Positive intermediate Ricci curvature on products of homogeneous spaces

Abstract

We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci curvature, i.e. $\mathrm{Ric}_2>0$, 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 $\mathrm{Ric}_2>0$ setting. These observations indicate that the class of manifolds with $\mathrm{Ric}_2>0$ is vastly different from the class of positively curved manifolds.