The geometry of a bi-Lagrangian manifold

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian distributions). We show that many different geometric structures can be attached to these manifolds and we carefully analyse the associated connections. Moreover, we introduce the problem of the intersection of two leaves, one of each foliation, through a point and show a lot of significative examples.Comment: 30 page

The dimension function of holomorphic spaces of a real submanifold of an almost complex manifold

summary:Let $M$ be a real submanifold of an almost complex manifold $(\overline{M},\overline{J})$ and let $H_{x}=T_{x}M\cap \overline{J}(T_{x}M)$ be the maximal holomorphic subspace, for each $x\in M$. We prove that $c\:M\rightarrow \mathbb{N}$, $c(x)=\dim _{\mathbb{R}} H_{x}$ is upper-semicontinuous