  S. Kov\'acs proposed a conjecture on rigidity results induced by ample
subsheaves of some exterior power of tangent bundles for projective manifolds.
We verify the conjecture in the case of second exterior products under a rank
condition. Besides, we prove a structure theorem satisfied by projective
manifolds whose third exterior power of tangent bundle is nef. Additionally, we
prove a weaker version of log Campana-Peternell conjecture for fourfolds.
Finally, we give the structure of manifolds with a regular foliation whose
exterior powers are strictly nef.
