| Title:
|
Invariant prolongation of BGG-operators in conformal geometry (English) |
| Author:
|
Hammerl, Matthias |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
44 |
| Issue:
|
5 |
| Year:
|
2008 |
| Pages:
|
367-384 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
conformal geometry |
| Keyword:
|
invariant differential operators |
| Keyword:
|
overdetermined systems |
| Keyword:
|
prolongation |
| Keyword:
|
tractor calculus |
| MSC:
|
35C15 |
| MSC:
|
35N10 |
| MSC:
|
53A30 |
| MSC:
|
58J70 |
| idZBL:
|
Zbl 1212.53014 |
| idMR:
|
MR2501573 |
| . |
| Date available:
|
2009-01-29T09:15:55Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127123 |
| . |
| Reference:
|
[1] Bailey, T. N., Eastwood, M. G., Gover, A. Rod: Thomas’s structure bundle for conformal, projective and related structures.Rocky Mountain J. Math. 24 (4) (1994), 1191–1217. MR 1322223, 10.1216/rmjm/1181072333 |
| Reference:
|
[2] Branson, T., Čap, A., Eastwood, M., Gover, A. R.: Prolongations of geometric overdetermined systems.Int. J. Math. 17 (6) (2006), 641–664. Zbl 1101.35060, MR 2246885, 10.1142/S0129167X06003655 |
| Reference:
|
[3] Calderbank, D. M. J., Diemer, T.: Differential invariants and curved Bernstein-Gelfand-Gelfand sequences.J. Reine Angew. Math. 537 (2001), 67–103. Zbl 0985.58002, MR 1856258 |
| Reference:
|
[4] Čap, A.: Infinitesimal automorphisms and deformations of parabolic geometries.J. Europ. Math. Soc., to appear. MR 2390330 |
| Reference:
|
[5] Čap, A.: 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. |
| Reference:
|
[6] Čap, A., Gover, A. R.: 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. MR 1822358 |
| Reference:
|
[7] Čap, A., Gover, A. R.: Tractor calculi for parabolic geometries.Trans. Amer. Math. Soc. 354 (4) (2002), 1511–1548, electronic. Zbl 0997.53016, MR 1873017, 10.1090/S0002-9947-01-02909-9 |
| Reference:
|
[8] Čap, A., Slovák, J., Souček, V.: Bernstein-Gelfand-Gelfand sequences.Ann. of Math. 154 (1) (2001), 97–113. Zbl 1159.58309, MR 1847589, 10.2307/3062111 |
| Reference:
|
[9] Eastwood, M.: Higher symmetries of the Laplacian.Ann. of Math. (2) 161 (3) (2005), 1645–1665. Zbl 1091.53020, MR 2180410, 10.4007/annals.2005.161.1645 |
| Reference:
|
[10] Gover, A. R.: Laplacian operators and $Q$-curvature on conformally Einstein manifolds.Math. Ann. 336 (2) (2006), 311–334. Zbl 1125.53032, MR 2244375, 10.1007/s00208-006-0004-z |
| Reference:
|
[11] Gover, A. R., Nurowski, P.: Obstructions to conformally Einstein metrics in $n$ dimensions.J. Geom. Phys. 56 (3) (2006), 450–484. Zbl 1098.53014, MR 2171895, 10.1016/j.geomphys.2005.03.001 |
| Reference:
|
[12] Gover, A. R., Šilhan, J.: The conformal Killing equation on forms – prolongations and applications.Diff. Geom. Appl., to appear. |
| Reference:
|
[13] Kashiwada, T.: On conformal Killing tensor.Natur. Sci. Rep. Ochanomizu Univ. 19 (1968), 67–74. Zbl 0179.26902, MR 0243458 |
| Reference:
|
[14] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem.Ann. of Math. (2) 74 (1961), 329–387. Zbl 0134.03501, MR 0142696, 10.2307/1970237 |
| Reference:
|
[15] Leitner, F.: Conformal Killing forms with normalisation condition.Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 279–292. Zbl 1101.53040, MR 2152367 |
| Reference:
|
[16] Penrose, R., Rindler, W.: Spinors and space-time.Two-spinor calculus and relativistic fields, vol. 1, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1987. Zbl 0663.53013, MR 0917488 |
| Reference:
|
[17] Semmelmann, U.: Conformal Killing forms on Riemannian manifolds.Math. Z. 245 (3) (2003), 503–527. Zbl 1061.53033, MR 2021568, 10.1007/s00209-003-0549-4 |
| Reference:
|
[18] Šilhan, J.: Invariant operators in conformal geometry.Ph.D. thesis, University of Auckland, 2006. |
| . |
