Affine hypersurfaces and superintegrable systems
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912313613746176 |
|---|---|
| author | Cortés, Vicente Vollmer, Andreas |
| author_facet | Cortés, Vicente Vollmer, Andreas |
| contents | It was recently shown that under mild assumptions second-order conformally superintegrable systems can be encoded in a $(0,3)$-tensor, called structure tensor. For abundant systems, this approach led to algebraic integrability conditions that essentially allow one to restore a system from the knowledge of its structure tensor in a point on the manifold. Here we study the geometric structure formalising such systems, which we call an abundant manifold. The underlying Riemannian manifold is necessarily conformally flat. We establish a correspondence between these superintegrable systems and the geometry of affine hypersurfaces. More precisely, we show that abundant manifolds correspond to certain non-degenerate relative affine hypersurfaces normalisations in $\mathbb R^{n+1}$ ($n\ge 2$). We also formulate the necessary and sufficient conditions non-degenerate relative affine hypersurface normalisations in $\mathbb R^{n+1}$ need to satisfy, if they arise from abundant manifolds. These relative affine hypersurface normalisations are called abundant hypersurface normalisations. Both for abundant manifolds and for relative affine hypersurface normalisations a natural concept of conformal equivalence can be defined. We prove that they are compatible, permitting us to identify conformal classes of abundant manifolds with abundant hypersurface immersions (without specified normalisation). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine hypersurfaces and superintegrable systems Cortés, Vicente Vollmer, Andreas Differential Geometry 53A15, 70H33, 70H06, 14R99 It was recently shown that under mild assumptions second-order conformally superintegrable systems can be encoded in a $(0,3)$-tensor, called structure tensor. For abundant systems, this approach led to algebraic integrability conditions that essentially allow one to restore a system from the knowledge of its structure tensor in a point on the manifold. Here we study the geometric structure formalising such systems, which we call an abundant manifold. The underlying Riemannian manifold is necessarily conformally flat. We establish a correspondence between these superintegrable systems and the geometry of affine hypersurfaces. More precisely, we show that abundant manifolds correspond to certain non-degenerate relative affine hypersurfaces normalisations in $\mathbb R^{n+1}$ ($n\ge 2$). We also formulate the necessary and sufficient conditions non-degenerate relative affine hypersurface normalisations in $\mathbb R^{n+1}$ need to satisfy, if they arise from abundant manifolds. These relative affine hypersurface normalisations are called abundant hypersurface normalisations. Both for abundant manifolds and for relative affine hypersurface normalisations a natural concept of conformal equivalence can be defined. We prove that they are compatible, permitting us to identify conformal classes of abundant manifolds with abundant hypersurface immersions (without specified normalisation). |
| title | Affine hypersurfaces and superintegrable systems |
| topic | Differential Geometry 53A15, 70H33, 70H06, 14R99 |
| url | https://arxiv.org/abs/2504.05200 |