Rank andk-nullity of contact manifolds

We prove that the dimension of the1-nullity distributionN(1)on a closed Sasakian manifoldMof ranklis at least equal to2l−1provided thatMhas 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 ofN(1)is less than or equal ton+1orN(1)is the entire tangent bundleTM. In the latter case, the Sasakian manifoldMis isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions betweenk-nullity, Weinstein conjecture, and minimal unit vector fields.