The completeness problem on the pseudo-homothetic Lie group
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916943665037312 |
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| author | Chaib, Salah Ferreira, Ana Cristina Zeghib, Abdelghani |
| author_facet | Chaib, Salah Ferreira, Ana Cristina Zeghib, Abdelghani |
| contents | Let us call pseudo-homothetic group the non-unimodular 3-dimensional Lie group that is the semi-direct product of $\mathbb{R}$ acting non-semisimply on $\mathbb{R}^2$. In this article, we solve the geodesic completeness problem on this Lie group. In particular, we exhibit a family of complete metrics such that all geodesics have bounded velocity. As an application, we show that the set of complete metrics is not closed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20612 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The completeness problem on the pseudo-homothetic Lie group Chaib, Salah Ferreira, Ana Cristina Zeghib, Abdelghani Differential Geometry 53C22, 53C30, 53C50 Let us call pseudo-homothetic group the non-unimodular 3-dimensional Lie group that is the semi-direct product of $\mathbb{R}$ acting non-semisimply on $\mathbb{R}^2$. In this article, we solve the geodesic completeness problem on this Lie group. In particular, we exhibit a family of complete metrics such that all geodesics have bounded velocity. As an application, we show that the set of complete metrics is not closed. |
| title | The completeness problem on the pseudo-homothetic Lie group |
| topic | Differential Geometry 53C22, 53C30, 53C50 |
| url | https://arxiv.org/abs/2410.20612 |