Asymptotic lines on the pseudo-spherical surfaces. (Russian. English summary) Zbl 1463.53021

Kostin, Andreĭ Viktorovich

Vladikavkaz. Mat. Zh. 21, No. 1, 16-26 (2019).

Summary: Consider the three-dimensional extended Lobachevsky space. In a proper area of Lobachevsky space take the ‘complete’ pseudosphere, that is, a surface of rotation of a straight line around a given parallel straight line. One part of it is embedded into Euclidean space in the form of the Beltrami-Minding funnel, the other one into three-dimensional Minkowski space as an analogue of the pseudosphere in this space. The interpretations of imaginary asymptotic lines on this pseudospherical surface with the Lobachevsky metric in Minkowski space are considered. Imaginary asymptotic lines on the pseudo-Euclidean continuation of the pseudosphere can be interpreted as real asymptotic lines on the surface of constant curvature with indefinite metric. These surfaces are other pseudo-Euclidean analogs of the Beltrami-Minding pseudosphere. The properties of the asymptotic lines on the pseudospheres with de Sitter metric in the three-dimensional Minkowsky space are studied. The considered properties of asymptotic lines on pseudospheres of pseudo-Euclidean space (Minkowski space) are similar to that of asymptotic lines on the Beltrami-Minding pseudosphere in Euclidean space. Areas of quadrangles of the asymptotic net on a surface of constant negative curvature in Euclidean space can be found by the Hazzidakis formula. These results are transferred to surfaces of constant curvature with indefinite metric in Minkowski space.

MSC:

53A35	Non-Euclidean differential geometry
53B30	Local differential geometry of Lorentz metrics, indefinite metrics

Keywords:

pseudosphere; Lobachevsky plane; de Sitter plane; asymptotic line; Chebyshev net; Minkowski space

References:

