Rank and k‐nullity of contact manifolds

Abstract

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 2l − 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.