Seminar talk, 18 February 2009

Speaker: Valentin Lychagin

Title: Geodesic webs of hypersurfaces

Joint work with Vladislav Goldberg, arXiv:0812.2126 (in Russian)

Absract:

Geometric structures associated with foliations (or webs) of hypersurfaces are studied. We show that with any geodesic $(n+2)$-web on $n$-dimensional manifold there is a unique projective structure, naturally associated with the web, and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.