Twistor Theory of Dancing Paths
Given a path geometry on a surface , we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on . This causal structure corresponds to a conformal structure if and only if is a real projective plane, and the paths are lines. We give th...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2022 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2022
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211641 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Twistor Theory of Dancing Paths. Maciej Dunajski. SIGMA 18 (2022), 027, 13 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862722933417312256 |
|---|---|
| author | Dunajski, Maciej |
| author_facet | Dunajski, Maciej |
| citation_txt | Twistor Theory of Dancing Paths. Maciej Dunajski. SIGMA 18 (2022), 027, 13 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Given a path geometry on a surface , we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on . This causal structure corresponds to a conformal structure if and only if is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds to an SL(2, ℝ)-invariant projective structure where the paths are ellipses of area π centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane.
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| first_indexed | 2026-03-21T05:15:25Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211641 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T05:15:25Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dunajski, Maciej 2026-01-07T13:43:26Z 2022 Twistor Theory of Dancing Paths. Maciej Dunajski. SIGMA 18 (2022), 027, 13 pages 1815-0659 2020 Mathematics Subject Classification: 32L25; 53A20 arXiv:2201.04717 https://nasplib.isofts.kiev.ua/handle/123456789/211641 https://doi.org/10.3842/SIGMA.2022.027 Given a path geometry on a surface , we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on . This causal structure corresponds to a conformal structure if and only if is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds to an SL(2, ℝ)-invariant projective structure where the paths are ellipses of area π centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane. This project originated from discussions with Gil Bor. I am very grateful to Gil for explaining the work [3] to me, and for sharing his geometric insight on Kepler and Hook ellipses. I thank the anonymous reviewers for their careful reading of the manuscript and many insightful suggestions. I also thank the Mathematics Research Center (CIMAT) in Guanajuato for its hospitality, where some of this research was done. My research has been partially supported by STFC grants ST/P000681/1 and ST/T000694/1. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Twistor Theory of Dancing Paths Article published earlier |
| spellingShingle | Twistor Theory of Dancing Paths Dunajski, Maciej |
| title | Twistor Theory of Dancing Paths |
| title_full | Twistor Theory of Dancing Paths |
| title_fullStr | Twistor Theory of Dancing Paths |
| title_full_unstemmed | Twistor Theory of Dancing Paths |
| title_short | Twistor Theory of Dancing Paths |
| title_sort | twistor theory of dancing paths |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211641 |
| work_keys_str_mv | AT dunajskimaciej twistortheoryofdancingpaths |