Manifolds with harmonic Weyl tensor and nonnegative curvature operator of the second kind

Abstract

We show that a compact manifold with harmonic Weyl tensor and nonnegative curvature operator of the second kind is globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. Moreover, we obtain that a compact manifold with harmonic curvature and nonnegative curvature operator of the second kind is isometric to a space of constant curvature. These improve Kashiwada's result that Riemannian manifolds with harmonic curvature tensor for which the curvature operator of the second kind is nonnegative and positive are locally symmetric spaces and constant curvature spaces, respectively.