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Abelian Equations and Rank Problems for Planar Webs

Vladislav V. Goldberg·Valentin V. Lychagin
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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's 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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Vladislav V. Goldberg, Valentin V. Lychagin. 2006-05-04. Abelian Equations and Rank Problems for Planar Webs. https://arxiv.org/abs/math/0605124

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