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On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
Репозиторій DSpace/Manakin
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- Фізико-технічні та математичні науки
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2013, том 9
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On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
Інший ID:
2010 Mathematics Subject Classification: 53A55; 53B25
DOI: http://dx.doi.org/10.3842/SIGMA.2013.036
DOI: http://dx.doi.org/10.3842/SIGMA.2013.036
Посилання:
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.
Підтримка:
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.
Дата:
2013
Переглядів:
502
Завантажень:
195
Анотація:
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.
