Torsion-free connections of second-order maximally superintegrable systems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915276527763456 |
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| author | Vollmer, Andreas |
| author_facet | Vollmer, Andreas |
| contents | Second-order (maximally) conformally superintegrable systems play an important role as models of mechanical systems, including systems such as the Kepler-Coulomb system and the isotropic harmonic oscillator. The present paper is dedicated to understanding non- and semi-degenerate systems. We obtain "projective flatness" results for two torsion-free connections naturally associated to such systems. This viewpoint sheds some light onto the interrelationship of properly and conformally (second-order maximally) superintegrable systems from a geometrical perspective. It is shown that the semi-degenerate secondary structure tensor can be viewed as the Ricci curvature of a natural torsion-free connection defined by the primary structure tensor (and similarly in the non-degenerate case). It is also shown that properly semi-degenerate systems are characterised, similar to the non-degenerate case, by the vanishing of the secondary structure tensor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion-free connections of second-order maximally superintegrable systems Vollmer, Andreas Differential Geometry 53A20, 53B50, 70H33 Second-order (maximally) conformally superintegrable systems play an important role as models of mechanical systems, including systems such as the Kepler-Coulomb system and the isotropic harmonic oscillator. The present paper is dedicated to understanding non- and semi-degenerate systems. We obtain "projective flatness" results for two torsion-free connections naturally associated to such systems. This viewpoint sheds some light onto the interrelationship of properly and conformally (second-order maximally) superintegrable systems from a geometrical perspective. It is shown that the semi-degenerate secondary structure tensor can be viewed as the Ricci curvature of a natural torsion-free connection defined by the primary structure tensor (and similarly in the non-degenerate case). It is also shown that properly semi-degenerate systems are characterised, similar to the non-degenerate case, by the vanishing of the secondary structure tensor. |
| title | Torsion-free connections of second-order maximally superintegrable systems |
| topic | Differential Geometry 53A20, 53B50, 70H33 |
| url | https://arxiv.org/abs/2403.08509 |