Magnitude of metric measure spaces and integrals over geodesics
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910247746011136 |
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| author | Hashimoto, Yoshinori |
| author_facet | Hashimoto, Yoshinori |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23485 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Magnitude of metric measure spaces and integrals over geodesics Hashimoto, Yoshinori Differential Geometry Metric Geometry 53C20 (Primary), 51F99 (Secondary) 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. |
| title | Magnitude of metric measure spaces and integrals over geodesics |
| topic | Differential Geometry Metric Geometry 53C20 (Primary), 51F99 (Secondary) |
| url | https://arxiv.org/abs/2605.23485 |