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 I is at least equal to 21-1 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, 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. Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.