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On the geometric interpretation of solutions of a system generalizing the Sine-Gordon equation

Abstract

We propose a geometric interpretation of solutions of the system generalizing the well-known sine-Gordon equation. We prove that to any solution of the Efimov-Poznyak system in a simply-connected domain, a C 3-smooth singular surface with given first fundamental bilinear form corresponds.

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References

  1. 1.

    N. V. Efimov and E. G. Poznyak, “Some transformations of the main equations of surface theory,” Dokl. Akad. Nauk SSSR, 137, No. 1, 25–27 (1961).

    MathSciNet  Google Scholar 

  2. 2.

    N. V. Efimov and E. G. Poznyak, “Surfaces with slowly varying negative curvature,” Usp. Mat. Nauk, 21, No. 5, 3–58 (1966).

    Google Scholar 

  3. 3.

    V. A. Il’in and E. G. Poznyak, Linear Algebra [in Russian], Nauka, Moscow (1984).

    MATH  Google Scholar 

  4. 4.

    M. M. Postnikov, Riemannian Geometry, Springer-Verlag (2002).

  5. 5.

    E. G. Poznyak, “Geometric interpretation of regular solutions of the equation Z xy = sin Z,” Differ. Equations, 15, No. 7, 1332–1336 (1979).

    MathSciNet  Google Scholar 

  6. 6.

    P. K. Rashevskii, Riemannian Geometry and Tensor Analysis [in Russian], Moscow (1953).

  7. 7.

    E. R. Rozendorn, “Main equations of surface theory in asymptotic coordinates,” Mat. Sb., 70, No. 4, 490–507 (1966).

    MathSciNet  Google Scholar 

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Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.

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Bad’in, A.V. On the geometric interpretation of solutions of a system generalizing the Sine-Gordon equation. J Math Sci 141, 970–1003 (2007). https://doi.org/10.1007/s10958-007-0026-4

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Keywords

  • Manifold
  • Bilinear Form
  • Geometric Interpretation
  • Geometric Object
  • Isometric Immersion