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–Weitzenböck–Lichnerowicz type formulas for the Weyl tensor, which are generalizations of identities in dimension four.
|Journal||Advances in Mathematics|
|State||Published - Nov 13 2017|