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Differential Invariants of Conformal and Projective Surfaces
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.
Keywords: conformal differential geometry; projective differential geometry; differential invariants; moving frame; syzygy; differential algebra. 
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Evelyne Hubert; Peter J. Olver. Differential Invariants of Conformal and Projective Surfaces. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a96/

[1] Akivis M. A., Goldberg V. V., Projective differential geometry of submanifolds, North-Holland Mathematical Library, 49, North-Holland Publishing Co., Amsterdam, 1993 | MR | Zbl

[2] Akivis M. A., Goldberg V. V., Conformal differential geometry and its generalizations, Pure and Applied Mathematics (New York), John Wiley Sons Inc., New York, 1996 | MR | Zbl

[3] Bailey T. N., Eastwood M. G., Graham C. R., “Invariant theory for conformal and CR geometry”, Ann. Math., 139 (1994), 491–552 | DOI | MR | Zbl

[4] Boulier F., Hubert E., Diffalg: description, help pages and examples of use, Symbolic Computation Group, University of Waterloo, Ontario, Canada, 1998; http://www.inria.fr/cafe/Evelyne.Hubert/diffalg

[5] Cartan E., La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobil, Cahiers scientifiques. Number 18, Gauthier-Villars, Paris, 1937

[6] Fels M., Olver P. J., “Moving coframes. II. Regularization and theoretical foundations”, Acta Appl. Math., 55 (1999), 127–208 | DOI | MR | Zbl

[7] Fefferman C., Graham C. R., “Conformal invariants”, Élie Cartan et les Mathématiques d'Aujourd'hui, hors série, Astérisque, Soc. Math. France, Paris, 1985, 95–116 | MR

[8] Fubini G., Čech E., Introduction à la Géométrie Projective Différentielle des Surfaces, Gauthier-Villars, Paris, 1931

[9] Green M. L., “The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces”, Duke Math. J., 45 (1978), 735–779 | DOI | MR | Zbl

[10] 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 | DOI | MR | Zbl

[11] Guggenheimer H. W., Differential geometry, McGraw-Hill Book Co., Inc., New York, 1963 | MR | Zbl

[12] Hubert E., “Notes on triangular sets and triangulation-decomposition algorithms. II: Differential systems”, Symbolic and Numerical Scientific Computing, Lecture Notes in Computer Science, 2630, eds. F. Winkler and U. Langer, Springer Verlag, Heidelberg, 2003, 40–87 | MR | Zbl

[13] Hubert E., Diffalg: extension to non commuting derivations,, INRIA, Sophia Antipolis, 2005; http://www.inria.fr/cafe/Evelyne.Hubert/diffalg

[14] Hubert E., “Differential algebra for derivations with nontrivial commutation rules”, J. Pure Applied Algebra, 200 (2005), 163–190 | DOI | MR | Zbl

[15] Hubert E., The maple package ALDA – algebraic invariants and their differential algebras, , INRIA, 2007 http://www.inria.fr/cafe/Evelyne.Hubert/aida

[16] Hubert E., Differential invariants of a Lie group action: syzygies on a generating set, in preparation

[17] Hubert E., Generation properties of Maurer–Cartan invariants, in preparation

[18] Hubert E., Kogan I. A., “Rational invariants of a group action. Construction and rewriting”, J. Symbolic Comput., 42 (2007), 203–217 | DOI | MR | Zbl

[19] Hubert E., Kogan I. A., “Smooth and algebraic invariants of a group action. Local and global constructions”, Found. Comput. Math., 7:4 (2007), 455–493 | DOI | MR | Zbl

[20] Ivey T. A., Landsberg J. M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, 61, American Mathematical Society, Providence, RI, 2003 | MR | Zbl

[21] Jensen G., Higher order contact of submanifolds of homogeneous spaces, Lecture Notes in Mathematics, 610, Springer-Verlag, Berlin – New York, 1977 | MR | Zbl

[22] Kogan I. A., Olver P. J., “Invariant Euler–Lagrange equations and the invariant variational bicomplex”, Acta Appl. Math., 76 (2003), 137–193 | DOI | MR | Zbl

[23] Lie S., Vorlesungen über Continuierliche Gruppen mit Geometrischen und anderen Anwendungen,, Chelsea Publishing Co., Bronx, New York, 1971 | MR | Zbl

[24] Marí Beffa G., “The theory of differential invariants and KdV Hamiltonian evolutions”, Bull. Soc. Math. France, 127 (1999), 363–391 | MR | Zbl

[25] Marí Beffa G., “Relative and absolute differential invariants for conformal curves”, J. Lie Theory, 13 (2003), 213–245 | MR | Zbl

[26] Marí Beffa G., “Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds”, Ann. Institut Fourier, 58:4 (2008), 1295–1335 | MR | Zbl

[27] Marí Beffa G., Olver P. J., “Differential invariants for parametrized projective surfaces”, Comm. Anal. Geom., 7 (1999), 807–839 | MR | Zbl

[28] Mansfield E. L., “Algorithms for symmetric differential systems”, Found. Comput. Math., 1 (2001), 335–383 | MR | Zbl

[29] Olver P. J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986 | MR | Zbl

[30] Olver P. J., Equivalence, invariants and symmetry, Cambridge University Press, 1995 | MR | Zbl

[31] Olver P. J., “Moving frames and singularities of prolonged group actions”, Selecta Math. (N.S.), 6 (2000), 41–77 | DOI | MR | Zbl

[32] Olver P. J., “A survey of moving frames”, Computer Algebra and Geometric Algebra with Applications, Lecture Notes in Computer Science, 3519, eds. H. Li, P. J. Olver and G. Sommer, Springer–Verlag, New York, 2005, 105–138 | Zbl

[33] Olver P. J., “Generating differential invariants”, J. Math. Anal. Appl., 333 (2007), 450–471 | DOI | MR | Zbl

[34] Olver P. J., Differential invariants of surfaces, Preprint, University of Minnesota, 2007 | MR

[35] Ovsiannikov L. V., Group analysis of differential equations, Academic Press, New York, 1982 | MR | Zbl

[36] Simon U., “The Pick invariant in equiaffine differential geometry”, Abh. Math. Sem. Univ. Hamburg, 53 (1983), 225–228 | DOI | MR | Zbl

[37] Spivak M., A comprehensive introduction to differential geometry, Vol. III, 2nd ed., Publish or Perish Inc., Wilmington, Del., 1979 | MR

[38] Tresse A., “Sur les invariants des groupes continus de transformations”, Acta Math., 18 (1894), 1–88 | DOI | MR

[39] Vessiot E., “Contribution à la géométrie conforme. Théorie des surfaces. I”, Bull. Soc. Math. France, 54 (1926), 139–179 | MR | Zbl