New sphere theorems under curvature operator of the second kind

Abstract We investigate Riemannian manifolds whose curvature operator of the second kind satisfies the condition where are the eigenvalues of , is their average, and . Under such curvature conditions with optimal (depending on and ), we prove differentiable sphere theorems and their rigidity results in dimensions 3 and 4, a homological sphere theorem in higher dimensions, and a curvature characterization of Kähler space forms. These results generalize recent works corresponding to (namely, ‐positivity of ) of Cao–Gursky–Tran, Nienhaus–Petersen–Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of most results.