Positive $\mathrm{Ric}_2$ curvature on products of spheres and their quotients via intermediate fatness
Jason DeVito, Miguel Domínguez-Vázquez, David González-Álvaro, Alberto Rodríguez-Vázquez
TL;DR
The paper advances the study of positive intermediate Ricci curvature by constructing new $\mathrm{Ric}_2>0$ metrics on products of spheres and their quotients in dimensions $10$–$14$, including infinite families in dimension $13$ and non-simply connected examples in dimensions $13$ and $14$. It introduces the fatness coindex $f(H<K<G)$ and develops a submersion-geometry framework that extends Wallach-type constructions to yield $\mathrm{Ric}_k>0$ on total spaces of homogeneous bundles, notably when $k=2$ under suitable conditions. Central to the results are explicit homogeneous models on $\mathsf{Spin}_7/\mathsf{SU}_3$ and $\mathsf{Spin}_8/\mathsf{G}_2$ (diffeomorphic to $\mathbb{S}^6\times\mathbb{S}^7$ and $\mathbb{S}^7\times\mathbb{S}^7$ respectively), together with a complete classification of free circle and $\mathsf{SU}_2$ actions yielding diverse biquotients with $\mathrm{Ric}_2>0$. The work also analyzes the isometry groups, submersions to $\mathbb{S}^4$, totally geodesic embeddings, and the topology (cohomology and Pontryagin classes) of the quotients, demonstrating that these spaces are topologically distinct from known $\sec>0$ examples while sharing strong curvature properties. Overall, the results broaden the landscape of manifolds with $\mathrm{Ric}_2>0$, illustrate the utility of intermediate fatness in curvature construction, and provide new obstructions distinguishing them from positively curved manifolds.
Abstract
We construct metrics of positive $2^{\rm nd}$ intermediate Ricci curvature, $\mathrm{Ric}_2>0$, on closed manifolds of dimensions 10, 11, 12, 13 and 14, including $\mathbb{S}^6\times\mathbb{S}^7$, $\mathbb{S}^7\times\mathbb{S}^7$ and all their simply connected isometric quotients. In particular, we obtain infinitely many examples in dimension 13. We also produce infinitely many non-simply connected spaces with $\mathrm{Ric}_2>0$ in dimensions 13 and 14, including $\mathbb{RP}^6\times \mathbb{RP}^7$ and $\mathbb{RP}^7\times \mathbb{RP}^7$, which cannot admit a metric of positive sectional curvature. The main new idea is a generalization of the concept of fatness which ensures the existence of $\mathrm{Ric}_2>0$ metrics on the total space of certain homogeneous bundles.