Positive Ric₂ curvature on products of spheres and their quotients via intermediate fatness

2024/10/24 by Jason DeVito, DeVito, Jason, Miguel Domínguez-Vázquez +5 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2410.18846

openalex publication_date 2024/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct metrics of positive 2^{\rm nd} intermediate Ricci curvature, Ric₂>0, on closed manifolds of dimensions 10, 11, 12, 13 and 14, including \mathbbS⁶×\mathbbS⁷, \mathbbS⁷×\mathbbS⁷ 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 Ric₂>0 in dimensions 13 and 14, including \mathbbRP⁶× \mathbbRP⁷ and \mathbbRP⁷× \mathbbRP⁷, 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 Ric₂>0 metrics on the total space of certain homogeneous bundles.

Cited by

• Classification for smooth manifolds looking like \mathbbCP³× S⁷
• Positive intermediate Ricci curvature on cohomogeneity one manifolds in low dimensions

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