Geodesic Webs of Hypersurfaces

TL;DR

This paper explores the geometric structures of hypersurface webs, establishing unique projective and affine structures associated with geodesic webs, and applies these to solve classical web theory problems.

Contribution

It introduces a method to associate unique projective and affine structures with geodesic hypersurface webs, enabling new insights into their invariants and classical problems.

Findings

Existence of a unique projective structure for any geodesic (n+2)-web.

Existence of a unique affine structure when a web foliation is pointed.

Application to web linearization and Gronwall theorem.

Abstract

In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure 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.