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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.15679 |
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| _version_ | 1866914047079743488 |
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| author | Bautista, A. Ibort, A. Lafuente, J. |
| author_facet | Bautista, A. Ibort, A. Lafuente, J. |
| contents | This paper describes the theory of Jacobi curves, a far reaching extension of the spaces of Jacobi fields along Riemannian geodesics, developed by Agrachev and Zelenko. Jacobi curves are curves in the Lagrangian Grassmannian of a symplectic space satisfying appropriate regularity conditions. It is shown that they are fully characterised in terms of a family of conformal symplectic invariant curvatures. In addition to a new derivation of the Ricci curvature tensor of a Jacobi curve, a Cartan-like theory of Jacobi curves is presented that allows to associate to any admissible Jacobi curve a reduced normal Cartan matrix. A reconstruction theorem proving that an admissible Jacobi curve is characterised, up to conformal symplectic transformations, by a reduced normal Cartan matrix and a geometric parametrization is obtained. The theory of cycles is studied proving that they correspond to flat Jacobi curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the classification of Jacobi curves and their conformal curvatures Bautista, A. Ibort, A. Lafuente, J. Differential Geometry 53 This paper describes the theory of Jacobi curves, a far reaching extension of the spaces of Jacobi fields along Riemannian geodesics, developed by Agrachev and Zelenko. Jacobi curves are curves in the Lagrangian Grassmannian of a symplectic space satisfying appropriate regularity conditions. It is shown that they are fully characterised in terms of a family of conformal symplectic invariant curvatures. In addition to a new derivation of the Ricci curvature tensor of a Jacobi curve, a Cartan-like theory of Jacobi curves is presented that allows to associate to any admissible Jacobi curve a reduced normal Cartan matrix. A reconstruction theorem proving that an admissible Jacobi curve is characterised, up to conformal symplectic transformations, by a reduced normal Cartan matrix and a geometric parametrization is obtained. The theory of cycles is studied proving that they correspond to flat Jacobi curves. |
| title | On the classification of Jacobi curves and their conformal curvatures |
| topic | Differential Geometry 53 |
| url | https://arxiv.org/abs/2509.15679 |