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

Abstract

We consider a smooth \begin{document}$ 2n $\end{document}-manifold \begin{document}$ M $\end{document} endowed with a bi-Lagrangian structure \begin{document}$ (\omega,\mathcal{F}_{1},\mathcal{F}_{2}) $\end{document}. That is, \begin{document}$ \omega $\end{document} is a symplectic form and \begin{document}$ (\mathcal{F}_{1},\mathcal{F}_{2}) $\end{document} is a pair of transversal Lagrangian foliations on \begin{document}$ (M, \omega) $\end{document}. 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 \begin{document}$ M $\end{document} can be lifted as a bi-Lagrangian structure on its trivial bundle \begin{document}$ M\times\mathbb{R}^n $\end{document}. Moreover, the lifting of an affine bi-Lagrangian structure is also affine. We define a dynamic on the symplectomorphism group and the set of bi-Lagrangian structures (that is an action of the symplectomorphism group on the set of bi-Lagrangian structures). This dynamic is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted on \begin{document}$ M\times\mathbb{R}^n $\end{document}. This lifting can be lifted again on \begin{document}$ \left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n} $\end{document}, and coincides with the initial dynamic (in our sense) on \begin{document}$ M\times\mathbb{R}^n $\end{document}. By replacing \begin{document}$ M\times\mathbb{R}^{2n} $\end{document} with the tangent bundle \begin{document}$ TM $\end{document} or cotangent bundle \begin{document}$ T^{*}M $\end{document} of \begin{document}$ M $\end{document}, results still hold when \begin{document}$ M $\end{document} is parallelizable.