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

SIGMA 9 (2013), 029, 43 pages      arXiv:1304.1616
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Solving Local Equivalence Problems with the Equivariant Moving Frame Method

Francis Valiquette
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada

Received July 21, 2012, in final form March 31, 2013; Published online April 05, 2013

Given a Lie pseudo-group action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the equivariant moving frame constructions extend to submanifold jets where the pseudo-group does not act freely at any order. Once this is done, we review the solution to the local equivalence problem of submanifolds within the equivariant moving frame framework. This offers an alternative approach to Cartan's equivalence method based on the theory of G-structures.

Key words: differential invariant; equivalence problem; Maurer-Cartan form; moving frame.

pdf (855 kb)   tex (227 kb)


  1. Akivis M.A., Rosenfeld B.A., Élie Cartan (1869-1951), Translations of Mathematical Monographs, Vol. 123, Amer. Math. Soc., Providence, RI, 1993.
  2. 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.
  3. Cartan E., La structure des groupes infinis, in Oeuvres Complètes, Part. II, Vol. 2, Gauthier-Villars, Paris, 1953, 1335-1384.
  4. Cartan E., Sur la structure des groupes infinis de transformations, in Oeuvres Complètes, Part. II, Vol. 2, Gauthier-Villars, Paris, 1953, 571-714.
  5. Cartan E., Sur les variétés à connexion projective, in Oeuvres Complètes, Part. III, Vol. 1, Gauthier-Villars, Paris, 1955, 825-861.
  6. Cheh J., Olver P.J., Pohjanpelto J., Algorithms for differential invariants of symmetry groups of differential equations, Found. Comput. Math. 8 (2008), 501-532.
  7. Cheh J., Olver P.J., Pohjanpelto J., Maurer-Cartan equations for Lie symmetry pseudogroups of differential equations, J. Math. Phys. 46 (2005), 023504, 11 pages.
  8. Cox D., Little J., O'Shea D., Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007.
  9. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  10. Gardner R.B., The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
  11. Gardner R.B., Shadwick W.F., An equivalence problem for a two-form and a vector field on R3, in Differential Geometry, Global Analysis, and Topology (Halifax, NS, 1990), CMS Conf. Proc., Vol. 12, Amer. Math. Soc., Providence, RI, 1991, 41-50.
  12. Guillemin V., Sternberg S., Deformation theory of pseudogroup structures, Mem. Amer. Math. Soc. 64 (1966), 80 pages.
  13. Hubert E., Differential invariants of a Lie group action: syzygies on a generating set, J. Symbolic Comput. 44 (2009), 382-416, arXiv:0710.4318.
  14. Johnson H.H., Classical differential invariants and applications to partial differential equations, Math. Ann. 148 (1962), 308-329.
  15. Kamran N., Contributions to the study of the equivalence problem of Élie Cartan and its applications to partial and ordinary differential equations, Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8o (2) 45 (1989), no. 7, 122 pages.
  16. Kogan I.A., Olver P.J., Invariant Euler-Lagrange equations and the invariant variational bicomplex, Acta Appl. Math. 76 (2003), 137-193.
  17. Kruglikov B., Lychagin V., Global Lie-Tresse theorem, arXiv:1111.5480.
  18. Kruglikov B., Lychagin V., Invariants of pseudogroup actions: homological methods and finiteness theorem, Int. J. Geom. Methods Mod. Phys. 3 (2006), 1131-1165, math.DG/0511711.
  19. Kumpera A., Invariants différentiels d'un pseudogroupe de Lie. I, J. Differential Geometry 10 (1975), 289-345.
  20. Kuranishi M., On the local theory of continuous infinite pseudo groups. I, Nagoya Math. J 15 (1959), 225-260.
  21. Lie S., Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestatten I-IV, in Gesammelte Abhandlungen, Vol. 5, B.G. Teubner, Leipzig, 1924, 240-310, 362-427, 432-448.
  22. Lie S., Scheffers G., Vorlesungen über Continuierliche Guppen mit Geometrischen und Anderen Anwendungen, B.G. Teubner, Leipzig, 1893.
  23. Mackenzie K., Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, Vol. 124, Cambridge University Press, Cambridge, 1987.
  24. Milson R., Valiquette F., Point equivalence of second-order odes: maximal classifying order, arXiv:1208.1014.
  25. Muñoz J., Muriel F.J., Rodríguez J., On the finiteness of differential invariants, J. Math. Anal. Appl. 284 (2003), 266-282.
  26. Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
  27. Olver P.J., Moving frames and singularities of prolonged group actions, Selecta Math. (N.S.) 6 (2000), 41-77.
  28. Olver P.J., Pohjanpelto J., Differential invariant algebras of Lie pseudo-groups, Adv. Math. 222 (2009), 1746-1792.
  29. Olver P.J., Pohjanpelto J., Maurer-Cartan forms and the structure of Lie pseudo-groups, Selecta Math. (N.S.) 11 (2005), 99-126.
  30. Olver P.J., Pohjanpelto J., Moving frames for Lie pseudo-groups, Canad. J. Math. 60 (2008), 1336-1386.
  31. Olver P.J., Pohjanpelto J., Persistence of freeness for Lie pseudogroup actions, Ark. Mat. 50 (2012), 165-182, arXiv:0912.4501.
  32. Olver P.J., Recursive moving frames, Results Math. 60 (2011), 423-452.
  33. Olver P.J., Pohjanpelto J., Valiquette F., On the structure of Lie pseudo-groups, SIGMA 5 (2009), 077, 14 pages, arXiv:0907.4086.
  34. Pohjanpelto J., Reduction of exterior differential systems with infinite dimensional symmetry groups, BIT Num. Math. 48 (2008), 337-355.
  35. Seiler W.M., Involution. The formal theory of differential equations and its applications in computer algebra, Algorithms and Computation in Mathematics, Vol. 24, Springer-Verlag, Berlin, 2010.
  36. Shemyakova E., Mansfield E.L., Moving frames for Laplace invariants, in Proceedings ISSAC 2008, Editor D. Jeffrey, ACM, New York, 2008, 295-302.
  37. Singer I.M., Sternberg S., The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1-114.
  38. Thompson R., Valiquette F., Group foliation of differential equations using moving frames, Preprint, Dalhousie University, 2012.
  39. Thompson R., Valiquette F., On the cohomology of the invariant Euler-Lagrange complex, Acta Appl. Math. 116 (2011), 199-226.
  40. Tresse A., Détermination des invariants ponctuels de l'équation différentielle ordinaire de second ordre y''=ω(x,y,y'), Hirzel, Leipzig, 1896.
  41. Tresse A., Sur les invariants différentiels des groupes continus de transformations, Acta Math. 18 (1894), 1-88.
  42. Valiquette F., Structure equations of Lie pseudo-groups, J. Lie Theory 18 (2008), 869-895.

Previous article  Next article   Contents of Volume 9 (2013)