BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor connection which yields an invariant prolongation of the first operator. The explicit calculation is presented in the case of conformal Killing forms.
@article{127123,
author = {Matthias Hammerl},
title = {Invariant prolongation of BGG-operators in conformal geometry},
journal = {Archivum Mathematicum},
volume = {044},
year = {2008},
pages = {367-384},
zbl = {1212.53014},
mrnumber = {2501573},
language = {en},
url = {http://dml.mathdoc.fr/item/127123}
}
Hammerl, Matthias. Invariant prolongation of BGG-operators in conformal geometry. Archivum Mathematicum, Tome 044 (2008) pp. 367-384. http://gdmltest.u-ga.fr/item/127123/
Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (4) (1994), 1191–1217. (1994) | Article | MR 1322223
Prolongations of geometric overdetermined systems, Int. J. Math. 17 (6) (2006), 641–664. (2006) | Article | MR 2246885 | Zbl 1101.35060
Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math. 537 (2001), 67–103. (2001) | MR 1856258 | Zbl 0985.58002
Infinitesimal automorphisms and deformations of parabolic geometries, J. Europ. Math. Soc., to appear. | MR 2390330
Overdetermined systems, conformal geometry, and the BGG complex, Symmetries and Overdetermined Systems of Partial Differential Equations (Eastwood, M. G., Millor, W., eds.), vol. 144, The IMA Volumes in Mathematics and its Applications, Springer, 2008, pp. 1–25. (2008)
Tractor bundles for irreducible parabolic geometries, Global analysis and harmonic analysis (Marseille-Luminy, 1999), vol. 4 of Sémin. Congr., Soc. Math. France, Paris, 2000, pp. 129–154. (2000) | MR 1822358
Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (4) (2002), 1511–1548, electronic. (2002) | Article | MR 1873017 | Zbl 0997.53016
Bernstein-Gelfand-Gelfand sequences, Ann. of Math. 154 (1) (2001), 97–113. (2001) | Article | MR 1847589 | Zbl 1159.58309
Higher symmetries of the Laplacian, Ann. of Math. (2) 161 (3) (2005), 1645–1665. (2005) | Article | MR 2180410 | Zbl 1091.53020
Laplacian operators and $Q$-curvature on conformally Einstein manifolds, Math. Ann. 336 (2) (2006), 311–334. (2006) | Article | MR 2244375 | Zbl 1125.53032
Obstructions to conformally Einstein metrics in $n$ dimensions, J. Geom. Phys. 56 (3) (2006), 450–484. (2006) | Article | MR 2171895 | Zbl 1098.53014
The conformal Killing equation on forms – prolongations and applications, Diff. Geom. Appl., to appear.
On conformal Killing tensor, Natur. Sci. Rep. Ochanomizu Univ. 19 (1968), 67–74. (1968) | MR 0243458 | Zbl 0179.26902
Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. (1961) | Article | MR 0142696 | Zbl 0134.03501
Conformal Killing forms with normalisation condition, Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 279–292. (2005) | MR 2152367 | Zbl 1101.53040
Spinors and space-time, Two-spinor calculus and relativistic fields, vol. 1, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1987. (1987) | MR 0917488 | Zbl 0663.53013
Conformal Killing forms on Riemannian manifolds, Math. Z. 245 (3) (2003), 503–527. (2003) | Article | MR 2021568 | Zbl 1061.53033
Invariant operators in conformal geometry, Ph.D. thesis, University of Auckland, 2006. (2006)