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


SIGMA 3 (2007), 084, 14 pages      arXiv:0708.4172      http://dx.doi.org/10.3842/SIGMA.2007.084
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Monogenic Functions in Conformal Geometry

Michael Eastwood a and John Ryan b
a) Department of Mathematics, University of Adelaide, SA 5005, Australia
b) Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA

Received August 29, 2007; Published online August 30, 2007

Abstract
Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition.

Key words: Clifford analysis; monogenic functions; Dirac operator; conformal invariance.

pdf (262 kb)   ps (187 kb)   tex (19 kb)

References

  1. Baston R.J., Eastwood M.G., The Penrose transform: its interaction with representation theory, Oxford University Press, 1989.
  2. 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.
  3. Brackx F., Delanghe R., Sommen F., Clifford analysis, Research Notes in Math., Vol. 76, Pitman, 1982.
  4. Branson T.P., Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293-345.
  5. Bures J., The Rarita-Schwinger operator and spherical monogenic forms, Complex Variables Theory Appl. 43 (2000), 77-108.
  6. Bures J., Sommen F., Soucek V., Van Lancker P., Symmetric analogues of Rarita-Schwinger equations, Ann. Global Anal. Geom. 21 (2002), 215-240.
  7. Calderbank D.M.J., Dirac operators and Clifford analysis on manifolds with boundary, Preprint no. 53, Department of Mathematics and Computer Science, University of Southern Denmark, 1997, available at http://bib.mathematics.dk/preprint.php?lang=en&id=IMADA-PP-1997-53.
  8. Cap A., Slovák J., Soucek V., Invariant operators on manifolds with almost Hermitian symmetric structures, III. Standard operators, Diff. Geom. Appl. 12 (2000), 51-84, math.DG/9812023.
  9. Eastwood M.G., Graham C.R., Invariants of conformal densities, Duke Math. J. 63 (1991), 633-671.
  10. Fegan H.D., Conformally invariant first order differential operators, Quart. J. Math. Oxford (2) 27 (1976), 371-378.
  11. Gover A.R., Hirachi K, Conformally invariant powers of the Laplacian - a complete nonexistence theorem, J. Amer. Math. Soc. 17 (2004), 389-405, math.DG/0304082.
  12. Graham C.R., Conformally invariant powers of the Laplacian II: nonexistence, J. London Math. Soc. (2) 46 (1992), 566-576.
  13. Harvey F.R., Spinors and calibrations, Academic Press, 1990.
  14. Kosmann-Schwarzbach Y., Propriétés des dérivations de l'algèbre des tenseurs-spineurs, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A355-A358.
  15. Kraußhar R.S., Ryan J., Clifford and harmonic analysis on cylinders and tori, Rev. Mat. Iberoamericana 21 (2005), 87-110.
  16. Kraußhar R.S., Ryan J., Some conformally flat spin manifolds, Dirac operators and automorphic forms, J. Math. Anal. Appl. 325 (2007), 359-376, math.AP/0212086.
  17. Penrose R., Rindler W., Spinors and space-time, Vol. 1, Cambridge University Press, 1984.
  18. Woodhouse N.M.J., Geometric quantization, Oxford University Press, 1980.

Previous article   Next article   Contents of Volume 3 (2007)