The geometry of $E$-manifolds

Abstract

Motivated by the study of symplectic Lie algebroids, we focus on a type of algebroid (called an $E$-tangent bundle) which is particularly well-suited to the study of singular differential forms and their cohomology. This setting generalizes $b$-symplectic manifolds, foliated manifolds, and a wide class of Poisson manifolds. We generalize Moser’s theorem to this setting, and use it to construct symplectomorphisms between singular symplectic forms. We give applications of this machinery (including the study of Poisson cohomology), and study specific examples of a few of them in depth.