Asymptotic lines on pseudospheres and the angle of parallelism

A. V. Kostin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2021), pp. 25-34.

Résumé

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, to interpret it as the doubled angle of parallelism of the Lobachevsky plane or its ideal region, locally having the geometry of the de Sitter plane, respectively.

Détail

Keywords: asymptotic line, Lobachevsky plane, de Sitter plane, Minkowski space, pseudosphere.

Bibliographie

[1] Hesse L. O., “Über ein Übertragungsprinzip”, J. für reine und angew. Math., 66 (1866), 15–21 | MR | Zbl

[2] Rozenfeld B. A., Neevklidovy prostranstva, Nauka, M., 1969

[3] Buyalo S. V., “Mebiusovy struktury i prichinno-sledstvennye prostranstva so vremenem na okruzhnosti”, Algebra i analiz, 29:5 (2018), 1–50 | MR | Zbl

[4] Yaglom I.M, Rozenfeld B. A., Yasinskaya E. U., “Proektivnye metriki”, UMN, 19:5:119 (1964), 51–113 | MR | Zbl

[5] Artikbaev A., Saitova S. S., “Interpretatsiya geometrii na mnogoobraziyakh kak geometrii v prostranstve s proektivnymi metrikami”, SMFN, 65, no. 1, 2019, 1–10 | MR

[6] Norden A. P., Prostranstva affinnoi svyaznosti, Nauka, M., 1976 | MR

[7] Shirokov A. P., Neevklidovy prostranstva, Izd-vo KGU, Kazan, 1997

[8] Brusilovskii G. K., “Integrirovanie s pomoschyu giperbolicheskikh funktsii i gudermaniana”, Matem. prosv.: Ser. 1, 13, 1938, 33–46

[9] Romakina L. N., “Analogi formuly Lobachevskogo dlya ugla parallelnosti na giperbolicheskoi ploskosti polozhitelnoi krivizny”, Sib. elektronnye matem. izv., 10 (2013), 393–407 | Zbl

[10] Shirokov P. A., “Interpretatsiya i metrika kvadratichnykh geometrii”, Izb. raboty po geom., Izd-vo Kazansk. un-ta, Kazan, 1966, 15–179 | MR

[11] Blanusha D., “$ C^\infty$ — Isometric Imbeddings of the Hiperbolic Plane and of Cylinders with Hiperbolic Metric in Spherical Spaces”, Ann.Math. Pura Appl., 57 (1962), 321–337 | DOI | MR

[12] Blanusha D., “$ C^\infty$ — Isometric Imbeddings of cylinders with hyperbolic metric in Euclidean $7$-space”, Glas. Mat.-Fiz. i Astron., 11:3–4 (1956), 243–246 | MR

[13] Mischenko A. S., Fomenko A. T., Kratkii kurs differentsialnoi geometrii i topologii, Izd-vo Moskovsk. un-ta, M., 1980 | MR

[14] Hilbert D., “Über Flächen von konstanten Gau$\beta$scher Krümmung”, Trans. Amer. Math. Soc., 2 (1901), 87–99 | MR | Zbl

[15] Poznyak E. G., Popov A. G., “Geometriya uravneniya sin-Gordona”, Itogi nauki i tekhn. Ser. Probl. geom., 23, 1991, 99–130 | MR | Zbl

[16] Popov A. G., Maevskii E. V., “Analiticheskie podkhody k issledovaniyu uravneniya sin-Gordona”, Sovr. matem. i ee prilozh., 31 (2005), 13–52

[17] Gribkov I. V., “Postroenie nekotorykh regulyarnykh reshenii uravneniya “sinus-Gordon” s pomoschyu poverkhnostei postoyannoi otritsatelnoi krivizny”, Vestn. Moskovsk. un-ta. Ser. 1: Matem. Mekhan., 5 (1977), 78–83 | MR

[18] Chern S. S., “Geometrical interpretation of sinh-Gordon equation”, Ann. Polon. Math., 39 (1981), 63–69 | DOI | MR | Zbl

[19] Galeeva R. F., Sokolov D. D., “O geometricheskoi interpretatsii reshenii nekotorykh uravnenii matematicheskoi fiziki”, Issledov. po teorii poverkhnostei v rimanovykh prostranstvakh, LGPI, L., 1984, 8–22 | MR

[20] Kostin A. V., “Ob asimptoticheskikh liniyakh na psevdosfericheskikh poverkhnostyakh”, Vladikavk. matem. zhurn., 21:1 (2019), 16–26 | MR | Zbl

[21] Kostin A. V., “Evolyuty meridianov i asimptoticheskie na psevdosferakh”, Itogi nauki i tekhn. Ser. Sovrem. matem. i ee prilozh. Temat. obz., 169, 2019, 24–31