Product manifolds and the curvature operator of the second kind

Abstract: We investigate the curvature operator of the second kind on product Riemannian manifolds and obtain some optimal rigidity results. For instance, we prove that the universal cover of an $n$-dimensional non-flat complete locally reducible Riemannian manifold with $(n+\frac{n-2}{n})$-nonnegative (respectively, $(n+\frac{n-2}{n})$-nonpositive) curvature operator of the second kind must be isometric to $\mathbb{S}^{n-1}\times \mathbb{R}$ (respectively, $\mathbb{H}^{n-1}\times \mathbb{R}$) up to scaling. We also prove analogous optimal rigidity results for $\mathbb{S}^{n_1}\times \mathbb{S}^{n_2}$ and $\mathbb{H}^{n_1}\times \mathbb{H}^{n_2}$, $n_1,n_2 \geq 2$, among product Riemannian manifolds, as well as for $\mathbb{CP}^{m_1}\times \mathbb{CP}^{m_2}$ and $\mathbb{CH}^{m_1}\times \mathbb{CH}^{m_2}$, $m_1,m_2\geq 1$, among product K\"ahler manifolds. Our approach is pointwise and algebraic.