Geodesic
On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 $k^{\mathrm{th}}$ 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 $\mathbb{R}^3$ subject to rotations.
Keywords: congruence; nonhomogeneous space; equivariant moving frame; constant-structure invariant coframe field. 
@article{SIGMA_2013_9_a35,
     author = {Jeongoo Cheh},
     title = {On {Local} {Congruence} of {Immersions} in {Homogeneous} or {Nonhomogeneous} {Spaces}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2013},
     volume = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a35/}
}
TY  - JOUR
AU  - Jeongoo Cheh
TI  - On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2013
VL  - 9
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a35/
LA  - en
ID  - SIGMA_2013_9_a35
ER  - 
%0 Journal Article
%A Jeongoo Cheh
%T On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
%J Symmetry, integrability and geometry: methods and applications
%D 2013
%V 9
%U http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a35/
%G en
%F SIGMA_2013_9_a35
Jeongoo Cheh. On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a35/

[1] Anderson I. M., “Introduction to the variational bicomplex”, Mathematical Aspects of Classical Field Theory (Seattle, WA, 1991), Contemp. Math., 132, Amer. Math. Soc., Providence, RI, 1992, 51–73 | DOI | MR

[2] Fels M., Olver P. J., “Moving coframes. I: A practical algorithm”, Acta Appl. Math., 51 (1998), 161–213 | DOI | MR

[3] Fels M., Olver P. J., “Moving coframes. II: Regularization and theoretical foundations”, Acta Appl. Math., 55 (1999), 127–208 | DOI | MR | Zbl

[4] Green M. L., “The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces”, Duke Math. J., 45 (1978), 735–779 | DOI | MR | Zbl

[5] 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 | DOI | MR | Zbl

[6] Hubert E., Olver P. J., “Differential invariants of conformal and projective surfaces”, SIGMA, 3 (2007), 097, 15 pp., arXiv: 0710.0519 | DOI | MR | Zbl

[7] Ivey T. A., Landsberg J. M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, 61, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[8] Kogan I. A., Olver P. J., “Invariant Euler–Lagrange equations and the invariant variational bicomplex”, Acta Appl. Math., 76 (2003), 137–193 | DOI | MR | Zbl

[9] Olver P. J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, 107, 2nd ed., Springer-Verlag, New York, 1993 | MR | Zbl

[10] Olver P. J., Differential invariants of surfaces, 27 (2009), 230–239 | DOI | MR | Zbl

[11] Olver P. J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[12] Olver P. J., “Moving frames and differential invariants in centro-affine geometry”, Lobachevskii J. Math., 31 (2010), 77–89 | DOI | MR | Zbl

[13] Olver P. J., Pohjanpelto J., “Differential invariant algebras of Lie pseudo-groups”, Adv. Math., 222 (2009), 1746–1792 | DOI | MR | Zbl

[14] Olver P. J., Pohjanpelto J., “Moving frames for Lie pseudo-groups”, Canad. J. Math., 60 (2008), 1336–1386 | DOI | MR | Zbl

[15] Sternberg S., Lectures on differential geometry, 2nd ed., Chelsea Publishing Co., New York, 1983 | MR | Zbl

[16] Warner F. W., Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94, Springer-Verlag, New York, 1983 | MR | Zbl