Tachibana-type theorems on complete manifolds

Abstract

We prove that a compact Riemannian manifold of dimension m ≥ 3 with harmonic curvature and ⌊ m−1 2 ⌋-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for manifolds with positive curvature operator. The condition of ⌊ m−1 2 ⌋-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 ⌊ m−1 2 ⌋ 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.

Manifolds with nonnegative curvature operator of the second kind

“…Nienhaus, Petersen and Wink [11] show that n-dimensional compact Einstein manifolds with k(< 3n(n+2) 2(n+4) )-nonnegative curvature operators of the second kind are either rational homology spheres or flat. We refer to [5,14] for details on compact Riemannian manifolds with k-positive curvature operator of the first kind by using Bochner technique.…”