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.