SpringerLink
Log in

4-Webs in the Plane and Their Linearizability

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

We investigate the linearizability problem for different classes of 4-webs in the plane. In particular, we apply the linearizability conditions, recently found by Akivis, Goldberg and Lychagin, to confirm that a 4-web MW (Mayrhofer's web) with equal curvature forms of its 3-subwebs and a nonconstant basic invariant is always linearizable (this result was first obtained by Mayrhofer in 1928; it also follows from the papers of Nakai). Using the same conditions, we further prove that such a 4-web with a constant basic invariant (Nakai's web) is linearizable if and only if it is parallelizable. Next we study four classes of the so-called almost parallelizable 4-webs APW a ,a=1,2,3,4 (for them the curvature K=0 and the basic invariant is constant on the leaves of the web foliation X a ), and prove that a 4-web APW a is linearizable if and only if it coincides with a 4-web MW a of the corresponding special class of 4-webs MW. The existence theorems are proved for all the classes of 4-webs considered in the paper.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Akivis, M. A. and Goldberg, V. V.: Conformal Differential Geometry and Its Generalizations, Wiley, New York, 1996 (MR 98a:53023; Zbl. 863.53002).

    Google Scholar 

  2. Akivis, M. A., Goldberg, V. V. and Lychagin, V. V.: Linearizability of d-webs, d _4, on 2-dimensional manifolds, Preprint, 2002, ar**v: math.DG/0209290.

  3. Blaschke, W.: Einführung in die Geometrie der Waben, Birkhäuser, Basel, 1955 (MR 17, 780; Zbl. 68, 365).

    Google Scholar 

  4. Blaschke, W. and Bol, G.: Geometrie der Gewebe, Springer-Verlag, Berlin, 1938 (MR 6, p. 19; Zbl. 20, p. 67).

    Google Scholar 

  5. Bryant, R. L., Chern, S. S., Gardner, R. B., Goldsmith, H. L. and Griffiths, P. A.: Exterior Differential Systems, Springer-Verlag, New York, 1991 (MR 92h:58007; Zbl. 726.58002).

    Google Scholar 

  6. Cartan, É.: Sur les variétés à connexion affine et la théorie de la relativité généralisée, Ann. École Norm. 40 (1923), 325–412 (JFM 50, p. 542); OEuvres complètes: Partie III, Divers, gèométrie, différentielle, vol. 3, Gauthier-Villars, Paris, 1955, pp. 659-746 (MR 17, 697; Zbl. 15, 83).

    Google Scholar 

  7. Goldberg, V. V.: On the theory of four-webs of multidimensional surfaces on a differ-entiable manifold X 2, Izv. Vyssh. Uchebn. Zaved. Mat. 1977(11), 15–22 (Russian), English transl., Soviet Math. (Iz. VUZ) 21(11) (1977), 97-100 (MR 58 #30859; Zbl. 398.53009 & 453.53010).

    Google Scholar 

  8. Goldberg, V. V.: On the theory of four-webs of multidimensional surfaces on a dif-ferentiable manifold X 2, Serdica 6(2) (1980), 105–119 (Russian) (MR 82f:53023; Zbl. 453.53010).

    Google Scholar 

  9. Goldberg, V. V.: Theory of Multicodimensional (n +1) -Webs, Kluwer Acad. Publ., Dordrecht, 1988 (MR 89h:53021; Zbl. 668.53001).

    Google Scholar 

  10. Goldberg, V. V.: Curvilinear 4-webs with equal curvature forms of its 3-subwebs, In: Webs and Quasigroups, Tver State Univ., Tver, 1993, pp. 9–19 (MR 94g:53012; Zbl. 792.53010).

    Google Scholar 

  11. Goldberg, V. V.: Special classes of curvilinear 4-webs with equal curvature forms of its 3-subwebs, In: Webs and Quasigroups, Tver State Univ., Tver, 1996–1997, pp. 24-39 (MR 2001f:53037; Zbl. 903.53013).

    Google Scholar 

  12. Hènaut, A.: Caractérisation des tissus de C 2 dont le rang est maximal et qui sont linéarisables, Compositio Math. 94(3) (1994), 247–268 (MR 96a:32057; Zbl. 877.53013).

    Google Scholar 

  13. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry,vol.1,Wiley-Interscience, New York, 1963 (MR 27 2945; Zbl. 119, p. 375).

    Google Scholar 

  14. Mayrhofer, K.: Kurvensysteme auf Flächen, Math. Z. 28 (1928), 728–752 (JFM 54, p. 745).

    Google Scholar 

  15. Mayrhofer, K.: Ñber Sechsensysteme, Abh. Math. Sem. Univ. Hamburg 7 (1929), 1–10 (JFM 55, p. 1013).

    Google Scholar 

  16. Nakai, I.: Curvature of curvilinear 4-webs and pencils of one forms, In: Topology of holomorphic dynamical systems and related topics (Kyoto, 1995), S¯urikaisekikenky¯ usho K¯ oky¯ uroku 955 (1996), no. 8, 109–132 (MR 98a:53026).

    Google Scholar 

  17. Nakai, I.: Curvature of curvilinear 4-webs and pencils of one forms: Variation on the theme a theorem of Poincaré, Mayrhofer and Reidemeister on curvilinear 4-webs, Comment. Math. Helv. 73 (1998), 177–205 (MR 99c:53013; Zbl. 926.53011).

    Google Scholar 

  18. Reidemeister, K.: Gewebe und Gruppen, Math. Z. 29 (1928), 427–435 (JFM 54, p. 746).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ, 07102, U.S.A.

    Vladislav V. Goldberg

Authors
  1. Vladislav V. Goldberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldberg, V.V. 4-Webs in the Plane and Their Linearizability. Acta Applicandae Mathematicae 80, 35–55 (2004). https://doi.org/10.1023/B:ACAP.0000013251.38211.88

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:ACAP.0000013251.38211.88

Search

Navigation