Characterization of the projective space Pr₃ by a contact form

Abstract

In this paper we present a theorem, which is a specific case of a very general problem, that could be described as follows: Given a differentiable manifoldV, “What are the elements of the exterior algebra ofV which describe the topology and differentiable structure ofV?” Classic examples of this situation are: (A) De Rham’s theorems, which describe the cohomology of a manifold by means of closed forms. (B) Reeb’s theorem, which describes the topology of a sphere by means of a function having only two critical points. In the present paper, we prove the following:

Theorem.The projective space Pr₃ and the sphere S ³ are the only three-dimensional, compact, manifold which has a contact form with a global expression:$\omega = f_1 df_2 - f_2 df_1 + f_3 df_4 - f_4 df_3 + f_5 df_6 - f_6 df_5$ where the f _i are global functions satisfying certain integrability conditions.