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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.08186 |
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| _version_ | 1866913807102640128 |
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| author | Bravo-Doddoli, Alejandro |
| author_facet | Bravo-Doddoli, Alejandro |
| contents | In the framework of sub-Riemannian Manifolds, a relevant question is: what are the \enquote{metric lines} (i.e., the isometric embedding of the real line)? This article presents a conjecture classifying the metric lines in Carnot groups and takes the first steps in answering this question for \enquote{arbitrary rank} Carnot groups. We classify the metric lines of the Engel-type groups $\Eng(n)$ (Theorem 1.2), whose sub-Riemannian structure is defined on a non-integrable distribution of rank $n+1$. Our approach is a new method, called the sequence method, which we began to develop to study metric lines in the jet space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08186 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric Lines in Engel-type Groups Bravo-Doddoli, Alejandro Differential Geometry Optimization and Control In the framework of sub-Riemannian Manifolds, a relevant question is: what are the \enquote{metric lines} (i.e., the isometric embedding of the real line)? This article presents a conjecture classifying the metric lines in Carnot groups and takes the first steps in answering this question for \enquote{arbitrary rank} Carnot groups. We classify the metric lines of the Engel-type groups $\Eng(n)$ (Theorem 1.2), whose sub-Riemannian structure is defined on a non-integrable distribution of rank $n+1$. Our approach is a new method, called the sequence method, which we began to develop to study metric lines in the jet space. |
| title | Metric Lines in Engel-type Groups |
| topic | Differential Geometry Optimization and Control |
| url | https://arxiv.org/abs/2405.08186 |