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 $ (\omega,\mathcal{F}_{1},\mathcal{F}_{2}) $. That is, $ \omega $ is a symplectic form and $ (\mathcal{F}_{1},\mathcal{F}_{2}) $ is a pair of transversal Lagrangian foliations on $ (M, \omega) $. 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 $ M\times\mathbb{R}^n $. 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 $ M\times\mathbb{R}^n $. This lifting can be lifted again on $ \left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n} $, and coincides with the initial dynamic (in our sense) on $ M\times\mathbb{R}^n $. By replacing $ M\times\mathbb{R}^{2n} $ with the tangent bundle $ TM $ or cotangent bundle $ T^{*}M $ of $ M $, results still hold when $ M $ is parallelizable.