Symmetries of $(2,3,5)$-distributions and associated Legendrian cone structures
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arXiv
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| Format: | Preprint |
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2024
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| author | Hwang, Jun-Muk The, Dennis |
| author_facet | Hwang, Jun-Muk The, Dennis |
| contents | We exploit a natural correspondence between holomorphic $(2,3,5)$-distributions and nondegenerate lines on holomorphic contact manifolds of dimension $5$ to present a new perspective in the study of symmetries of $(2,3,5)$-distributions. This leads to a number of new results in this classical subject, including an unexpected relation between the multiply-transitive families of models having $7$- and $6$-dimensional symmetries, and a one-to-one correspondence between equivalence classes of nontransitive $(2,3,5)$-distributions with $6$-dimensional symmetries and nonhomogeneous nondegenerate Legendrian curves in $\mathbb{P}^3$. An ingredient for establishing the former is an explicit classification of homogeneous nondegenerate Legendrian curves in $\mathbb{P}^3$, which we present. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11459 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetries of $(2,3,5)$-distributions and associated Legendrian cone structures Hwang, Jun-Muk The, Dennis Differential Geometry Primary: 58A30, 58J70, Secondary: 32L25, 34C41, 53A55 We exploit a natural correspondence between holomorphic $(2,3,5)$-distributions and nondegenerate lines on holomorphic contact manifolds of dimension $5$ to present a new perspective in the study of symmetries of $(2,3,5)$-distributions. This leads to a number of new results in this classical subject, including an unexpected relation between the multiply-transitive families of models having $7$- and $6$-dimensional symmetries, and a one-to-one correspondence between equivalence classes of nontransitive $(2,3,5)$-distributions with $6$-dimensional symmetries and nonhomogeneous nondegenerate Legendrian curves in $\mathbb{P}^3$. An ingredient for establishing the former is an explicit classification of homogeneous nondegenerate Legendrian curves in $\mathbb{P}^3$, which we present. |
| title | Symmetries of $(2,3,5)$-distributions and associated Legendrian cone structures |
| topic | Differential Geometry Primary: 58A30, 58J70, Secondary: 32L25, 34C41, 53A55 |
| url | https://arxiv.org/abs/2408.11459 |