[1]	Minding F., “Wie sich entscheiden lässt, ob zwei gegebene krumme Flächen auf einander abwickelbar sind oder nicht; nebst Bemerkungen über die Flächen von unveränderlichem Krümmungsmasse”, #J. für die Reine und Angewandte Mathematik, #1839:19 (1839), 370-387 · ERAM 019.0625cj · doi:10.1515/crll.1839.19.370
[2]	Blanusha D., “\( C^\infty \)-isometric imbeddings of the hiperbolic plane and of cylinders with hiperbolic metric in spherical spaces”, #Ann. Math. Pura Appl., #57:1 (1962), 321-337 · Zbl 0106.36203 · doi:10.1007/BF02417747
[3]	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 · Zbl 0073.16905
[4]	Rosenfeld B. A., #Non-Euclidean Space, Nauka, M., 1969, 548 pp. (in Russian)
[5]	Hesse L. O., “Über ein übertragungsprinzip”, #J. für die Reine und Angewandte Mathematik, #66 (1866), 15-21 · ERAM 066.1713cj · doi:10.1515/crll.1866.66.15
[6]	Tchebychev P. L., “Sur la coupe de vêtements”, #Association Francaise pour l’Avancement de Sciences, Congres de Paris, 1878, 154-155
[7]	Chebyshev P. L., “On the Cutting of Garments”, #Uspekhi Mat. Nauk, #1:2(12) (1946), 38-42 (in Russian) · Zbl 0063.00746
[8]	Hazzidakis J. N., “Über einige Eigenschaften der Flächen mit konstantem Krümmungsmasz”, #J. für die Reine und Angewandte Mathematik, #88 (1880), 68-73 · JFM 11.0527.03 · doi:10.1515/crll.1880.88.68
[9]	Hilbert D., “Über Flächen von konstanten Gaußscher Krümmung”, #Trans. Amer. Math. Soc., #2 (1901), 87-99 · JFM 32.0608.01
[10]	Shirokov P. A., “Interpretation and Metric of Quadratic Geometries”, #Selected Works on Geometry, Kazan, 1966, 15-179 (in Russian)
[11]	Kostin A. V., Sabitov I. K., “Smarandache theorem in hyperbolic geometry”, #J. of Math. Physics, Analysis, Geometry, #10:2 (2014), 221-232 · Zbl 1323.51007 · doi:10.15407/mag10.02.221
[12]	Poznyak É. G., Popov A. G., “The Geometry of the sine-Gordon Equation”, #Journal of Mathematical Sciences, #70:2 (1994), 1666-1684 · Zbl 0835.35123 · doi:10.1007/BF02110595
[13]	Chern S. S., “Geometrical interpretation of sinh-Gordon equation”, #Annales Polonici Mathematici, #39 (1981), 63-69 · Zbl 0497.53056 · doi:10.4064/ap-39-1-63-69
[14]	Galeeva R. F., Sokolov D. D., “On the Geometric Interpretation of Solutions of Some Nonlinear Equations of Mathematical Physics”, #Research on the Theory of Surfaces in Riemann Spaces, L., 1984, 8-22 (in Russian) · Zbl 0616.35074
[15]	Klotz-Milnor T., “Harmonic maps and classical surface theory in Minkowski 3-Space”, #Trans. Amer. Math. Soc., #280:1 (1983), 161-185 · Zbl 0532.53047 · doi:10.2307/1999607
[16]	Rosenfeld B. A., Maryukova N. E., “Surfaces of constant curvature and Geometric interpretation of the Klein-Gordon, sin-Gordon and sinh-Gordon equation”, #Publications de L’Institut Mathématique, #61(75) (1997), 119-132 · Zbl 0885.53015
[17]	Lopez R., “Differential geometry of curves and surfaces in Lorentz-Minkowski space”, #Int. Eletron. J. of Geometry, #7:1 (2014), 44-107 · Zbl 1312.53022
[18]	Barros M., Caballero M., Ortega M., “Rotational surfaces in \(L^3\) and solutions of the nonlinear sigma model”, #Communication in Math. Physics, #290:2 (2009), 437-477 · Zbl 1182.53057 · doi:10.1007/s00220-009-0850-0
[19]	Albujer A. L., Caballero M., “Geometric properties of same mean curvature in \(R^3\) and \(L^3\)”, #J. of Math. Anal. Appl., #445:1 (2017), 1013-1024 · Zbl 1348.53006 · doi:10.1016/j.jmaa.2016.07.062
[20]	Lopez R., Kaya S., “New examples of maximal surfaces in Lorentz-Minkowski space”, #Kyoshu J. of Math., #71:2 (2017), 311-327 · Zbl 1392.53070 · doi:10.2206/kyushujm.71.311
[21]	Poznyak É. G., Shikin E. V., #Differential Geometry, Moscow State Univ. Publ., M., 1990, 384 pp. (in Russian) · Zbl 0724.53002
[22]	Vygodskii M. Ya., #Differential Geometry, M.-L., 1949, 512 pp. (in Russian) · Zbl 0041.28902
[23]	Kostin A. V., “The Regularity of Asymptotic Lines on the de Sitter Pseudosphere”, #Geometry Days in Novosibirsk - 2012, Abstracts of the Inter. Conf. dedicated to 100th anniversary of A. D. Aleksandrov, Sobolev Institute of Mathematics of Siberian Branch of the RAS, Novosibirsk, 2012, 48-49 (in Russian)
[24]	Kostin A. V., Kostina N. N., “On the Evolutes of some Curves on the Pseudo-Euclidean Plane”, #Proceedings of the Participants of the International School-Seminar on Geometry and Analysis in Memory of N. V. Efimov (Abrau-Durso, 2004), 34-35 (in Russian)
[25]	Kostin A. V., “On Asymptotic Nets on Pseudospheres”, #Geometry Days in Novosibirsk - 2014, Abstracts of the Inter. Conf. dedicated to 85th anniversary of academian Yu. G. Reshetnyak, Sobolev Institute of Mathematics of Siberian Branch of the RAS, Novosibirsk, 2014, 41 (in Russian)
[26]	Kostin A. V., Kostina N. N., “On the Interpretation of Asymptotic Directions”, #Proceedings International Youth School-Seminar «Modern Geometry and its Applications», International Scientific Conference «Modern Geometry and its Applications», Kazan Univ. Publ., Kazan, 2017, 75-76 (in Russian)