Skip to main content
SpringerLink
Log in

Classification of regular circle three-webs up to circular transformations

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

Abstract

A curvilinear three-web formed by three pencils of circles is called a circle web. Generally speaking, the circle three-web is not regular, i.e., it is not locally diffeomorphic to a web formed by three families of parallel straight lines. In this paper, all regular circle three-webs are classified up to circular transformations. The main result is as follows: There exist 48 nonequivalent (with respect to circular transformations) types of regular three-webs. Five of them contain ∞3 nonequivalent webs each, 11 types contain ∞2 nonequivalent webs each, and 12 types contain ∞1 nonequivalent webs each; 5 webs admit a one-parameter group of automorphisms.

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. R. S. Balabanova, “Hexagonal three-webs of circle pencils two of which are conjugated,” Nauch. Tr. Plovdiv. Univ., Mat., 11, No. 4, 128–141 (1973).

    MathSciNet  Google Scholar 

  2. W. Blaschke, Einfürung in die Geometrie der Waben, Birkhäuser, Basel (1955).

    Google Scholar 

  3. H. I. Erdogan, Düzlemde 6-gen doku teşkil eden čember demety 3-üzleri, Ph.D. Thesis, Istanbul Teknik Ueniversitesi (1974).

  4. H. I. Erdogan, “Triples of circle-pencils forming a hexagonal three-web in E 2,” J. Geom., 35, Nos. 1–2, 39–65 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  5. V. B. Lazareva, “Three-webs formed by circle families in a plane,” in: Differential Geometry [in Russian], Kalinin State Univ., Kalinin (1977), pp. 49–64.

  6. V. B. Lazareva, “Three-webs on a two-dimensional surface in a triaxial space,” Differential Geometry of Manifolds of Figures, No. 10, 54–59 (1979).

  7. V. B. Lazareva, “Parallelizable three-webs formed by pencils of circles,” in: Webs and Quasigroups [in Russian], Kalinin State Univ., Kalinin (1988), pp. 74–77.

  8. V. B. Lazareva and O. V. Orlova, “On one class of hexagonal three-webs formed by circle pencils,” in: Webs and Quasigroups [in Russian], Kalinin State Univ., Kalinin (1986), pp. 115–119.

  9. V. B. Lazareva and A. M. Shelekhov, “Around a Blaschke problem in the web theory,” in: Webs and Quasigroups, 1996–1997, Tver State Univ., Tver (1997), pp. 65–73.

  10. V. B. Lazareva and A. M. Shelekhov, “Configurations and webs generated by sphere pencils,” Vestn. Chuvash. Gos. Ped. Univ., 87–95 (2006).

  11. V. B. Lazareva and A. M. Shelekhov, “On triangulations of a plane by pencils of conics,” Mat. Sb., 198, No. 11, 107–134 (2007).

    MathSciNet  Google Scholar 

  12. V. B. Lazareva and A. M. Shelekhov, “To the problem of classification of regular 4-webs formed by sphere pencils,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 12, p. 70–76 (2007).

  13. V. B. Lazareva and A. M. Shelekhov, “On triangulations of a plane by pencils of second-order curves,” Deposited at VINITI (2009).

  14. A. M. Shelekhov, “On three-webs formed by circle pencils,” Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh., 32, 7–28 (2005).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tver State University, Tver, Russia

    V. B. Lazareva

Authors
  1. V. B. Lazareva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. B. Lazareva.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 1, pp. 95–107, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lazareva, V.B. Classification of regular circle three-webs up to circular transformations. J Math Sci 177, 579–588 (2011). https://doi.org/10.1007/s10958-011-0483-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-011-0483-7

Keywords

Search

Navigation