The Entropic Geometry of Manifolds
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| Natura: | Recurso digital |
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Zenodo
2025
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| _version_ | 1866901503613075456 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the nascent field of entropic geometry, a framework integrating information theory, particularly entropy, with the study of geometric structures on manifolds. We propose that entropic functionals provide novel insights into fundamental geometric properties, offering an alternative or complementary perspective to classical Riemannian and differential geometry. The work situates entropic geometry within mathematical physics and information theory, highlighting its potential to bridge disparate fields. We delve into how various entropy measures (Shannon, Rényi, and Tsallis) can define intrinsic geometric properties like curvature, distance, and volume. The paper reviews foundational work, notably Perelman's entropy functional for the Ricci flow, and extends these ideas. We outline a methodology for constructing entropic metrics and investigating their associated geometric flows and variational principles. Expected results include a deeper understanding of how information content influences intrinsic geometry, potentially leading to new geometric inequalities and characterizations of manifold topology. The discussion addresses implications for theoretical physics (e.g., quantum gravity, statistical mechanics) and emerging areas like machine learning and data science. Ultimately, this work aims to establish entropic geometry as a robust and fruitful area of inquiry, opening new avenues for research into the intrinsic nature of space. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17687747 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Entropic Geometry of Manifolds SÉRGIO DE ANDRADE, PAULO This paper explores the nascent field of entropic geometry, a framework integrating information theory, particularly entropy, with the study of geometric structures on manifolds. We propose that entropic functionals provide novel insights into fundamental geometric properties, offering an alternative or complementary perspective to classical Riemannian and differential geometry. The work situates entropic geometry within mathematical physics and information theory, highlighting its potential to bridge disparate fields. We delve into how various entropy measures (Shannon, Rényi, and Tsallis) can define intrinsic geometric properties like curvature, distance, and volume. The paper reviews foundational work, notably Perelman's entropy functional for the Ricci flow, and extends these ideas. We outline a methodology for constructing entropic metrics and investigating their associated geometric flows and variational principles. Expected results include a deeper understanding of how information content influences intrinsic geometry, potentially leading to new geometric inequalities and characterizations of manifold topology. The discussion addresses implications for theoretical physics (e.g., quantum gravity, statistical mechanics) and emerging areas like machine learning and data science. Ultimately, this work aims to establish entropic geometry as a robust and fruitful area of inquiry, opening new avenues for research into the intrinsic nature of space. |
| title | The Entropic Geometry of Manifolds |
| url | https://doi.org/10.5281/zenodo.17687747 |