Recovering Riemannian Geometry from Diffusion
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866909999761981440 |
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| author | Sangha, Amandip |
| author_facet | Sangha, Amandip |
| contents | We present an intrinsic reconstruction of Riemannian geometry from a symmetric, strongly local diffusion semigroup. Starting from a diffusion operator and its associated first- and second-order diffusion calculus, we recover the full weighted Riemannian structure of the underlying manifold. In particular, we show that the carre du champ determines a unique smooth Riemannian metric, that the iterated carre du champ encodes curvature, and that the symmetry of the diffusion fixes the Levi-Civita connection and reference measure. As a consequence, the diffusion semigroup determines the global Riemannian manifold uniquely up to isometry. The results provide an information-theoretic perspective on differential geometry in which geometric structure emerges from the intrinsic behavior of diffusion, without assuming any prior metric or coordinate description. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17166 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Recovering Riemannian Geometry from Diffusion Sangha, Amandip Differential Geometry Analysis of PDEs Probability We present an intrinsic reconstruction of Riemannian geometry from a symmetric, strongly local diffusion semigroup. Starting from a diffusion operator and its associated first- and second-order diffusion calculus, we recover the full weighted Riemannian structure of the underlying manifold. In particular, we show that the carre du champ determines a unique smooth Riemannian metric, that the iterated carre du champ encodes curvature, and that the symmetry of the diffusion fixes the Levi-Civita connection and reference measure. As a consequence, the diffusion semigroup determines the global Riemannian manifold uniquely up to isometry. The results provide an information-theoretic perspective on differential geometry in which geometric structure emerges from the intrinsic behavior of diffusion, without assuming any prior metric or coordinate description. |
| title | Recovering Riemannian Geometry from Diffusion |
| topic | Differential Geometry Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2601.17166 |