Minimal Projectivity Condition for a Smooth Mapping and the Gronwall Problem

In this paper, the following assertion is proved: let $GW$ and $\widetilde{GW}$ be the Grassmannian three-webs defined respectively in domains $D$ and $\tilde D$ of the Grassmannian manifold of straight lines of the projective space $P^{r+1}$; $\Phi: D\rightarrow \tilde D$ be a local diffeomorphism that maps foliations of the web $GW$ to foliations of the web $\widetilde{GW}$. Then $\Phi$ 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 the 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 $\widetilde{LW}$ 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 $\theta$ and $\tilde{\theta}$ are local diffeomorphisms that map the three-web $W$ to some rectilinear three-webs, then $\tilde{\theta}=\pi \circ \theta$, where $\pi$ is a projective transformation.