Tachibana-type theorems on complete manifolds (Q6609500)

A classical result of \textit{S.-i. Tachibana} [Proc. Japan Acad. 50, 301--302 (1974; Zbl 0299.53031)] asserts that a closed Einstein manifold (or more generally with harmonic curvature) with positive curvature operator has constant sectional curvature and a closed Einstein manifold with nonnegative curvature operator is locally symmetric. The proof uses the Bochner technique on the elliptic equation satisfied by the Riemann curvature tensor. Over the past few decades, Tachibana's result has been generalized by many mathematicians, including \textit{C. Böhm} and \textit{B. Wilking} [Ann. Math. (2) 167, No. 3, 1079--1097 (2008; Zbl 1185.53073)] (a more general Ricci flow convergence result under two-positive curvature operator), \textit{S. Brendle} [Duke Math. J. 151, No. 1, 1--21 (2010; Zbl 1189.53042)] (under positive isotropic curvature), and \textit{P. Petersen} and \textit{M. Wink} [Ann. Global Anal. Geom. 61, No. 4, 847--868 (2022; Zbl 1495.32060)] (under \(\lfloor \frac{m-1}{2} \rfloor\)-positive curvature operator), just to name a few.\N\NThis paper proves several generalizations of Tachibana's results. First, it proves that a closed Riemannian manifold of dimension \(m\geq 3\) with harmonic curvature and \(\lfloor \frac{m-1}{2} \rfloor\)-positive curvature operator has constant sectional curvature, thus weakening the Einstein condition to harmonic curvature in the work of Petersen and Wink.\N\NSecond, it shows 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. The observation is that \(|W|\) is a subharmonic function under the curvature assumptions and it vanishes under appropriate assumptions on complete manifolds. Then one can apply the classification theorem for locally conformally flat complete Riemannian manifolds with non-negative Ricci curvature and rule out the cases that cannot happen.\N\NThird, the positivity of the curvature operator can be relaxed to the positivity of the Ricci tensor for 3-manifolds. This relies on the fact that in dimension three the eigenvalues of the curvature operator can be expressed in terms of the eigenvalues of Ricci curvature.\N\NThe paper is well-written. The authors provide ample detail and clear explanations of their ideas, making the paper highly accessible to early-career researchers.