On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in G-spaces, whether homogeneous or not, provided that a certain kth order jet bundle over the G-space admits a G-invariant local coframe field of constant...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149192 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces / J. Cheh // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862718430375837696 |
|---|---|
| author | Cheh, J. |
| author_facet | Cheh, J. |
| citation_txt | On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces / J. Cheh // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 16 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in G-spaces, whether homogeneous or not, provided that a certain kth order jet bundle over the G-space admits a G-invariant local coframe field of constant structure. As a corollary, we note that the differential order of a minimal complete set of congruence invariants is bounded by k+1. We demonstrate the method by rediscovering the speed and curvature invariants of Euclidean planar curves, the Schwarzian derivative of holomorphic immersions in the complex projective line, and equivalents of the first and second fundamental forms of surfaces in R³ subject to rotations.
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| first_indexed | 2025-12-07T18:14:57Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149192 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:14:57Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cheh, J. 2019-02-19T18:27:40Z 2019-02-19T18:27:40Z 2013 On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces / J. Cheh // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A55; 53B25 DOI: http://dx.doi.org/10.3842/SIGMA.2013.036 https://nasplib.isofts.kiev.ua/handle/123456789/149192 We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in G-spaces, whether homogeneous or not, provided that a certain kth order jet bundle over the G-space admits a G-invariant local coframe field of constant structure. As a corollary, we note that the differential order of a minimal complete set of congruence invariants is bounded by k+1. We demonstrate the method by rediscovering the speed and curvature invariants of Euclidean planar curves, the Schwarzian derivative of holomorphic immersions in the complex projective line, and equivalents of the first and second fundamental forms of surfaces in R³ subject to rotations. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants
 and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
 This work has benefited from the discussions held in the Dif ferential Geometry and Lie Theory
 seminars at the University of Toledo; the author would like to thank the organizers and participants of the seminars. Also, the anonymous referees’ critical and yet helpful comments have
 contributed significantly in the process of revising and improving the paper; the author is very
 grateful to the referees.
 It is hoped that this work serves to reflect, although only to a small extent limited by
 the author’s meager knowledge, the author’s appreciation of the introduction by Professor Peter
 Olver to the marvelous unifying philosophy and technology of symmetry, invariance, and equivalence. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces Article published earlier |
| spellingShingle | On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces Cheh, J. |
| title | On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces |
| title_full | On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces |
| title_fullStr | On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces |
| title_full_unstemmed | On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces |
| title_short | On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces |
| title_sort | on local congruence of immersions in homogeneous or nonhomogeneous spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149192 |
| work_keys_str_mv | AT chehj onlocalcongruenceofimmersionsinhomogeneousornonhomogeneousspaces |