Volume of Tubes and Concentration of Measure in Riemannian Geometry

2023/07/31 by Sergio L. Cacciatori, Cacciatori, S. L., Pietro Ursino +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #General Topology (math.GN) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2308.00080

openalex publication_date 2023/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the notion of concentration locus introduced in \citeCacUrs22, in the case of Riemann manifolds sequences and its relationship with the volume of tubes. After providing a general formula for the volume of a tube around a Riemannian submanifold of a Riemannian manifold, we specialize it to the case of totally geodesic submanifolds of compact symmetric spaces. In the case of codimension one, we prove explicitly concentration. Then, we investigate for possible characterizations of concentration loci in terms of Wasserstein and Box distances.

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