  We give some rigidity theorems for an n$(\geq4)$-dimensional compact
Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and
positive constant $\sigma_2$. Moreover, when $n=4,$ we prove that a
4-dimensional compact locally conformally flat Riemannian manifold with
positive scalar curvature and positive constant $\sigma_2$ is isometric to a
quotient of the round $\mathbb{S}^4$.
