2024/09/30 by Heiko Gimperlein, Gimperlein, Heiko, Magnus Goffeng +1
Computer Science · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2409.19969
openalex publication_date 2024/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the geometric significance of Leinster's magnitude invariant. For closed manifolds we find a precise relation with Brylinski's beta function and therefore with classical invariants of knots and submanifolds. In the special case of compact homogeneous spaces we obtain an elementary proof that the residues of the beta function contain the same geometric information as the asymptotic expansion of the magnitude function. For general closed manifolds we use the recent pseudodifferential analysis of the magnitude operator to relate these via an interpolating polynomial family. Beyond manifolds, the relation with the Brylinski beta function allows to deduce unexpected properties of the magnitude function for the p-adic integers.