutions to the sine-Gordon equation. This article shows that the angle between the asymptotic lines on the pseudospheres of Euclidean and pseudo-Euclidean spaces can be interpreted differently, namely, as the doubled angle of parallelism of the Lobachevsky plane or its ideal region, locally having the geometry of the de Sitter plane, respectively.

Author information
  • Elabuga Institute of Kazan Federal University, 89 Kazanskaya str., 423600, Elabuga, RussiaA. V. Kostin
References
  1. Hesse L.O. "Über ein Übertragungsprinzip", J. für reine und angew. Math. 66, 15-21 (1866).
  2. Rosenfeld B.A. Non-Euclidean Spaces (Nauka, Moscow, 1969) [in Russian].
  3. Buyalo S.V. "Möbius structures and timed causal spaces on the circle", St. Petersburg Math. J. 29 (5), 715-747 (2018).
  4. Yaglom I.M., Rosenfeld B.A., Yasinskaya E.U. "Projective metrics", UMN 19 (5:119), 51-113 (1964) [in Russian].
  5. Artikbaev A., Saitova S.S. "Interpretation of Geometry on Manifolds as a Geometry in a Space with Projective Metric", Contemporary Mathematics. Fundamental Directions 65 (1), 1-10 (2019).
  6. Norden A.P. Spaces with affine connection (Nauka, Moscow, 1976) [in Russian].
  7. Shirokov A.P. Non-Euclidean Spaces (Izd-vo KGU, Kazan, 1977) [in Russian].
  8. Brusilobskii G.K. "Integration with the help of hyperbolic functions and the Gudermannian", Matem. prosv.: Ser. 1 13, 33-46 (1938).
  9. Romakina L. N. "Analogs of a formula of Lobachevsky for the angle of parallelism on the hyperbolic plane of positive curvature", Siberian Electronic Math. Reports 10, 393-407 (2013) [in Russian].
  10. Shirokov P.A. "Interpretation and metric of quadratic geometries", in: Izbr. raboty po geom., 15-179 (Izd-vo Kazan. un-ta, Kazan, 1966) [in Russian].
  11. Blanusha D. "\(C^\infty\)-isometric imbeddings of the hyperbolic plane and of cylinders with hyperbolic metric in spherical spaces", Ann. Math. Pura Appl. 57, 321-337 (1962).
  12. Blanusha D. "\(C^\infty\)-isometric imbeddings of cylinders with hyperbolic metric in Euclidean 7-space", Glas. Mat.-Fiz. i Astron. 11 (3–4), 243-246 (1956).
  13. Mishchenko A.S., Fomenko A.T. Course of differential geometry and topology (Izd-vo Moskovsk. un-ta, Moscow, 1980) [in Russian].
  14. Hilbert D. "Über Flächen von konstanter Gau{ß}scher Krümmung", Trans. Amer. Math. Soc. 2, 87-99 (1901).
  15. Poznyak E.G., Popov A.G. "Geometry of sin-Gordon equations", Itog nauki i tekhn. Ser. Probl. geom. 23, 99-130 (1991).
  16. Popov A.G., Maevskii E.V. "Analytical approaches to the study of the sine-Gordon equation and pseudospherical surfaces", Journal of Mathematical Sciences 142, 2377-2418 (2007).
  17. Grobkov I.V. "Construction of certain regular solutions of the “sine-Gordon” equation using surfaces of constant negative curvature", Vestnik Mosk. Univ., Ser. 1: Mat. Mech. 4, 78-83 (1977) [in Russian].
  18. Chern S.S. "Geometrical interpretation of sinh-Gordon equation", Ann. Polon. Math. 39, 63-69 (1981).
  19. Galeeva R.F., Sokolov D.D. "On geometric interpretation of solutions to some equations of mathematical physics", in: The study in the theory of surfaces in Riemannian spaces, 8-22 (LGPI, Leningrad, 1984) [in Russian].
  20. Kostin A.V. "Asymptotic Lines on the Pseudo-Spherical Surfaces", Vladikavkaz Math. J. 21 (1), 16-26 (2019).
  21. Kostin A.V. "Evolutes of meridians and asymptotic lines on pseudospheres", Itogi nauki i tekhn. Ser. Sovr. matem. Appl. Temat. obzor 169, 24-31 (2019).