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SIGMA 9 (2013), 036, 21 pages arXiv:1304.7430
https://doi.org/10.3842/SIGMA.2013.036
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”
On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
Jeongoo Cheh
Department of Mathematics & Statistics, The University of Toledo, Toledo, OH 43606, USA
Received May 14, 2012, in final form April 19, 2013; Published online April 28, 2013
Abstract
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.
Key words:
congruence; nonhomogeneous space; equivariant moving frame; constant-structure invariant coframe field.
pdf (441 kb)
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References
- Anderson I.M., Introduction to the variational bicomplex, in Mathematical
Aspects of Classical Field Theory (Seattle, WA, 1991), Contemp.
Math., Vol. 132, Amer. Math. Soc., Providence, RI, 1992, 51-73.
- Fels M., Olver P.J., Moving coframes. I. A practical algorithm,
Acta Appl. Math. 51 (1998), 161-213.
- Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical
foundations, Acta Appl. Math. 55 (1999), 127-208.
- Green M.L., The moving frame, differential invariants and rigidity theorems for
curves in homogeneous spaces, Duke Math. J. 45 (1978),
735-779.
- Griffiths P., On Cartan's method of Lie groups and moving frames as applied
to uniqueness and existence questions in differential geometry, Duke
Math. J. 41 (1974), 775-814.
- Hubert E., Olver P.J., Differential invariants of conformal and projective
surfaces, SIGMA 3 (2007), 097, 15 pages,
arXiv:0710.0519.
- Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via
moving frames and exterior differential systems, Graduate Studies in
Mathematics, Vol. 61, Amer. Math. Soc., Providence, RI, 2003.
- Kogan I.A., Olver P.J., Invariant Euler-Lagrange equations and the
invariant variational bicomplex, Acta Appl. Math. 76
(2003), 137-193.
- Olver P.J., Applications of Lie groups to differential equations, 2nd ed.,
Graduate Texts in Mathematics, Vol. 107, Springer-Verlag,
New York, 1993.
- Olver P.J., Differential invariants of surfaces, Differential Geom.
Appl. 27 (2009), 230-239.
- Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press,
Cambridge, 1995.
- Olver P.J., Moving frames and differential invariants in centro-affine
geometry, Lobachevskii J. Math. 31 (2010), 77-89.
- Olver P.J., Pohjanpelto J., Differential invariant algebras of Lie
pseudo-groups, Adv. Math. 222 (2009), 1746-1792.
- Olver P.J., Pohjanpelto J., Moving frames for Lie pseudo-groups,
Canad. J. Math. 60 (2008), 1336-1386.
- Sternberg S., Lectures on differential geometry, 2nd ed., Chelsea Publishing
Co., New York, 1983.
- Warner F.W., Foundations of differentiable manifolds and Lie groups,
Graduate Texts in Mathematics, Vol. 94, Springer-Verlag, New York,
1983.
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