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On the problem of classification of regular 4-webs formed by pencils of spheres

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Abstract

Shelehov’s theorem on bondary curves of a regular curvilinear 3-web is generalized to the case of an arbitrary regular codimension 1 (n + 1)-web; an example is given of a regular 4-web formed by pencils of spheres in the three-dimensional conformal space (W. Blaschke’s problem); it is proved that a spherical 4-web of the basic type cannot be regular.

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References

  1. W. Blaschke, Einfürung in die Geometrie der Waben (Birkhäuser, Basel-Stuttgart, 1955; GIFML, Moscow, 1959)

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  2. A. M. Shelekhov, “On Three-webs Formed by Pencils of Circles,” in Modern Mathematics and its Applications (VINITI, Moscow, 2005), Itogi Nauki i Tekhn. Sovremennaya Matematika i eyo Plilozheniya 32, pp. 7–28.

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  3. V. B. Lazareva, “Parallelizable Three-Webs Formed by Pencils of Circles,” in Webs and Quasigroups (Kalinin, Kalinin University, 1988), pp. 74–77.

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  4. V. V. Goldberg, Theory of Multicodimensional (n + 1)-Webs (Kluwer Academic Publishers, Dordrecht-Boston-London, 1988).

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Authors and Affiliations

  1. Tver State University, ul. Zhelyabova 33, Tver, 170100, Russia

    V. B. Lazareva & A. M. Shelekhov

Authors
  1. V. B. Lazareva
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  2. A. M. Shelekhov
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Correspondence to A. M. Shelekhov.

Additional information

Original Russian Text © V.B. Lazareva, A.M. Shelekhov, 2007, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2007, No. 12, pp. 70–76.

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Lazareva, V.B., Shelekhov, A.M. On the problem of classification of regular 4-webs formed by pencils of spheres. Russ Math. 51, 71–77 (2007). https://doi.org/10.3103/S1066369X07120055

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  • DOI: https://doi.org/10.3103/S1066369X07120055

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