Geodesic Webs on a Two-Dimensional Manifold and Euler Equations

TL;DR

This paper establishes a unique projective structure associated with any planar 4-web, characterizes conditions for symmetric affine connections, and extends results to higher webs with specific invariants, linking web geometry to Euler equations.

Contribution

It introduces a novel connection between planar webs and projective structures, providing conditions for geodesic webs and symmetric affine connections, extending to higher webs with invariant constraints.

Findings

Unique projective structure for any planar 4-web

Conditions for a web to be geodesic on an affine symmetric surface

Extension of results to higher webs with vanishing invariants

Abstract

We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d > 4, provided that additional d-4 second-order invariants vanish.