Contact Geometry of Curves
Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equ...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2009 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2009
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/149111 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Contact Geometry of Curves / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 30 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862741791105613824 |
|---|---|
| author | Vassiliou, P.J. |
| author_facet | Vassiliou, P.J. |
| citation_txt | Contact Geometry of Curves / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 30 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M,g) is described. For the special case in which the isometries of (M,g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M. The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space H3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.
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| first_indexed | 2025-12-07T20:21:45Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-149111 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T20:21:45Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Vassiliou, P.J. 2019-02-19T17:27:30Z 2019-02-19T17:27:30Z 2009 Contact Geometry of Curves / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 30 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53A35; 53A55; 58A15; 58A20; 58A30 https://nasplib.isofts.kiev.ua/handle/123456789/149111 Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M,g) is described. For the special case in which the isometries of (M,g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M. The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space H3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries. This paper is a contribution to the Special Issue “Elie Cartan and Differential Geometry”. I am indebted to the anonymous referees for insightful comments and for corrections which greatly improved the paper. Any remaining errors are mine. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Contact Geometry of Curves Article published earlier |
| spellingShingle | Contact Geometry of Curves Vassiliou, P.J. |
| title | Contact Geometry of Curves |
| title_full | Contact Geometry of Curves |
| title_fullStr | Contact Geometry of Curves |
| title_full_unstemmed | Contact Geometry of Curves |
| title_short | Contact Geometry of Curves |
| title_sort | contact geometry of curves |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149111 |
| work_keys_str_mv | AT vassilioupj contactgeometryofcurves |