Abelian Equations and Rank Problems for Planar Webs

Details Abelian Equations and Rank Problems for Planar Webs Abelian Equations and Rank Problems for Planar Webs

2006-05-04

arxiv.org/abs/math/0605124v1

Mathematics

Differential Geometry

43 pages

Scientific paper

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 Poincar\'{e}'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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