Infinite lifting of an action of symplectomorphism group on the set of bi-Lagrangian structures

We consider a smooth $ 2n $-manifold M endowed with a bi-Lagrangian structure (ω,F₁,F₂). That is, ω is a symplectic form and (F₁,F₂) is a pair of transversal Lagrangian foliations on (M, ω). Such a structure has an important geometric object called the Hess Connection. Among the many importance of Hess connections, they allow to classify affine bi-Lagrangian structures.In this work, we show that a bi-Lagrangian structure on M can be lifted as a bi-Lagrangian structure on its trivial bundle Mⁿ. Moreover, the lifting of an affine bi-Lagrangian structure is also affine. We define a dynamic on the symplectomorphism group and the set of bi-Lagrangian structures (that is an action of the symplectomorphism group on the set of bi-Lagrangian structures). This dynamic is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted on Mⁿ. This lifting can be lifted again on (M²ⁿ)⁴ⁿ, and coincides with the initial dynamic (in our sense) on Mⁿ. By replacing M²ⁿ with the tangent bundle $ TM $ or cotangent bundle T*M of M, results still hold when M is parallelizable.