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Two Algorithms for a Moving Frame Construction

Published online by Cambridge University Press:  20 November 2018

Irina A. Kogan
Irina A. Kogan*
Affiliation:
Department of Mathematics, Yale University, New Haven, Connecticut 06520, USA, e-mail: [email protected]
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Abstract

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The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.

Keywords

53A5558D1968U10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 2 , 01 April 2003 , pp. 266 - 291
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Anderson, I. M., The Vessiot Handbook; Technical Report. Utah Sate University, 2000.Google Scholar
[2] Anderson, I. M. and Fels, M. E., Symmetry reduction of variational bicomplexes and the principle of symmetric criticality. Amer. J. Math. 119 (1997), 609–670.Google Scholar
[3] Blaschke, W., Vorlesungen über Differentialgeometrie, vol. II. Springer, Berlin, 1923.Google Scholar
[4] Bredon, G. E., Introduction to compact transformation groups. Academic Press, New York, 1972.Google Scholar
[5] Cartan, É., La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés. Exposés de Géométrie 5, Hermann, Paris, 1935.Google Scholar
[6] Cartan, É., Groupes finis et continus et la géométrie différentielle traitées par la methode du repère mobile. Gauthier-Villars, Paris, 1937.Google Scholar
[7] Cartan, É., Leçons sur la géométrie projective complexe. Gauthier-Villars, Paris, 1950.Google Scholar
[8] Faugeras, O., Cartan's moving frame method and its application to the geometry and evolution of curves in the Euclidean, affine and projective planes. In: Application of Invariance in Computer Vision, (eds., J. L. Mundy, A. Zisserman, D. Forsyth), Springer-Verlag Lecture Notes in Computer Science 825, 11–46, 1994.Google Scholar
[9] Fels, M. and Olver, P. J., On relative invariants. Math. Ann. 308 (1997), 701–732.Google Scholar
[10] Fels, M. and Olver, P. J., Moving Coframes. I. A Practical Algorithm. Acta Appl. Math. 51 (1998), 161–213.Google Scholar
[11] Fels, M. and Olver, P. J., Moving Coframes. II. Regularization and Theoretical Foundations. Acta Appl. Math. 55 (1999), 127–208.Google Scholar
[12] Gardner, R. B., The method of equivalence and its applications. SIAM, Philadelphia, 1989.Google Scholar
[13] Gorbatsevich, V. V., Onishchik, A. L., and Vinberg, E. B., Foundations of Lie Theory and Lie Transformations Groups. Springer, 1993.Google Scholar
[14] Green, M. L., The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces. Duke Math. J. 45 (1978), 735–779.Google Scholar
[15] Griffiths, P. A., On Cartan's method of Lie groups as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41 (1974), 775–814.Google Scholar
[16] Guggenheimer, H. W., Differential Geometry. McGraw-Hill, New York, 1963.Google Scholar
[17] Kamran, N., Contributions to the study of the equivalence problem of Elie Cartan and its applications to partial and ordinary differential equations. Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8o (2) (7) 45(1989).Google Scholar
[18] Kogan, I. A. and Olver, P. J., The invariant variational bicomplex. ContemporaryMath. 285 (2001), 131–144.Google Scholar
[19] Olver, P. J., Equivalence Invariants and Symmetry. Cambridge University Press, 1995.Google Scholar
[20] Olver, P. J., Joint invariant signatures. Found. Comput. Math. 1 (2001), 3–67.Google Scholar
[21] Ovsiannikov, L. V., Group Analysis of Differential Equations. Academic Press, New York, 1982.Google Scholar
[22] Sapiro, G. and Tannenbaum, A., On affine plane curve evolution. J. Funct. Anal. 119 (1994), 79–120.Google Scholar
[23] Spivak, M., A comprehensive introduction to differential geometry, vol. V. Publish or Perish, 1975.Google Scholar
[24] Vinberg, E. B. and Popov, V. L., Invariant theory. Algebraic geometry 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, VINITI, Moscow, 1989.Google Scholar