Lifting Symplectomorphism Group Actions on Bi-Lagrangian Structures to the Whitney Sum

Let $M$ be a manifold endowed with a bi-Lagrangian structure $(\omega ,{\mathcal{F}}_1,{\mathcal{F}}_2)$. Thus, $\omega$ is a symplectic form, and $({\mathcal{F}}_1,{\mathcal{F}}_2)$ is a pair of transverse Lagrangian foliations on the symplectic manifold $(M,\omega )$. We prove that, if $M$ is parallelizable, then every bi-Lagrangian structure on $M$ naturally induces bi-Lagrangian structures on the tangent bundle $TM$ and on the cotangent bundle $T^{\ast }M$, and hence on the Whitney sum $W=TM\oplus T^{\ast }M$. We show that, if the bi-Lagrangian structures of $M$ can be lifted to $TM$ or $T^{\ast }M$, then the action of the symplectomorphism group on the set of bi-Lagrangian structures defined in \cite{TNB} admits natural lifts to $TM$, $T^{\ast }M$, and hence to $W=TM\oplus T^{\ast }M$.