  For the Bach-flat closed manifold with positive scalar curvature, we prove a
rigidity result under a given inequality involving the Weyl curvature and the
traceless Ricci curvature. Moveover, under an inequality involving
$L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and
the Yamabe invariant, we also provide a similar rigidity result. As an
application, we obtain some rigidity results on 4-dimensional manifolds.
