E-symplectic and almost regular Poisson manifolds

When considering manifolds with boundary it is common to only consider vector fields tangent to the boundary. This set of vector fields is called the b-foliation and it coincides with sections of a vector bundle B. This choice of vector fields allows us to consider smooth sections on E*, which do not correspond to smooth forms but they give smooth functions when evaluated on elements of the b-foliation. A well studied class of singular symplectic manifolds are b-symplectic manifolds which are given by a symplectic B form, i.e. a non degenerate closed section on B*^B*. In this talk we will compare 2 objects. On the one hand, we will not restrict ourselves to study the b-foliation case, we will consider any set of vector fields described as sections of some vector bundle E, and symplectic forms on E, E-symplectic manifolds. On the other hand, we will consider a similar object, Poisson manifolds whose symplectic foliation is also controlled by a vector bundle E, almost regular Poisson manifolds. These two are surprisingly not the same object but they are related in a natural way.