Couples contacto-symplectiques

Abstract:

We introduce a new geometric structure on differentiable manifolds. A contact-symplectic pair on a manifold $M$ is a pair $\left ( \alpha ,\eta \right )$ where $\alpha$ is a Pfaffian form of constant class $2k+1$ and $\eta$ a $2$-form of constant class$\ 2h$ such that $\alpha \wedge d\alpha ^{k}\wedge \eta ^{h}$ is a volume form. Each form has a characteristic foliation whose leaves are symplectic and contact manifolds respectively. These foliations are transverse and complementary. Some other differential objects are associated to it. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds and principal torus bundles. As a deep application of this theory, we give a negative answer to the famous Reeb’s problem which asks if every vector field without closed 1-codimensional transversal on a manifold having contact forms is the Reeb vector field of a contact form.