A method for determining Cartan geometries from the local behavior of automorphisms
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916715889164288 |
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| author | Erickson, Jacob W. |
| author_facet | Erickson, Jacob W. |
| contents | We introduce a construction for a Cartan geometry that captures the local behavior of a given geometric automorphism near a distinguished element. The result of this construction, which we call the sprawl generated by the automorphism, is uniquely characterized by a kind of universal property that allows us to compare different Cartan geometries that admit automorphisms with equivalent local behavior near a distinguished element. As example applications, we describe how to construct non-flat real projective structures admitting nontrivial automorphisms with higher-order fixed points and extend some known local automorphisms with higher-order fixed points on non-flat parabolic geometries to global automorphisms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_00561 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A method for determining Cartan geometries from the local behavior of automorphisms Erickson, Jacob W. Differential Geometry 58D19 (Primary) 57S25 (Secondary) We introduce a construction for a Cartan geometry that captures the local behavior of a given geometric automorphism near a distinguished element. The result of this construction, which we call the sprawl generated by the automorphism, is uniquely characterized by a kind of universal property that allows us to compare different Cartan geometries that admit automorphisms with equivalent local behavior near a distinguished element. As example applications, we describe how to construct non-flat real projective structures admitting nontrivial automorphisms with higher-order fixed points and extend some known local automorphisms with higher-order fixed points on non-flat parabolic geometries to global automorphisms. |
| title | A method for determining Cartan geometries from the local behavior of automorphisms |
| topic | Differential Geometry 58D19 (Primary) 57S25 (Secondary) |
| url | https://arxiv.org/abs/2303.00561 |