Rank and k -nullity of contact manifolds

We prove that the dimension of the 1 -nullity distribution N ( 1 ) on a closed Sasakian manifold M of rank l is at least equal to 2 l − 1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k -contact manifolds which are not Sasakian. On a closed, 2 n + 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 T M . 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.