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.
Categories: math.DG