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


SIGMA 3 (2007), 097, 15 pages      arXiv:0710.0519      http://dx.doi.org/10.3842/SIGMA.2007.097
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Differential Invariants of Conformal and Projective Surfaces

Evelyne Hubert a and Peter J. Olver b
a) INRIA, 06902 Sophia Antipolis, France
b) School of Mathematics, University of Minnesota, Minneapolis 55455, USA

Received August 15, 2007, in final form September 24, 2007; Published online October 02, 2007

Abstract
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.

Key words: conformal differential geometry; projective differential geometry; differential invariants; moving frame; syzygy; differential algebra.

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References

  1. Akivis M.A., Goldberg V.V., Projective differential geometry of submanifolds, North-Holland Mathematical Library, Vol. 49, North-Holland Publishing Co., Amsterdam, 1993.
  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.
  3. Bailey T.N., Eastwood M.G., Graham C.R., Invariant theory for conformal and CR geometry, Ann. Math. 139 (1994), 491-552.
  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.
  7. Fefferman C., Graham C.R., Conformal invariants, in Élie Cartan et les Mathématiques d'Aujourd'hui, Astérisque, hors série, Soc. Math. France, Paris, 1985, 95-116.
  8. Fubini G., Cech 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.
  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.
  11. Guggenheimer H.W., Differential geometry, McGraw-Hill Book Co., Inc., New York, 1963.
  12. Hubert E., Notes on triangular sets and triangulation-decomposition algorithms II: Differential systems, in Symbolic and Numerical Scientific Computing, Editors F. Winkler and U. Langer, Lecture Notes in Computer Science, no. 2630, Springer Verlag Heidelberg, 2003, 40-87.
  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.
  15. Hubert E., The MAPLE package AIDA - 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.
  19. Hubert E., Kogan I.A., Smooth and algebraic invariants of a group action. Local and global constructions, Found. Comput. Math., in press.
  20. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, American Mathematical Society, Providence, RI, 2003.
  21. Jensen G., Higher order contact of submanifolds of homogeneous spaces, Lecture Notes in Mathematics, Vol. 610, Springer-Verlag, Berlin - New York, 1977.
  22. Kogan I.A., Olver P.J., Invariant Euler-Lagrange equations and the invariant variational bicomplex, Acta Appl. Math. 76 (2003), 137-193.
  23. Lie S., Vorlesungen über Continuierliche Gruppen mit Geometrischen und anderen Anwendungen, Chelsea Publishing Co., Bronx, New York, 1971.
  24. Marí Beffa G., The theory of differential invariants and KdV Hamiltonian evolutions, Bull. Soc. Math. France 127 (1999), 363-391.
  25. Marí Beffa G., Relative and absolute differential invariants for conformal curves, J. Lie Theory 13 (2003), 213-245.
  26. Marí Beffa G., Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, Ann. Institut Fourier, to appear.
  27. Marí Beffa G., Olver P.J., Differential invariants for parametrized projective surfaces, Comm. Anal. Geom. 7 (1999), 807-839.
  28. Mansfield E.L., Algorithms for symmetric differential systems, Found. Comput. Math. 1 (2001), 335-383.
  29. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, no. 107, Springer-Verlag, New York, 1986.
  30. Olver P.J., Equivalence, invariants and symmetry, Cambridge University Press, 1995.
  31. Olver P.J., Moving frames and singularities of prolonged group actions, Selecta Math. (N.S.) 6 (2000), 41-77.
  32. Olver P.J., A survey of moving frames, in Computer Algebra and Geometric Algebra with Applications, Editors H. Li, P.J. Olver and G. Sommer, Lecture Notes in Computer Science, Vol. 3519, Springer-Verlag, New York, 2005, 105-138.
  33. Olver P.J., Generating differential invariants, J. Math. Anal. Appl. 333 (2007), 450-471.
  34. Olver P.J., Differential invariants of surfaces, Preprint, University of Minnesota, 2007.
  35. Ovsiannikov L.V., Group analysis of differential equations, Academic Press, New York, 1982.
  36. Simon U., The Pick invariant in equiaffine differential geometry, Abh. Math. Sem. Univ. Hamburg 53 (1983), 225-228.
  37. Spivak M., A comprehensive introduction to differential geometry, Vol. III, 2nd ed., Publish or Perish Inc., Wilmington, Del., 1979.
  38. Tresse A., Sur les invariants des groupes continus de transformations. Acta Math. 18 (1894), 1-88.
  39. Vessiot E., Contribution à la géométrie conforme. Théorie des surfaces. I, Bull. Soc. Math. France 54 (1926), 139-179.

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