Rank and \(k\)-nullity of contact manifolds (Q1774721)

Summary: 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.