Skip to main content
SpringerLink
Log in

A-deformations of a surface with stationary lengths of LGT-lines

  • Published:
Ukrainian Mathematical Journal Aims and scope

We investigate infinitesimal areal deformations (A-deformations) of the first order under which the lengths of LGT-lines of a surface are preserved in the E 3 -space. We prove that any regular surface of the class C 4 of nonzero Gaussian curvature without umbilical points admits nontrivial A-deformations with stationary lengths of LGT-lines.

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. N. Vekua, Generalized Analytic Functions [in Russian], Nauka, Moscow (1988).

    MATH  Google Scholar 

  2. P. Vincensini, “Sur les déformations équivalentes infinitésimales des surfaces,” Rev. Univ. Nac. Tucumán A, 14, No. 2, 177–188 (1962).

    MathSciNet  Google Scholar 

  3. A. D. Aleksandrov, Convex Polyhedrons [in Russian], Gostekhteorizdat, Moscow (1950).

    Google Scholar 

  4. V. T. Fomenko, “Some results on the theory of infinitesimal bending of surfaces,” Mat. Sb., 72 (114), No. 3, 388–411 (1967).

  5. P. G. Kolobov, “On infinitesimal deformations of a surface with preservation of the area,” Uch. Zap. Kabardino-Balkar. Univ., Ser. Mat., Issue 30, 65–68 (1966).

  6. L. L. Bezkorovaina, “On infinitesimal deformations that preserve the lengths of asymptotic lines,” in: Proceedings of the Scientific Conference of Young Scientists (Natural Sciences) [in Ukrainian], Odessa (1970), pp. 104–109.

  7. L. L. Bezkorovainaya, “Canonical A-deformations that preserve the lengths of curvature lines of a surface,” Mat. Sb., 97 (139), No. 2 (6), 163–176 (1975).

  8. N. V. Dermanets, On Extension of Infinitesimal First-Order Areal Deformations of a Surface of Positive Curvature with Edge to Analytic Ones [in Russian], Dep. UkrNIINTI, No. 813 Uk-85, Odessa (1985).

  9. L. L. Bezkorovaina, Infinitesimal Areal Deformations and Steady States of an Elastic Shell [in Ukrainian], Astroprint, Odessa (1999).

    Google Scholar 

  10. T. Yu. Vashpanova and L. L. Bezkorovaina, “Geodesic torsion and its extremal values,” in: Proceedings of the Mathematical Scientific Conference of Young Scientists and Students on Partial Differential Equations and Their Applications (Donetsk, December 6–7, 2006) [in Ukrainian], Donetsk (2006), pp. 28–29.

  11. A. V. Bitsadze, Boundary-Value Problems for Elliptic Equations of the Second Order [in Russian], Nauka, Moscow (1966).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechnikov Odessa National University, Odessa, Ukraine

    L. L. Bezkorovaina & T. Yu. Vashpanova

Authors
  1. L. L. Bezkorovaina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Yu. Vashpanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 7, pp. 878–884, July, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bezkorovaina, L.L., Vashpanova, T.Y. A-deformations of a surface with stationary lengths of LGT-lines. Ukr Math J 62, 1018–1027 (2010). https://doi.org/10.1007/s11253-010-0410-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-010-0410-y

Keywords

Search

Navigation