Geodesic Webs on a Two-Dimensional Manifold and Euler Equations

2008/10/30 by Vladislav V. Goldberg, Goldberg, Vladislav V., Valentin Lychagin +2
Mathematics · Physics and Astronomy · #53A60 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #math.DG #msc:53A60

paper · pdf · doi:10.48550/arxiv.0810.5392

15 pages

arxiv created 2008/10/30 · openalex publication_date 2008/10/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Geodesic Webs of Hypersurfaces
• Geodesic Webs and PDE Systems of Euler Equations
• Webs and projective structures on a plane
• Projective invariants of linear 3-webs and Gronwall's Conjecture
• Paraconformal structures, ordinary differential equations and totally geodesic manifolds