  Let G be a simple linear algebraic group defined over the complex numbers.
Fix a proper parabolic subgroup P of G and a nontrivial antidominant character
\chi of P. We prove that a holomorphic principal G-bundle E over a connected
complex projective manifold M is semistable and the second Chern class of its
adjoint bundle vanishes in rational cohomology if and only if the line bundle
over E/P defined by \chi is numerically effective. Similar results remain valid
for principal bundles with a reductive linear algebraic group as the structure
group. These generalize an earlier work of Y. Miyaoka where he gave a
characterization of semistable vector bundles over a smooth projective curve.
Using these characterizations one can also produce similar criteria for the
semistability of parabolic principal bundles over a compact Riemann surface.
