  A well known result of Miyaoka asserts that a complex projective manifold is
uniruled if its cotangent bundle restricted to a general complete intersection
curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle,
it can moreover be shown that the choice of such a curve gives rise to a
rationally connected foliation of the manifold. In this note we show that,
conversely, a movable curve can be found so that the maximal rationally
connected fibration of the manifold may be recovered as a term of the
associated Harder-Narasimhan filtration of the tangent bundle.
