Skip to main content
SpringerLink
Log in

Non-euclidean geometry: The Gauss formula and an interpretation of partial differential equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature Cited

  1. A. Artikbaev and D. D. Sokolov,Geometry in the Large in the Space-Time Plane [in Russian], Fan, Tashkent (1991).

  2. A. Barone and G. Paterno,The Josephson Effect [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  3. I. N. Vekua, “Remarks on the properties of solution of the equation Δ2=-2KE u,”Sib. Mat. Zh.,1, No. 3, 331–342 (1960).

    MATH  MathSciNet  Google Scholar 

  4. F. Galogero and A. Degasperis,Spectral Transform and Solitons [Russian translation], Mir, Moscow (1985).

    Google Scholar 

  5. G. L. Lamb,An Introduction to Soliton Theory [Russian translation], Mir, Moscow (1983).

    Google Scholar 

  6. V. S. Malakhovskii,An Introduction to the Theory of Exterior Forms [in Russian], Kaliningrad University Press, Kaliningrad (1980).

    Google Scholar 

  7. É. G. Posnyak and A. G. Popov, “The geometry of the sine-Gordon equation,” In:Problemy Geometrii, Vol. 23,Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1991), pp. 99–130.

    Google Scholar 

  8. É. G. Poznyak and A. G. Popov,The sine-Gordon Equation: Geometry and Physics [in Russian], Znanie, Moscow (1991).

    Google Scholar 

  9. É. G. Poznyak and A. G. Popov, “Lobachevski geometry and equations of mathematical physics,”Dokl. Akad. Nauk (Rossiisk.),332, No. 4, 41–421 (1993).

    Google Scholar 

  10. É. G. Poznyak and E. V. Shikin,Differential Geometry [In Russian], Mosk. Gos. Univ., Moscow (1990).

    Google Scholar 

  11. A. G. Popov, “The Chebyshev net as a geometrical invariant for evolutional equations,” In:All-Union Colloquium of Young Scientists on Differential Geometry Dedicated to the 80th Anniversary of the Birth of N. V. Efimov, Abstracts of Reports, Rostov-on-Don (1990), p. 25.

  12. A. G. Popov, “A geometrical approach to the interpretation of solutions of the sine-Gordon equation,”Dokl. Akad. Nauk SSSR,312, No. 5, 1109–1111 (1990).

    Google Scholar 

  13. A. G. Popov, “The exact formulas for the construction of solutions of the Liouville equation Δ2υ=e u by solutions of the Laplace equation Δ2υ=0,”Dokl. Akad. Nauk (Rossiisk.),333, No. 4 (1993).

    Google Scholar 

  14. I. Kh. Sabitov, “A list of the central-symmetric forms of two-dimensional metrics of a constant curvature,” In:International Conf. “Lobachevski and Modern Geometry,” Abstracts of Reports, Part I, Kazan' (1992), p. 87.

  15. G. Favar,A Course of Local Differential Geometry [Russian translation], IL, Moscow (1960).

    Google Scholar 

  16. R. Beals, M. Rabelo, and K. Tenenblat, “Bäcklund transformations and inverse scattering for some pseudospherical surface equations,”Stud. Appl. Math.,81, 125–151 (1989).

    MathSciNet  Google Scholar 

  17. F. Galogero and J. Xiaoda, “C-integrable partial differential equations. I,”J. Math. Phys.,32, No. 4, 875–887 (1991).

    MathSciNet  Google Scholar 

  18. F. Galogero and J. Xiaoda, “C-integrable PDEs. II,”J. Math. Phys.,32, No. 10, 2703–2717 (1991).

    MathSciNet  Google Scholar 

  19. S. S. Chern and K. Tenenblat, “Pseudospherical surfaces and evolutional equations,”Stud. Appl. Math.,74, No. 1, 55–83 (1986).

    MathSciNet  Google Scholar 

  20. M. Crampin and D. J. Saunders, “The sine-Gordon equations, Chebyshev nets, and harmonic maps,”Repts. Math. Phys.,23, No. 3, 327–340 (1986).

    MathSciNet  Google Scholar 

  21. N. Kamran and K. Tenenblat, “On differential equations describing pseudo-spherical surfaces,”J. Differ. Equat. (in press).

  22. H. Poincare, “Les fonctions Fuchsiennes et l'equation Δu=e u,”J. Math.,4, No. 5, 137–230 (1898).

    MATH  Google Scholar 

  23. M. Rabelo, “On evolution equations which describe a pseudospherical surfaces,”Stud. Appl. Math.,81, 221–248 (1989).

    MATH  MathSciNet  Google Scholar 

  24. A. Sanchez and L. Vasquez, “Nonlinear wave propagation in disordered media,”Int. J. Mod. Phys.,B5, No. 18, 2825–2882 (1991).

    Google Scholar 

  25. R. Sasaki, “Geometrization of soliton equations,”Phys. Lett. A.,71, 390–392 (1979).

    Article  MathSciNet  Google Scholar 

  26. R. Sasaki, “Soliton equations and pseudospherical surfaces,”Nucl. Phys.,154, 343–357 (1979).

    Article  Google Scholar 

  27. A. Sym, “Soliton surfaces,”Lett. Nuovo Cim.,33, No. 12, 394–400 (1982).

    MathSciNet  Google Scholar 

  28. P. Chebyshev, “Sur la couple des vetements,” In:Assos. Fran., Geuvres II (1878).

Download references

Authors
  1. É. G. Poznyak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. G. Popov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 11, Geometry-2, 1994.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Poznyak, É.G., Popov, A.G. Non-euclidean geometry: The Gauss formula and an interpretation of partial differential equations. J Math Sci 78, 241–252 (1996). https://doi.org/10.1007/BF02365190

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02365190

Keywords

Search

Navigation