Manifolds with nonnegative curvature operator of the second kind

We investigate the curvature operator of second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed manifold with three-positive is diffeomorphic to spherical space form, improving recent result Cao-Gursky-Tran assuming two-positivity. states three-nonnegative either or flat, isometric quotient compact irreducible symmetric space. This settles nonnegativity part Nishikawa's conjecture under weaker assumption.