Positive intermediate Ricci curvature on cohomogeneity one manifolds in low dimensions

TL;DR

The paper investigates the existence of invariant metrics with positive intermediate Ricci curvature $Ric_k>0$ on low-dimensional cohomogeneity one manifolds. It combines Cheeger deformations with Jacobi-field obstructions, via Wilking’s transverse Jacobi equation, to both construct curvature-positive metrics and prove symmetry obstructions. A principal result is the construction of a cohomogeneity one metric on $S^3\times\mathbb{C}\mathrm{P}^2$ with $Ric_4>0$ (invariant under a $(S^3\times S^3)$-action) and a corresponding obstruction showing no invariant metric with $Ric_3>0$. The paper then delineates dimension-by-dimension obstructions in dimensions $5$–$7$ for several manifolds, including Brieskorn varieties, product manifolds, and seven-dimensional families, highlighting a nuanced landscape where some manifolds admit $Ric_k>0$ while many symmetric constructions fail. These results map the boundary between curved examples and obstructions, advancing our understanding of intermediate curvature in low dimensions and suggesting directions for future constructions and classifications.

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.

Theorems & Definitions (37)

• Theorem 1
• Corollary 2
• Theorem 3
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Example 2.4
• Lemma 2.5
• Remark 2.6
• Lemma 2.7