Basic Results

Abstract

In this chapter we will present several classical results. We will prove Darboux’s theorem which states that all contact manifolds of the same dimension are locally diffeomorphic, hence the only local invariant of a contact manifold is its dimension. We will also prove the Legendre neighborhood theorem and its symplectic counterpart, the Lagrange neighborhood theorem. These results provide normal forms for neighborhoods of Legendrian submanifolds or Lagrangian submanifolds. We follow the presentation in [2] for these topics. Moreover we show that contact structures are stable on a compact manifold without boundary (which is called Gray’s theorem), i.e. smoothly homotopic contact structures are diffeomorphic. Further, we will give a characterization of those vector fields whose flow preserves a contact structure (‘contact vector fields’). Reeb vector fields are a special class of contact vector fields. Given a contact structure ξ, we will find a necessary and sufficient condition for a contact vector field to be the Reeb vector field of some contact form λ with $\ker \lambda =\xi$.