THE WEINSTEIN CONJECTURE IN THE PRESENCE OF SUBMANIFOLDS HAVING

Abstract. Helmut Hofer introduced in ’93 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3–manifolds (M, ξ) with pi2(M) 6 = 0. We modify Hofer’s argument to prove the Weinstein conjecture for some examples of higher dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit. Let (M, ξ) be a contact manifold with contact form α. The associated Reeb field Rα is the unique vector field that satisfies the equations α(Rα) = 1 and ιRαdα = 0 everywhere