On Closed Manifolds with Harmonic Weyl Curvature

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenb\"ock-Lichnerowicz type formulas for the Weyl tensor, which are generalizations of identities in dimension four.