2022/07/21 by Leonardo F. Cavenaghi, Cavenaghi, Leonardo F.
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Connective tissue disorders research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2207.10757
openalex publication_date 2022/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal G-bundle (P, g) → (B, h) with dim G = 3, and g representing a Riemannian submersion metric ensuring that the G-orbits are totally geodesic, can one modify h to render all vertical curvatures equal to 1? In this note, we establish a rigidity result for fat Riemannian foliations with bounded holonomy and a specific curvature constraint. Our result addresses Ziller's question for fat fiber bundles with compact structure groups, considering connected compact total spaces under a curvature constraint that holds on various examples, such as locally symmetric spaces. Additionally, we assume that all vertizontal curvatures coincide at a point.