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Standard Tractors and the Conformal Ambient Metric Construction

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Abstract

In this paper we relate the Fefferman–Graham ambientmetric construction for conformal manifolds to the approach toconformal geometry via the canonical Cartan connection. We show thatfrom any ambient metric that satisfies a weakening of the usualnormalisation condition, one can construct the conformal standardtractor bundle and the normal standard tractor connection, which areequivalent to the Cartan bundle and the Cartan connection. This resultis applied to obtain a procedure to get tractor formulae for allconformal invariants that can be obtained from the ambient metricconstruction. We also get information on ambient metrics whichare Ricci flat to higher order than guaranteed by the results ofFefferman–Graham.

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References

  1. Bailey, T. N., Eastwood, M. G. and Gover, A. R.: Thomas's structure bundle for conformal, projective and related structures, Rocky Mountain J. 24 (1994), 1191–1217.

    Google Scholar 

  2. Branson, T. and Gover, A. R.: Conformally Invariant Non-Local Operators, Pacific J. Math. 201 (2001), 19–60.

    Google Scholar 

  3. Cartan, E.: Les espaces à connexion conforme, Ann. Soc. Pol. Math. 2 (1923), 171–202.

    Google Scholar 

  4. Čap, A. and Gover, A. R.: Tractor bundles for irreducible parabolic geometries, SMF Séminaires et congrès 4 (2000), 129–154, electronically available at http://smf.emath.fr/SansMenu/Publications/SeminairesCongres/

    Google Scholar 

  5. Čap, A. and Gover, A. R.: Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2002), 1511–1548, electronically available as Preprint ESI 792 at http://www.esi.ac.at

    Google Scholar 

  6. Čap, A. and Schichl, H.: Parabolic geometries and canonical Cartan connections, Hokkaido Math. J. 29(3) (2000), 453–505.

    Google Scholar 

  7. Eastwood, M. G.: Notes on conformal differential geometry, Supp. Rend. Circ.Matem. Palermo 43 (1996), 57–76.

    Google Scholar 

  8. Fefferman, C. and Graham, C. R.: Conformal invariants, in Élie Cartan et les Mathématiques d'Aujourd'hui, (Astérisque, hors serie) (1985), 95–116.

  9. Fefferman, C. and Graham, C. R.: Q-curvature and Poincaré metrics, Preprint, math.DG/0110271.

  10. Gover, A. R.: Aspects of parabolic invariant theory, Supp. Rend. Circ. Matem. Palermo, Ser. II, Suppl. 59 (1999), 25–47.

    Google Scholar 

  11. Gover, A. R.: Invariants and calculus for conformal geometry, Adv. Math. 163 (2001), 206–257.

    Google Scholar 

  12. Gover, A. R. and Peterson, L. J.: Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus, Comm. Math. Phys. 235(2) (2003), 339–378, electronically available as preprint math-ph/0201030.

    Google Scholar 

  13. Graham, C. R.: Conformally invariant powers of the Laplacian, II: Nonexistence, J. London Math. Soc. 46 (1992), 566–576.

    Google Scholar 

  14. Graham, C. R., Jenne, R., Mason, L. J. and Sparling, G. A.: Conformally invariant powers of the Laplacian, I: Existence, J. London Math. Soc. 46 (1992), 557–565.

    Google Scholar 

  15. Graham, C. R. and Zworski, M.: Scattering matrix in conformal geometry, Preprint, math.DG/0109089.

  16. Graham, C. R. and Witten, E.: Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nuclear Phys. B 546 (1999), 52–64.

    Google Scholar 

  17. Henningson, M. and Skenderis, K.: Holography and theWeyl anomaly, Proceedings of the 32nd International Symposium Ahrenshoop on the Theory of Elementary Particles (Buckow, 1998), Fortschr. Phys. 48 (2000), 125–128.

    Google Scholar 

  18. Thomas, T. Y.: On conformal geometry, Proc. Nat. Acad. Sci. USA 12 (1926), 352–359.

    Google Scholar 

  19. Thomas, T. Y.: The Differential Invariants of Generalized Spaces, Cambridge University Press, Cambridge, 1934.

    Google Scholar 

  20. Witten, E.: Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998), 253–291.

    Google Scholar 

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Čap, A., Gover, A.R. Standard Tractors and the Conformal Ambient Metric Construction. Annals of Global Analysis and Geometry 24, 231–259 (2003). https://doi.org/10.1023/A:1024726607595

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