Abstract. The dimension of a characteristic closure on a closed K-contact 2n+1-manifold $M$ is at most the smaller of n+1 and 2n+1 minus the rank of the vector space of harmonic vector fields on $M$. Moreover, there are at least n+1 closed characteristics on $M$ and if their number is finite, then each of the Betti numbers of $M$ is at most equal to 1. If in addition, $M$ is simply connected, then it is homeomorphic to a sphere.
AMSclassification. Primary 58F22, 58F18; Secondary 53C15
Keywords. K-contact, Betti number, Morse Theory