Horizontal diameter of unit spheres with polar foliations and infinitesimally polar actions

2019/07/29 by Yi Shi, Shi, Yi
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1907.12442

openalex publication_date 2019/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a singular Riemannian foliation F on a Riemannian manifold, a curve is called horizontal if it meets the leaves of F perpendicularly. For a singular Riemannian foliation F on a unit sphere \mathbbSⁿ, we show that if F is a polar foliation or if F is given by the orbits of an infinitesimally polar action, then the horizontal diameter of \mathbbSⁿ is π, i.e., any two points in \mathbbSⁿ can be connected by a horizontal curve of length ≤π.

Related

• Transverse sphere theorems for Riemannian foliations
• Infinitesimal adjunction and polar curves
• Low dimensional Singular Riemannian Foliations in spheres
• The dual foliation of polar actions on nonnegatively curved manifolds
• Great sphere foliations and manifolds with curvature bounded above