Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 036, 21 pages      arXiv:1304.7430      http://dx.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.

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References

  1. 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.
  2. Fels M., Olver P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
  3. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  4. Green M.L., The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J. 45 (1978), 735-779.
  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.
  6. Hubert E., Olver P.J., Differential invariants of conformal and projective surfaces, SIGMA 3 (2007), 097, 15 pages, arXiv:0710.0519.
  7. 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.
  8. Kogan I.A., Olver P.J., Invariant Euler-Lagrange equations and the invariant variational bicomplex, Acta Appl. Math. 76 (2003), 137-193.
  9. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1993.
  10. Olver P.J., Differential invariants of surfaces, Differential Geom. Appl. 27 (2009), 230-239.
  11. Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
  12. Olver P.J., Moving frames and differential invariants in centro-affine geometry, Lobachevskii J. Math. 31 (2010), 77-89.
  13. Olver P.J., Pohjanpelto J., Differential invariant algebras of Lie pseudo-groups, Adv. Math. 222 (2009), 1746-1792.
  14. Olver P.J., Pohjanpelto J., Moving frames for Lie pseudo-groups, Canad. J. Math. 60 (2008), 1336-1386.
  15. Sternberg S., Lectures on differential geometry, 2nd ed., Chelsea Publishing Co., New York, 1983.
  16. 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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