On a problem of Chern-Akivis-Shelekhov on hexagonal three-webs.
We investigate real analytic Levi-flat hypersurfaces tangent to holomorphic webs. We introduce the notion of first integrals for local webs. In particular, we prove that a -web with finitely many invariant subvarieties through the origin tangent to a Levi-flat hypersurface has a holomorphic first integral.
Plane -webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two -webs, are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable -webs, however, there always exists a nonmeasurable conjugacy; still,...
A 3-web on a smooth -dimensional manifold can be regarded locally as a triple of integrable -distributions which are pairwise complementary, [5]; that is, we can work on the tangent bundle only. This approach enables us to describe a -web and its properties by invariant -tensor fields and where is a projector and id. The canonical Chern connection of a web-manifold can be introduced using this tensor fields, [1]. Our aim is to express the torsion tensor of the Chern connection through...