The paper is devoted to study two problems: (1) Determine when the canonical connection of a bi-Lagrangian manifold is flat and (2) determine for which pairs of disjoint hypersurfaces of \(\mathbb{R}^{n}\) the bi-Lagrangian structure induced on the space of oriented affine lines of \(\mathbb{R}^{n}\) is flat.
A bi-Lagrangian manifold is a symplectic manifold endowed with two transversal Lagrangian foliations. It has a canonical symplectic connection, which is the unique torsionless connection parallelizing the symplectic structure and preserving the leaves of both foliations. As is well known, a (semi)-Riemmannian manifold such that all the sectional curvatures vanish is a flat manifold. A similar result in the bi-Lagrangian context is obtained in the present paper considering appropriate 2-dimensional bi-Lagrangian submanifolds.
For the second problem, remember that there exists an identification between the space of oriented affine lines (rays) in \(\mathbb{R}^{n}\) and the tangent bundle \(TS^{n-1}\) of the sphere \(S^{n-1}\subset \mathbb{R}^{n}\): the point \(p\in S^{n-1}\) of the sphere corresponds to the unit vector defining the direction of the oriented line, and the tangent vector \(q\in T_{p}S^{n-1}\) corresponds to the perpendicular vector from the origin of coordinates to the line. The authors obtain 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.
The paper is very well written.