Abstract
The primary goal of this paper is to provide a rigorous theoretical justification of Cartan’s method of moving frames for arbitrary finite-dimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential operators on jet bundles, as well as equivalence, symmetry, and rigidity theorems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential invariants, explicit commutation formulae for the associated invariant differential operators, and a general classification theorem for syzygies of the higher order differentiated differential invariants. A variety of illustrative examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations.
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References
Anderson, I. M.: Introduction to the variational bicomplex, Contemp. Math. 132 (1992), 51-73.
Anderson, I. M. and Torre, C. G.: Two component spinors and natural coordinates for the prolonged Einstein equation manifolds, Preprint, Utah State University, 1997.
Atiyah, M. F. and Bott, R.: The moment map and equivariant cohomology, Topology 23 (1984), 1-28.
Bryant, R. L., Chern, S.-S., Gardner, R. B., Goldschmidt, H. L. and Griffiths, P. A.: Exterior Differential Systems, Math. Sci. Res. Inst. Publ., Vol. 18, Springer-Verlag, New York, 1991.
Cartan, É.: La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés, Exposés de Géométrie No. 5, Hermann, Paris, 1935.
Cartan, É.: Leçons sur la théorie des espaces à connexion projective, Cahiers Scientifiques, Vol. 17, Gauthier-Villars, Paris, 1937.
Cartan, É.: La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Cahiers Scientifiques, Vol. 18, Gauthier-Villars, Paris, 1937.
Cartan, É.: Les problèmes d'équivalence, in: Oeuvres complètes, Part. II, Vol. 2, Gauthier-Villars, Paris, 1953, pp. 1311-1334.
Fels, M. and Olver, P. J.: Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
Gardner, R. B.: The Method of Equivalence and Its Applications, SIAM, Philadelphia, 1989.
Green, M. L.: The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J. 45 (1978), 735-779.
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.
Husemoller, D.: Fiber Bundles, McGraw-Hill, New York, 1966.
Jensen, G. R.: Higher Order Contact of Submanifolds of Homogeneous Spaces, Lecture Notes in Math. 610, Springer-Verlag, New York, 1977.
Lisle, I.: Equivalence transformations for classes of differential equations, PhD Thesis, University of British Columbia, Vancouver, 1992.
Olver, P. J.: Symmetry groups and group invariant solutions of partial differential equations, J. Differential Geom. 14 (1979), 497-542.
Olver, P. J.: Applications of Lie Groups to Differential Equations, 2nd edn, Graduate Texts in Math. 107, Springer-Verlag, New York, 1993.
Olver, P. J.: Equivalence, Invariants, and Symmetry, Cambridge Univ. Press, Cambridge, 1995.
Olver, P. J.: Non-associative local Lie groups, J. Lie Theory 6 (1996), 23-51.
Olver, P. J.: Singularities of prolonged group actions on jet bundles, Preprint, University of Minnesota, 1998.
Ondich, J.: A differential constraints approach to partial invariance, Europ. J. Appl. Math. 6 (1995), 631-638.
Ovsiannikov, L. V.: Group Analysis of Differential Equations, Academic Press, New York, 1982.
Sternberg, S.: Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, 1964.
Thomas, T. Y.: The Differential Invariants of Generalized Spaces, Chelsea Publ. Co., New York, 1991.
Tresse, A.: Sur les invariants différentiels des groupes continus de transformations, Acta Math. 18 (1894), 1-88.
Tsujishita, T.: On variational bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), 311-363.
Weyl, H.: Cartan on groups and differential geometry, Bull. Amer. Math. Soc. 44 (1938), 598-601.
Weyl, H.: Classical Groups, Princeton Univ. Press, Princeton, NJ, 1946.
Zharinov, V. V.: Geometrical Aspects of Partial Differential Equations, World Scientific, Singapore, 1992.
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Fels, M., Olver, P.J. Moving Coframes: II. Regularization and Theoretical Foundations. Acta Applicandae Mathematicae 55, 127–208 (1999). https://doi.org/10.1023/A:1006195823000
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DOI: https://doi.org/10.1023/A:1006195823000