Symplectic Manifolds

In this chapter, we are going to see the definition of a symplectic manifold, several examples, and constructions of symplectic manifolds. We will see that cotangent bundle of a manifold is equipped with a symplectic structure. In fact, cotangent bundle is the most important example in the theory, since symplectic geometry started from solving equations of motion in a classical mechanical system in cotangent bundle. We study reduction, a procedure to obtain a symplectic manifold. Then, as an example of reduction, we see how complex projective manifolds are equipped with symplectic structure.