TABLE OF CONTENTS This Chapter summarizes the relevant material on geometry of Poisson manifolds and classical Hamiltonian systems quantized in the sequel.
Subsections: A. Symplectic manifolds, 91; B. Presymplectic manifolds, 96; C. Poisson manifolds, 97; D. Symplectic and Poisson reductions, 103; E. Koszul-Brylinski-Poisson homology, 108; F. Lichnerowicz-Poisson cohomology, 109.
We start with symplectic manifolds. Every symplectic manifold is a regular non-degenerate Poisson manifold, and vice versa.
Let Z be a smooth manifold. Any exterior two-form ? on Z yields the linear bundle morphism
| (2.1.1) |  |
One says that ?, is of rank r if the morphism (2.1.1) has the rank r. The kernel Ker ? of ? is defined as the kernel of the morphism (2.1.1). If ? is of constant rank, its kernel is a subbundle of the tangent bundle TZ. In particular, Ker ? contains the canonical zero section 
of TZ ? Z. If Ker ? = 0 (one customarily writes Ker 
), a two-form ? is said to be non-degenerate. It is called an almost symplectic form. Equipped with such a form, a manifold Z becomes an almost symplectic manifold. It is never odd-dimensional. Unless otherwise stated, we put dim Z = 2 m. In accordance with Theorem 10.10.5, a 2 m-dimensional manifold Z admits an almost symplectic form if and...
