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

Abstract

$(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$, where $\omega$ is symplectic and $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian foliations. Such structures admit a natural geometric object, the Hess connection, which is central to the classification of affine bi-Lagrangian structures. We show that any bi-Lagrangian structure on $M$ lifts to a bi-Lagrangian structure on the trivial bundle $M\times\mathbb{R}^{2n}$. Moreover, the lift of an affine bi-Lagrangian structure is again affine. We then define a dynamical system in terms of an action of the symplectomorphism group on the space of bi-Lagrangian structures. This dynamics is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted to $M\times\mathbb{R}^{n}$, and iterated to $\left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n}$. The resulting lifted dynamics agrees with the initial one on $M\times\mathbb{R}^{n}$ for suitable structures. Finally, the same results hold when $M\times\mathbb{R}^{2n}$ is replaced by $TM$ or $T^{*}M$, provided that $M$ is parallelizable.