Tachibana-type theorems on complete manifolds

TL;DR

This paper extends Tachibana's classical theorem by proving that certain complete manifolds with harmonic curvature and specific positivity conditions on their curvature operator must have constant sectional curvature, broadening the scope of rigidity results.

Contribution

It generalizes Tachibana's theorem to complete manifolds with harmonic curvature under weaker positivity conditions, including integral bounds and parabolicity assumptions.

Findings

Manifolds with harmonic curvature and loor((m-1)/2)-positive curvature operator are of constant sectional curvature.

The rigidity result applies to complete manifolds under conditions like parabolicity, integral bounds, or positive lower bounds on eigenvalues.

For 3-manifolds, positivity of the Ricci tensor suffices to ensure constant curvature.

Abstract

We prove that a compact Riemannian manifold of dimension $m \geq 3$ with harmonic curvature and $\lfloor\frac{m-1}{2}\rfloor$-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for manifolds with positive curvature operator. The condition of $\lfloor\frac{m-1}{2}\rfloor$-positivity originates from recent work of Petersen and Wink, who proved a similar Tachibana-type theorem under the stronger condition that the manifold be Einstein. We show that the same rigidity property holds for complete manifolds assuming either parabolicity, an integral bound on the Weyl tensor or a stronger pointwise positive lower bound on the average of the first $\lfloor\frac{m-1}{2}\rfloor$ eigenvalues of the curvature operator. For 3-manifolds, we show that positivity of the curvature operator can be relaxed to positivity of the Ricci tensor.