in 1986, nishikawa conjectured that a closed riemannian manifold with positive curvature operator of the second kind is diffeomorphic to a spherical space form and a closed riemannian manifold with nonnegative curvature operator of the second kind is diffeomorphic to a riemannian locally symmetric space. recently, the positive case of nishikawa's conjecture was proved by cao-gursky-tran and the nonnegative case was settled by myself. in this talk, i will first talk about curvature operators of the second kind and then present a proof of nishikawa's conjecture under weaker assumptions.