Rank and k-nullity of contact manifolds

We prove that the dimension of the 1-nullity distributionN(1) on a closed Sasakianmanifold M of rank l is at least equal to 2l−1 provided thatM has an isolated closed characteristic. The result is then used to provide some examples ofK-contact manifolds which are not Sasakian. On a closed, 2n+ 1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM . In the latter case, the Sasakian manifold M is isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields.