Abstract:We propose a definition of magnitude for a length space with a Borel measure, which involves integrals over the set of geodesics. This quantity agrees with the magnitude of finite metric spaces, up to re-scaling the metric to ensure the convergence, when we use the counting measure on them. We also prove a version of the homogeneous magnitude theorem, by showing that the new definition agrees with the volume when we use the weight measure on a compact homogeneous Riemannian manifold. We compute various examples, which suggest that this quantity can capture information of non-uniqueness of geodesics, such as the injectivity radius, corresponding to the generating degrees of the magnitude homology.
| Comments: | 43 pages |
| Subjects: | Differential Geometry (math.DG); Metric Geometry (math.MG) |
| MSC classes: | 53C20 (Primary), 51F99 (Secondary) |
| Cite as: | arXiv:2605.23485 [math.DG] |
| (or arXiv:2605.23485v1 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.23485 arXiv-issued DOI via DataCite (pending registration) |
From: Yoshinori Hashimoto [view email]
[v1]
Fri, 22 May 2026 10:44:24 UTC (40 KB)
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