Bi-Lagrangian and transverse Dirac structures induced on the Whitney sum

Abstract

Let $(M,\omega ,{\mathcal{F}}_1,{\mathcal{F}}_2)$ be a bi-Lagrangian manifold. For a distribution $\mathcal{D}\subseteq TM$, we characterize when the subbundle $N^{\ast t}\mathcal{D}:=\mathcal{D}\oplus N^{\ast }\mathcal{D}$ of the Whitney sum $W=TM\oplus T^{\ast }M$ is a Dirac structure for both the untwisted and $(H,\theta )$-twisted Courant brackets. Consequently, every bi-Lagrangian structure canonically determines a pair of transverse Dirac structures. the associated bi-Lagrangian connection, also called the Hess connection $\nabla$ induces a linear connection on $W$ with respect to which both Dirac subbundles are parallel, thereby establishing a natural link between bi-Lagrangian geometry and Courant Dirac geometry. We also construct induced bi-Lagrangian structures on $TM$, $T^{\ast }M$, and $W$, and study the relation between their affineness and that of the original structure. We also lift to these bundles the canonical action of the symplectomorphism group on bi-Lagrangian structures.

21 pages