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

Abstract

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 $(M, \omega)$. A bi-Lagrangian structure is said to be \textbf{affine} if the associated linear connection is curvature-free. We prove that, if $M$ is parallelizable, then every bi-Lagrangian structure on $M$ naturally induces two bi-Lagrangian structures on the tangent bundle $TM$ and on the cotangent bundle $T^*M$, and hence on the Whitney sum $W = TM \oplus T^*M$. The first way to lift a bi-Lagrangian structure yields an affine bi-Lagrangian structure. For the second way, we prove that the lifted bi-Lagrangian structure is affine if and only if the initial one is affine. We also show that, if the bi-Lagrangian structures on $M$ can be lifted to $TM$ or $T^*M$, then the action of the symplectomorphism group on the set of bi-Lagrangian structures defined in \cite{TNB} admits natural lifts to $TM$, $T^*M$, and hence to $W = TM \oplus T^*M$.