2025/07/23 by Hasan M. El-Hasan, El-Hasan, Hasan M., Russell Phelan +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Myofascial pain diagnosis and treatment
paper · pdf · doi:10.48550/arxiv.2507.17839
openalex publication_date 2025/07/23 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
Pro and the third author showed that there are Riemannian submersions π: M → B with M a compact manifold with positive Ricci curvature, whose base B, has Ricci curvatures with both signs. Thus, Riemannian submersions need not preserve positive Ricci curvature. In this note we establish the degree to which this result extends into the setting of positive intermediate Ricci curvature. It is an immediate consequence of the Gray--O'Neill Horizontal curvature equation that if π: M→ B is a Riemannian submersion whose base is b-dimensional and Rick(M) >0 for any k ∈ \ 1,2,⋯, b-1\, then Rick(B) is also positive. Here we show that this observation is optimal in the following strong sense: For k ≥ dim(B), let π: (M,gM) → (B,gB) be a Riemannian submersion from a complete Riemannian manifold with Rick(M) >0. We show how to perturb gM in the C1-topology to produce a Riemannian submersion π: (M,gM) → (B,gB) whose total space has Rick >0, but whose base has Ricci curvature of both signs. In particular, this shows that Riemannian submersions that do not preserve positive Ricci curvature are dense in the C1-topology among the complete metrics on M with Ric>0 for which a given submersion π: M→ B is Riemannian.