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

SIGMA 3 (2007), 113, 10 pages      arXiv:0711.3746      https://doi.org/10.3842/SIGMA.2007.113
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Symmetries and Invariant Differential Pairings

Michael G. Eastwood
Department of Mathematics, University of Adelaide, SA 5005, Australia

Received November 14, 2007; Published online November 23, 2007

Abstract
The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings, explains how they naturally arise, and formulates various associated problems.

Key words: conformal invariance; differential pairing; symmetry.

pdf (209 kb)   ps (151 kb)   tex (14 kb)

References

  1. Baston R.J., Eastwood M.G., Invariant operators, in Twistors in Mathematics and Physics, London Math. Soc. Lecture Note Ser., Vol. 156, Cambridge University Press, 1990, 129-163.
  2. Baird P., Wood J.C., Harmonic morphisms between Riemannian manifolds, Oxford University Press, 2003.
  3. Boyer C.P., Kalnins E.G., Miller W. Jr., Symmetry and separation of variables for the Helmholtz and Laplace equations, Nagoya Math. J. 60 (1976), 35-80.
  4. Branson T.P., Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293-345.
  5. Branson T.P., Cap A., Eastwood M.G., Gover A.R., Prolongations of geometric overdetermined systems, Internat. J. Math. 17 (2006), 641-664, math.DG/0402100.
  6. Calderbank D.M.J., Diemer T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math. 537 (2001), 67-103, math.DG/0001158.
  7. Cap A., Slovák J., Soucek V., Invariant operators on manifolds with almost Hermitian symmetric structures. III. Standard operators, Differential Geom. Appl. 12 (2000), 51-84, math.DG/9812023.
  8. Duval C., Ovsienko V., Conformally equivariant quantum Hamiltonians, Selecta Math. 7 (2001), 291-230.
  9. Eastwood M.G., Notes on conformal differential geometry, Rendi. Circ. Mat. Palermo Suppl. 43 (1996), 57-76.
  10. Eastwood M.G., Higher symmetries of the Laplacian, Ann. Math. 161 (2005), 1645-1665, hep-th/0206233.
  11. Eastwood M.G., Leistner T., Higher symmetries of the square of the Laplacian, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Volumes, no. 144, Springer Verlag, 2007, 319-338, math.DG/0610610.
  12. Eastwood M.G., Rice J.W., Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys. 109 (1987), 207-228, Erratum, Comm. Math. Phys. 144 (1992), 213.
  13. Eastwood M.G., Somberg P., Soucek V., Symmetries of the Dirac operator, in preparation.
  14. Fegan H.D., Conformally invariant first order differential operators, Quart. J. Math. 27 (1976), 371-378.
  15. Fox D.J.F., Projectively invariant star products, Int. Math. Res. Not. 2005 (2005), no. 9, 461-510, math.DG/0504596.
  16. Kroeske J., Invariant differential pairings, Acta Math. Univ. Comenian., to appear, math.DG/0703866.
  17. Miller W. Jr., Symmetry and separation of variables, Addison-Wesley, 1977.
  18. Penrose R., Rindler W., Spinors and space-time, Vol. 1, Cambridge University Press, 1984.
  19. Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179-239.

Previous article   Next article   Contents of Volume 3 (2007)