  The goal of this paper is to investigate the rigidity of 4-dimensional
manifolds involving some pinching curvature conditions. To this end, we make
use of the approach of biorthogonal curvature which is weaker than the
sectional curvature. Here, we prove a rigidity result for 4-dimensional compact
manifolds under a suitable lower bound condition on the minimum of the
biorthogonal curvature. From this, we improve the pinching constants considered
by some preceding works on a rigidity result for 4-dimensional manifolds.
