Positive $\mathrm{Ric}_2$ curvature on products of spheres and their quotients via intermediate fatness

TL;DR

This paper constructs new metrics with positive second intermediate Ricci curvature on various high-dimensional manifolds, including products of spheres and their quotients, expanding the known examples beyond those with positive sectional curvature.

Contribution

It introduces a generalized notion of fatness to produce $ ext{Ric}_2>0$ metrics on homogeneous bundles, including infinitely many examples in dimensions 13 and 14.

Findings

Constructed $ ext{Ric}_2>0$ metrics on spheres and their quotients.

Produced infinitely many non-simply connected examples with $ ext{Ric}_2>0$.

Extended the concept of fatness for homogeneous bundles.

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.