| _version_ | 1866901557023342592 |
|---|---|
| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the intricate relationship between the spectral properties of geometric operators and the intrinsic curvature of underlying spaces, presenting a comprehensive investigation into the concept of spectral curvature duality. We delve into how the eigenvalues and eigenfunctions of operators like the Laplace-Beltrami operator encode fundamental information about the geometry of a manifold, particularly its curvature. Our analysis extends across both Euclidean and various non-Euclidean geometries, including spherical, hyperbolic, and more general Riemannian manifolds. We examine classical results such as Weyl's law and heat kernel asymptotics, demonstrating their profound implications for inferring global and local curvature characteristics from spectral data. Furthermore, we propose a unified framework for understanding how spectral invariants manifest curvature duality in diverse geometric contexts, highlighting similarities and distinctions in how spectral signatures reflect scalar curvature, Ricci curvature, and sectional curvature. The study emphasizes the potential for spectral methods to characterize complex geometries and offers new perspectives on the interplay between analysis and geometry, with implications for fields ranging from theoretical physics to data analysis on curved spaces. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17682285 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spectral Curvature Duality Across Euclidean and Non-Euclidean Spaces SÉRGIO DE ANDRADE, PAULO This paper explores the intricate relationship between the spectral properties of geometric operators and the intrinsic curvature of underlying spaces, presenting a comprehensive investigation into the concept of spectral curvature duality. We delve into how the eigenvalues and eigenfunctions of operators like the Laplace-Beltrami operator encode fundamental information about the geometry of a manifold, particularly its curvature. Our analysis extends across both Euclidean and various non-Euclidean geometries, including spherical, hyperbolic, and more general Riemannian manifolds. We examine classical results such as Weyl's law and heat kernel asymptotics, demonstrating their profound implications for inferring global and local curvature characteristics from spectral data. Furthermore, we propose a unified framework for understanding how spectral invariants manifest curvature duality in diverse geometric contexts, highlighting similarities and distinctions in how spectral signatures reflect scalar curvature, Ricci curvature, and sectional curvature. The study emphasizes the potential for spectral methods to characterize complex geometries and offers new perspectives on the interplay between analysis and geometry, with implications for fields ranging from theoretical physics to data analysis on curved spaces. |
| title | Spectral Curvature Duality Across Euclidean and Non-Euclidean Spaces |
| url | https://doi.org/10.5281/zenodo.17682285 |