4 problems

Nishikawa's conjecture. If $M$ has positive curvature operator of the second kind, then $M$ is diffeomorphic to a spherical space form; if $M$ has nonnegative curvature operator of…

Let $m$ be a positive integer, and let $(M^m,g)$ be a compact Riemannian manifold with diameter $\operatorname{diam}(M)=1$. Write $\mathcal{R}m$ for its curvature operator, and let…

Let $(M^n,g,f)$ be an $n$-dimensional steady gradient Ricci soliton with positive curvature operator. Here, $O(2)$-symmetry means that the soliton admits an isometric $O(2)$ action…

Nishikawa's conjecture.