On Closed Manifolds with Harmonic Weyl Curvature

TL;DR

This paper establishes rigidity and gap results for closed manifolds with harmonic Weyl curvature, generalizing known theorems and introducing new formulas for the Weyl tensor across dimensions.

Contribution

It introduces new Bochner-Weitzenb"ock-Lichnerowicz formulas for the Weyl tensor and generalizes Tachibana's theorem to higher dimensions with non-negative curvature operator.

Findings

Rigidity results for manifolds with harmonic Weyl curvature

Generalized Tachibana's theorem for non-negative curvature operator

New formulas for the Weyl tensor in various dimensions

Abstract

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.