Infinite Lifting of an Action of Symplectomorphism Group on the set of Bi-Lagrangian Structures

Abstract

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 structures have an important geometric object called the Hess Connection. Among the many importance of these 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 an affine bi-Lagrangian structure. 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 $M\times\mathbb{R}^n$ for some bi-Lagrangian structures. Results still hold by replacing $M\times\mathbb{R}^{2n}$ with the tangent bundle $TM$ of $M$ or its cotangent bundle $T^{*}M$ for some manifolds $M$.