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Abelian equations and rank problems for planar webs

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Abstract

We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré theorem: A planar 4-web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem “to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3.” Finally, we find invariant characterization of planar 5-webs of maximum rank and prove than in general such webs are not linearizable.

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Authors and Affiliations

  1. New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA

    V. V. Goldberg

  2. University of Tromsø, N-9037, Tromsø, Norway

    V. V. Lychagin

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  1. V. V. Goldberg
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  2. V. V. Lychagin
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Correspondence to V. V. Goldberg.

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The text was submitted by the authors in English.

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Goldberg, V.V., Lychagin, V.V. Abelian equations and rank problems for planar webs. Russ Math. 51, 39–75 (2007). https://doi.org/10.3103/S1066369X07100039

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