New Sphere Theorems under Curvature Operator of the Second Kind

We investigate Riemannian manifolds (Mⁿ, g) whose curvature operator of the second kind R satisfies the condition equation* ^-1 (₁ + +_) > -, equation* where ₁ (₍-₁) (₍+₂) /₂ are the eigenvalues of R, is their average, and > -1. Under such conditions with optimal depending on n and, we prove two differentiable sphere theorems in dimensions three and four, a homological sphere theorem in higher dimensions, and a curvature characterization of K\"ahler space forms. These results generalize recent works corresponding to =0 of Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of all results.