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Asymptotic Lines on Pseudospheres and the Angle of Parallelism

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Abstract

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.

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ACKNOWLEDGMENTS

The author expresses his gratitude to the anonymous reviewer for his careful reading of the manuscript and his fruitful remarks.

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  1. Elabuga Institute of Kazan Federal University, 89 Kazanskaya str., 423600, Elabuga, Russia

    A. V. Kostin

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  1. A. V. Kostin
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Correspondence to A. V. Kostin.

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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 6, pp. 25–34.

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Kostin, A.V. Asymptotic Lines on Pseudospheres and the Angle of Parallelism. Russ Math. 65, 21–28 (2021). https://doi.org/10.3103/S1066369X21060037

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