Couples contacto-symplectiques

Abstract

We introduce a new geometric structure on differentiable manifolds. A contact-symplectic pair on a manifold M is a pair (α, η) where α is a Pfaffian form of constant class 2k + 1 and η a 2-form of constant class 2h such that α Λ dα k Λ η 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.