Characterization of the projective space \(\Pr _ 3\) by a contact form (Q1081907)

We present a theorem, which is a specific case of a very general problem, that could be described as follows: Given a differentiable manifold V, ''What are the elements of the exterior algebra of V which describe the topology and differentiable structure of V ?'' 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 \(\Pr_ 3\) 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.