Positive intermediate Ricci curvature on products of homogeneous spaces

We establish metrics of positive $2^{\mathrm{n}\mathrm{d}}$-intermediate Ricci curvature, i.e. ${\mathrm{R}\mathrm{i}\mathrm{c}}_2>0$, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and Hsiang-Kleiner theorem for positively curved manifolds do not hold in the ${\mathrm{R}\mathrm{i}\mathrm{c}}_2>0$ setting. These observations indicate that the class of manifolds with ${\mathrm{R}\mathrm{i}\mathrm{c}}_2>0$ is vastly different from the class of positively curved manifolds.