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


SIGMA 5 (2009), 098, 27 pages      arXiv:0910.3646      https://doi.org/10.3842/SIGMA.2009.098
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Contact Geometry of Curves

Peter J. Vassiliou
Faculty of Information Sciences and Engineering, University of Canberra, 2601 Australia

Received May 07, 2009, in final form October 16, 2009; Published online October 19, 2009

Abstract
Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M,g) is described. For the special case in which the isometries of (M,g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M. The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space H3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.

Key words: moving frames; Goursat normal forms; curves; Riemannian manifolds.

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References

  1. Alekseev D.V., Vinogradov A.M., Lychagin V.V., Geometry I, Encycl. Math. Sci., Vol. 28, Spinger-Verlag, Berlin, 1991.
  2. Bryant R.L., Some aspects of the local and global theory of Pfaffian systems, PhD thesis, University of North Carolina, Chapel-Hill, 1979.
  3. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
  4. Cartan É., La méthode du repère mobile, la théorie des groupes continus et les espaces généralisés Hermann & Cie, Paris, 1935.
  5. Cartan É., La théorie des groupes finis et continus et la géométrie différentielle traitees par la méthode du repère mobile, Gautier-Villars, Paris, 1937.
  6. Chern S.S., Moving frames, in The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérique, Vol. 1985, Numero Hors Serie, 1985, 67-77.
  7. Chou K.-S., Qu C.-Z., Integrable equations arising from motions of plane curves, Phys. D 162 (2002), 9-33.
  8. Favard J., Cours de géométrie différentielle locale, Gauthier-Villars, Paris, 1957.
  9. Fels M., Olver P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
  10. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  11. Green M.L., The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J. 45 (1978), 735-779.
  12. Griffiths P.A., 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.
  13. Ivey T., Integrable geometric evolution equations for curves, in The Geometrical Study of Differential Equations (Washington, DC, 2000), Contemp. Math., Vol. 285, Amer. Math. Soc., Providence, RI, 2001, 71-84.
  14. Jensen G.R., Higher order contact of submanifolds of homogeneous spaces, Lecture Notes in Mathematics, Vol. 610, Springer-Verlag, Berlin - New York, 1977.
  15. Kogan I.A., Inductive approach to moving frames and applications in classical invariant theory, PhD Thesis, University of Minesota, 2000.
  16. Mansfield E.L., A guide to the symbolic invariant calculus, Cambridge University Press, Cambridge, to appear.
  17. Mansfield E.L., van der Kamp P.E., Evolution of curve invariants and lifting integrability, J. Geom. Phys. 56 (2006), 1294-1325.
  18. Olver P.J., Moving frames - in geometry, algebra, computer vision and numerical analysis, in Foundations of Computational Mathematics (Oxford, 1999), Editors R. DeVore, A. Iserles and E. Süli, London Math. Soc. Lecture Note Ser., Vol. 284, Cambridge University Press, Cambridge, 2001, 267-297.
  19. Olver P.J., Invariant submanifold flows, J. Phys. A: Math. Theor. 41 (2008), 344017, 22 pages.
  20. Shadwick W.F., Sluis W.M., Dynamic feedback for the classical geometries, in Differential Geometry and Mathematical Physics (Vancouver, BC, 1993), Contemp. Math., Vol. 170, Amer. Math. Soc., Providence, RI, 1994, 207-213.
  21. Sharpe R., Differential geometry. Cartan's generalisation of Klein's erlangen program, Graduate Texts in Mathematics, Springer-Verlag, New York, 1997.
  22. Spivak M., A comprehensive introduction to differential geometry, Vol. 1, Publish or Perish Press, 1970.
  23. Spivak M., A comprehensive introduction to differential geometry, Vol. 2, Publish or Perish Press, 1979.
  24. Streltsova I.S., R-conformal invariants of curves, Izv. Vyssh. Uchebn. Zaved. Mat. 53 (2009), no. 5, 67-69.
  25. Stormark O., Lie's structural approach to PDE systems, Encyclopedia of Mathematics and its Applications, Vol. 80, Cambridge University Press, Cambridge, 2000.
  26. Sulanke R., On É. Cartan's method of moving frames, Colloq. Math. Soc. Janos Bolyai, Vol. 31, Differential Geometry, Budapest, 1979.
  27. Vassiliou P., A constructive generalised Goursat normal form, Differential Geom. Appl. 24 (2006), 332-350, math.DG/0404377.
  28. Vassiliou P., Efficient construction of contact coordinates for partial prolongations, Found. Comput. Math. 6 (2006), 269-308, math.DG/0406234.
  29. Vessiot E., Sur l'intégration des faisceaux de transformations infinitésimales dans le cas où, le degré du faisceau étant n, celui du faisceau derivée est n+1, Ann. Sci. École Norm. Sup. (3) 45 (1928), 189-253.
  30. Yamaguchi K., Contact geometry of higher order, Japan. J. Math. (N.S.) 8 (1982), 109-176.
    Yamaguchi K., Geometrization of jet bundles, Hokkaido Math. J. 12 (1983), 27-40.

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