Bi-Lagrangian structures and the space of rays

Abstract

This paper focuses on local curvature invariants associated with bi-Lagrangian structures. We establish several geometric conditions that determine when the canonical connection is flat, building on our previous findings regarding divergence-free webs (Domitrz and Zubilewicz 2023 Anal. Math. Phys. 13 4). Addressing questions raised by Tabachnikov (1993 Differ. Geom. Appl. 3 265–84), we provide complete solutions to two problems: the existence of flat bi-Lagrangian structures within the space of rays induced by a pair of hypersurfaces, and the existence of flat bi-Lagrangian structures induced by tangents to Lagrangian curves in the symplectic plane.