Positive intermediate Ricci curvature on cohomogeneity one manifolds in low dimensions

TL;DR

This paper investigates the existence of invariant metrics with positive intermediate Ricci curvature on low-dimensional cohomogeneity one manifolds, revealing both constructions and obstructions in specific cases.

Contribution

It constructs an invariant metric with positive 4th-intermediate Ricci curvature on a specific cohomogeneity one manifold and identifies obstructions to positive 3rd-intermediate Ricci curvature, advancing understanding of curvature conditions.

Findings

Constructed a metric with positive 4th-intermediate Ricci curvature on a specific manifold.

Proved non-existence of metrics with positive 3rd-intermediate Ricci curvature on the same manifold.

Identified symmetry-based obstructions to positive curvature on several other cohomogeneity one manifolds.

Abstract

We explore existence of invariant metrics with positive intermediate Ricci curvature on closed, low-dimensional cohomogeneity one manifolds. For a certain cohomogeneity one $\mathsf{Spin}(4)$-action on $S^3 \times \mathbb{C}\mathrm{P}^2$, we construct an invariant metric with positive 4th-intermediate Ricci curvature and show it cannot admit one with positive 3rd-intermediate Ricci curvature. We further establish similar symmetry obstructions to positive curvature for $S^3 \times S^3$, $S^3 \times S^4$, and several families of cohomogeneity one manifolds.