Skip to main content
SpringerLink
Log in

Differential-geometric structures defining higher order contact transformations

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

The paper is devoted to the study of differential-geometric structures generated by Lie-Bäcklund transformations (or, what is the same, higher order contact transformations), which are a special case of diffeomorphisms between two manifolds of holonomic jets of sections. We study the structure of the fundamental object of a second order contact diffeomorphism (2-diffeomorphism). We also consider the case when a 2-diffeomorphism is given by explicit equations connecting local coordinates of 2-jet manifolds and establish conditions under which 2-diffeomorphisms defined by explicit equations are contact diffeomorphisms.

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

Similar content being viewed by others

References

  1. I. Kh. Ibragimov, Transformation Groups in Mathematical Physics (Nauka, Moscow, 1983).

    MATH  Google Scholar 

  2. G. F. Laptev, “Invariant Analytic Theory of Differentiable Maps,” in Oeuvres du Congrès International des Mathématiciens, Nice, 1970 (Nice, 1970), pp. 84–85.

  3. G. F. Laptev, “On Invariant Theory of Differentiable Maps,” in Trudy Geom. Semin. (VINITI, Moscow, 1974), 6, pp. 37–42.

    Google Scholar 

  4. A. K. Rybnikov, “Differential-Geometric Structures Defining Lie-Bäcklund Transformations,” Dokl. Phys. 425(1), 25–30 (2009).

    Google Scholar 

  5. L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, and A. P. Shirokov, “Differential-Geometric Structures on Manifolds,” in Itogi Nauki i Tekhn. Problemy Geometrii (VINITI, Moscow, 1979), 9, pp. 5–246.

    Google Scholar 

  6. G. F. Laptev, “Differential Geometry of Immersed Manifolds. Group-theoretical Method of Differential-Geometric Investigations,” Trudy Mosk. Matem. Ob-va, № 2, 275–382 (1953).

  7. G. F. Laptev, “Group-Theoretical Method of Differential-Geometric Investigations,” in Proceedings of 3-rd All-Union Math. Congress. Moscow, 1956 (Akad. Nauk SSSR, Moscow, 1958), 3, pp. 409–418.

    Google Scholar 

  8. G. F. Laptev, “The Basic Infinitesimal Structures of Higher Order on a Smooth Manifold,” in Trudy Geom. Semin. (VINITI, Moscow, 1966), 1, pp. 139–189.

    Google Scholar 

  9. G. F. Laptev, “The Structure Equations of a Principal Fiber Bundle,” in Trudy Geom. Semin. (VINITI, Moscow, 1969), 2, pp. 161–178.

    Google Scholar 

  10. A. M. Vasiliev, The Theory of Differential-Geometric Structures (Moscow University, Moscow, 1987).

    Google Scholar 

  11. L. D. Polyanin, V. F. Zaitsev, and A. I. Zhurov, Methods for Solving Nonlinear Equations of Mathematical Physics and Mechanics (Fizmatlit, Moscow, 2005).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Leninskie Gory 1, GSP-1, Moscow, 119991, Russia

    A. K. Rybnikov

Authors
  1. A. K. Rybnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. Rybnikov.

Additional information

Original Russian Text © A.K. Rybnikov, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 9, pp. 70–89.

About this article

Cite this article

Rybnikov, A.K. Differential-geometric structures defining higher order contact transformations. Russ Math. 55, 58–75 (2011). https://doi.org/10.3103/S1066369X11090088

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X11090088

Keywords and phrases

Search

Navigation