By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-K¨ahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.

We give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities to hold.

In this article we show how the classical theory of Reeb and others extends to the case of codimension one singular foliations on closed oriented manifolds, provided that at the singular set sing(F), the foliation is locally defined by Bott-Morse functions which are transversely centers. We prove, in this setting, the equivalent of the local and the complete stability theorems of Reeb. We show that if F has a compact leaf with finite fundamental group, or if a component of sing(F) has codimension ≥ 3 and finite fundamental group, then all leaves of F are compact and diffeomorphic, sing(F) consists of two connected components, and there is a Bott-Morse function f : M → [0, 1] such that f : M \ sing(F) → (0, 1) is a fiber bundle defining F and sing(F) = f.1({0, 1}). This yields a topological description of the type of leaves that appear in these foliations, and also the type of manifolds admitting such foliations. These results unify and generalize well known results for cohomogeneity one isometric actions, and a theorem of Reeb for foliations with Morse singularities of center type.