Carrollian $\mathbb{R}^\times$-bundles: Connections and Beyond
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911158742548480 |
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| author | Bruce, Andrew James |
| author_facet | Bruce, Andrew James |
| contents | We propose an approach to Carrollian geometry using principal $\mathbb{R}^\times$-bundles ($\mathbb{R}^\times := \matthbb{R} \setminus \{0\}$) equipped with a degenerate metric whose kernel is the module of vertical vector fields. The constructions allow for non-trivial bundles, and a large class of Carrollian manifolds can be analysed in this formalism. A key result in this is that once a principal connection has been selected, there is a canonical non-degenerate metric that can be leveraged to circumvent the difficulties associated with a degenerate metric. Within this framework, we examine the Levi-Civita connection and null geodesics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21332 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Carrollian $\mathbb{R}^\times$-bundles: Connections and Beyond Bruce, Andrew James Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics 53B05, 53B15, 53C50, 53Z05, 58A30 We propose an approach to Carrollian geometry using principal $\mathbb{R}^\times$-bundles ($\mathbb{R}^\times := \matthbb{R} \setminus \{0\}$) equipped with a degenerate metric whose kernel is the module of vertical vector fields. The constructions allow for non-trivial bundles, and a large class of Carrollian manifolds can be analysed in this formalism. A key result in this is that once a principal connection has been selected, there is a canonical non-degenerate metric that can be leveraged to circumvent the difficulties associated with a degenerate metric. Within this framework, we examine the Levi-Civita connection and null geodesics. |
| title | Carrollian $\mathbb{R}^\times$-bundles: Connections and Beyond |
| topic | Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics 53B05, 53B15, 53C50, 53Z05, 58A30 |
| url | https://arxiv.org/abs/2505.21332 |