2026/07/28 by Bertuel Tangue Ndawa, Ferdinand Ngakeu, Nasser Saipele Nansidi
#math.DS
Let M be a manifold endowed with a bi-Lagrangian structure (ω, F1, F2). Thus, ω is a symplectic form, and (F1, F2) is a pair of transverse Lagrangian foliations on the symplectic manifold (M, ω). A bi-Lagrangian structure is said to be affine if the associated linear connection is curvature-free. We prove that, if M is parallelizable, then every bi-Lagrangian structure on M naturally induces two bi-Lagrangian structures on the tangent bundle TM and on the cotangent bundle T^*M, and hence on the Whitney sum W = TM ⊕ T^*M. The first way to lift a bi-Lagrangian structure yields an affine bi-Lagrangian structure. For the second way, we prove that the lifted bi-Lagrangian structure is affine if and only if the initial one is affine. We also show that, if the bi-Lagrangian structures on M can be lifted to TM or T^*M, then the action of the symplectomorphism group on the set of bi-Lagrangian structures defined in \citeTNB admits natural lifts to TM, T^*M, and hence to W = TM ⊕ T^*M.