Minimal Projectivity Condition for a Smooth Mapping and the Gronwall Problem

Abstract

In this paper, the following assertion is proved: Let GW and $$ \tilde{GW} $$ be Grassmannian three-webs defined respectively in domains D and $$ \tilde{D} $$ of a Grassmannian manifold of straight lines of the projective space Pr+1; Φ : D → $$ \tilde{D} $$ be a local diffeomorphism that maps foliations of the web GW to foliations of the web $$ \tilde{GW} $$. Then Φ maps bundles of lines to bundles of lines, i.e., induces a point transformation, which is a projective transformation. In the case where r = 1, the proof is much more complicated than in the multidimensional case. In the case where r = 1, the dual theorem is formulated as follows: Let LW be a rectilinear three-web on a plane, i.e., three families of lines in the general position, and let this web be not regular, i.e., not locally diffeomorphic to a three-web formed by three families of parallel straight lines. Then each local diffeomorphism that maps a three-web LW to another rectilinear three-web $$ \tilde{LW} $$ is a projective transformation. As a consequence, we obtain the positive solution of the Gronwall problem (Gronwall, 1912): If W is a linearizable irregular three-web and θ and $$ \tilde{\theta} $$ are local diffeomorphisms that map the three-web W to some rectilinear three-webs, then $$ \tilde{\theta} $$ = π ° θ, where π is a projective transformation.