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.

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Correspondence to A. M. Shelekhov.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 148, Proceedings of the International Conference “Geometric Methods in Control Theory and Mathematical Physics: Differential Equations, Integrability, and Qualitative Theory” (Ryazan, September 15–18, 2016), 2018.

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Shelekhov, A.M. Minimal Projectivity Condition for a Smooth Mapping and the Gronwall Problem. J Math Sci 248, 484–496 (2020). https://doi.org/10.1007/s10958-020-04889-9

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Keywords and phrases

  • three-web
  • rectilinear three-web
  • Grassmannian three-web
  • Gronwall problem

AMS Subject Classification

  • 53A60