Subtleties concerning conformal tractor bundles

C. Graham; Travis Willse

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1721-1732
  • ISSN: 2391-5455

Abstract

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The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures.

How to cite

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C. Graham, and Travis Willse. "Subtleties concerning conformal tractor bundles." Open Mathematics 10.5 (2012): 1721-1732. <http://eudml.org/doc/269690>.

@article{C2012,
abstract = {The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures.},
author = {C. Graham, Travis Willse},
journal = {Open Mathematics},
keywords = {Conformal geometry; Parabolic geometry; Tractor bundle; Cartan connection; Associated bundle; conformal geometry; parabolic geometry; tractor bundle; associated bundle},
language = {eng},
number = {5},
pages = {1721-1732},
title = {Subtleties concerning conformal tractor bundles},
url = {http://eudml.org/doc/269690},
volume = {10},
year = {2012},
}

TY - JOUR
AU - C. Graham
AU - Travis Willse
TI - Subtleties concerning conformal tractor bundles
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1721
EP - 1732
AB - The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures.
LA - eng
KW - Conformal geometry; Parabolic geometry; Tractor bundle; Cartan connection; Associated bundle; conformal geometry; parabolic geometry; tractor bundle; associated bundle
UR - http://eudml.org/doc/269690
ER -

References

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  2. [2] Čap A., Gover A.R., Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc., 2002, 354(4), 1511–1548 http://dx.doi.org/10.1090/S0002-9947-01-02909-9 Zbl0997.53016
  3. [3] Čap A., Gover A.R., Standard tractors and the conformal ambient metric construction, Ann. Global Anal. Geom., 2003, 24(3), 231–259 http://dx.doi.org/10.1023/A:1024726607595 Zbl1039.53021
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  7. [7] Fefferman C., Graham C.R., Conformal invariants, In: The Mathematical Heritage of Élie Cartan, Lyon, 1984, Astérisque, 1985, Numero Hors Serie, 95–116 
  8. [8] Fefferman C., Graham C.R., The Ambient Metric, Ann. of Math. Stud., 178, Princeton University Press, Princeton, 2012 Zbl1243.53004
  9. [9] Graham C.R., Willse T., Parallel tractor extension and ambient metrics of holonomy split G 2, J. Diff. Geom. (in press), preprint available at http://arxiv.org/abs/1109.3504 
  10. [10] Hammerl M., Sagerschnig K., Conformal structures associated to generic rank 2 distributions on 5-manifolds - characterization and Killing-field decomposition, SIGMA Symmetry Integrability Geom. Methods Appl., 2009, 5, #081 Zbl1191.53016
  11. [11] Morimoto T., Geometric structures on filtered manifolds, Hokkaido Math. J., 1993, 22(3), 263–347 Zbl0801.53019
  12. [12] Nurowski P., Differential equations and conformal structures, J. Geom. Phys., 2005, 55(1), 19–49 http://dx.doi.org/10.1016/j.geomphys.2004.11.006 Zbl1082.53024
  13. [13] Sharpe R.W., Differential Geometry, Grad. Texts in Math., 166, Springer, New York, 1997 
  14. [14] Tanaka N., On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J., 1979, 8(1), 23–84 Zbl0409.17013

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