Jet isomorphism for conformal geometry

Robin C. Graham

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 5, page 389-415
  • ISSN: 0044-8753

Abstract

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Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension is obstructed is described. A jet isomorphism theorem for even dimensional conformal geometry is formulated using the inhomogeneous ambient metrics recently introduced by the author and K. Hirachi.

How to cite

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Graham, Robin C.. "Jet isomorphism for conformal geometry." Archivum Mathematicum 043.5 (2007): 389-415. <http://eudml.org/doc/250179>.

@article{Graham2007,
abstract = {Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension is obstructed is described. A jet isomorphism theorem for even dimensional conformal geometry is formulated using the inhomogeneous ambient metrics recently introduced by the author and K. Hirachi.},
author = {Graham, Robin C.},
journal = {Archivum Mathematicum},
keywords = {conformal geometry; ambient metric; jet isomorphism; deformation complex; conformal geometry; ambient metric; jet isomorphism; deformation complex},
language = {eng},
number = {5},
pages = {389-415},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Jet isomorphism for conformal geometry},
url = {http://eudml.org/doc/250179},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Graham, Robin C.
TI - Jet isomorphism for conformal geometry
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 5
SP - 389
EP - 415
AB - Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension is obstructed is described. A jet isomorphism theorem for even dimensional conformal geometry is formulated using the inhomogeneous ambient metrics recently introduced by the author and K. Hirachi.
LA - eng
KW - conformal geometry; ambient metric; jet isomorphism; deformation complex; conformal geometry; ambient metric; jet isomorphism; deformation complex
UR - http://eudml.org/doc/250179
ER -

References

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  1. Bailey T. N., Eastwood M. G., Graham C. R., Invariant theory for conformal and CR geometry, Ann. of Math. (2) 139 (1994), 491–552. (1994) Zbl0814.53017MR1283869
  2. Branson T., Gover A. R., Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature, Comm. P.D.E. 30 (2005), 1611–1669, arXiv:math/0309085. Zbl1226.58011MR2182307
  3. Calderbank D. M. J., Diemer T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math. 537 (2001), 67–103, arXiv:math/0001158. Zbl0985.58002MR1856258
  4. Čap A., Gover A. R., Standard tractors and the conformal ambient metric construction, Ann. Global Anal. Geom. 24 (2003), 231–259, arXiv:math/0207016. Zbl1039.53021MR1996768
  5. Čap A., Slovák J., Souček V., Bernstein-Gelfand-Gelfand sequences, Ann. of Math. (2) 154 (2001), 97–113. MR1847589
  6. Eastwood M. G., Graham C. R., Invariants of conformal densities, Duke Math. J. 63 (1991), 633–671. (1991) Zbl0745.53007MR1121149
  7. Epstein D., Natural tensors on Riemannian manifolds, J. Differential Geom. 10 (1975), 631–645. (1975) Zbl0321.53039MR0415531
  8. Fefferman C.,, Parabolic invariant theory in complex analysis, Adv. Math. 31 (1979), 131–262. (1979) Zbl0444.32013MR0526424
  9. Fefferman C., Graham C. R., Conformal invariants, in The mathematical heritage of Élie Cartan (Lyon, 1984), Astérisque, 1985, Numero Hors Serie, 95–116. (1984) MR0837196
  10. Fefferman C., Graham C. R., The ambient metric, arXiv:0710.0919. 
  11. Gasqui J., Goldschmidt H., Déformations Infinitésimales des Structures Conformes Plates, Prog. Math. 52, Birkhäuser, 1984. (1984) Zbl0585.53001MR0776970
  12. Gover A. R., Invariant theory and calculus for conformal geometries, Adv. Math. 163 (2001), 206–257. Zbl1004.53010MR1864834
  13. Gover A. R., Peterson L. J., The ambient obstruction tensor and the conformal deformation complex, Pacific J. Math. 226 (2006), 309–351, arXiv:math/0408229. Zbl1125.53010MR2247867
  14. Graham C. R., Hirachi K., Inhomogeneous ambient metrics, IMA Vol. Math. Appl. 144: Symmetries and Overdetermined Systems of Partial Differential Equations, Springer, to appear, arXiv:math/0611931. Zbl1148.53023MR2384722
  15. Graham C. R., Hirachi K., Ambient realization of conformal jets and deformation complex, in preparation. 
  16. Hirachi K., Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. (2) 151 (2000), 151–191, arXiv:math/0010014. (191,) MR1745015
  17. Lepowsky J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra 49 (1977), 496–511. (1977) Zbl0381.17006MR0476813

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