×

zbMATH — the first resource for mathematics

Tractor calculi for parabolic geometries. (English) Zbl 0997.53016
Summary: Parabolic geometries may be considered as curved analogues of the homogeneous spaces \( G/P\) where \( G\) is a semisimple Lie group and \( P\subset G\) parabolic subgroup. Conformal geometries and CR geometries are examples of such structures. We present a uniform description of a calculus, called tractor calculus, based on natural bundles with canonical linear connections for all parabolic geometries. It is shown that from these bundles and connections one can recover the Cartan bundle and the Cartan connection. In particular we characterize the normal Cartan connection from this induced bundle/connection perspective. We construct explicitly a family of fundamental first order differential operators, which are analogous to a covariant derivative, iterable and defined on all natural vector bundles on parabolic geometries. For an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles, their canonical linear connections and the machinery for explicitly calculating via the tractor calculus.

MSC:
53B15 Other connections
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C05 Connections (general theory)
32V05 CR structures, CR operators, and generalizations
53A20 Projective differential geometry
53A30 Conformal differential geometry (MSC2010)
53A40 Other special differential geometries
53A55 Differential invariants (local theory), geometric objects
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Toby N. Bailey and Michael G. Eastwood, Complex paraconformal manifolds — their differential geometry and twistor theory, Forum Math. 3 (1991), no. 1, 61 – 103. · Zbl 0728.53005
[2] T. N. Bailey, M. G. Eastwood, and A. R. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), no. 4, 1191 – 1217. · Zbl 0828.53012
[3] Toby N. Bailey, Michael G. Eastwood, and C. Robin Graham, Invariant theory for conformal and CR geometry, Ann. of Math. (2) 139 (1994), no. 3, 491 – 552. · Zbl 0814.53017
[4] R. J. Baston, Almost Hermitian symmetric manifolds. I. Local twistor theory, Duke Math. J. 63 (1991), no. 1, 81 – 112. · Zbl 0724.53019
[5] T. Branson, A.R. Gover, Conformally Invariant Non-Local Operators, to appear in Pacific J. Math. · Zbl 1052.58026
[6] A. Cap, H. Schichl, Parabolic Geometries and Canonical Cartan Connections, Hokkaido Math. J. 29 No.3 (2000), 453-505. CMP 2001:04
[7] Andreas Čap and Jan Slovák, On local flatness of manifolds with AHS-structures, The Proceedings of the 15th Winter School ”Geometry and Physics” (Srní, 1995), 1996, pp. 95 – 101. · Zbl 1067.53501
[8] A. Čap, J. Slovák, and V. Souček, Invariant operators on manifolds with almost Hermitian symmetric structures. I. Invariant differentiation, Acta Math. Univ. Comenian. (N.S.) 66 (1997), no. 1, 33 – 69. A. Čap, J. Slovák, and V. Souček, Invariant operators on manifolds with almost Hermitian symmetric structures. II. Normal Cartan connections, Acta Math. Univ. Comenian. (N.S.) 66 (1997), no. 2, 203 – 220. Andreas Čap, Jan Slovák, and Vladimír Souček, Invariant operators on manifolds with almost Hermitian symmetric structures. III. Standard operators, Differential Geom. Appl. 12 (2000), no. 1, 51 – 84. · Zbl 0969.53004
[9] A. Cap, J. Slovák, V. Soucek, Bernstein-Gelfand-Gelfand Sequences to appear in Ann. of Math., extended version electronically available as Preprint ESI 722 at http://www.esi.ac.at · Zbl 1159.58309
[10] E. Cartan, Les espaces à connexion conforme, Ann. Soc. Pol. Math. 2 (1923), 171-202. · JFM 50.0493.01
[11] Kenza Dighton, An introduction to the theory of local twistors, Internat. J. Theoret. Phys. 11 (1974), 31 – 43.
[12] Michael Eastwood, Notes on conformal differential geometry, The Proceedings of the 15th Winter School ”Geometry and Physics” (Srní, 1995), 1996, pp. 57 – 76. · Zbl 0911.53020
[13] Michael G. Eastwood and John W. Rice, Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys. 109 (1987), no. 2, 207 – 228. M. G. Eastwood and J. W. Rice, Erratum: ”Conformally invariant differential operators on Minkowski space and their curved analogues” [Comm. Math. Phys. 109 (1987), no. 2, 207 – 228; MR0880414 (89d:22012)], Comm. Math. Phys. 144 (1992), no. 1, 213. · Zbl 0659.53047
[14] Charles L. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), no. 2, 395 – 416. , https://doi.org/10.2307/1970945 C. Fefferman, Correction to: ”Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains” (Ann. of Math. (2) 103 (1976), no. 2, 395 – 416), Ann. of Math. (2) 104 (1976), no. 2, 393 – 394. , https://doi.org/10.2307/1970961 Charles L. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), no. 2, 395 – 416. , https://doi.org/10.2307/1970945 C. Fefferman, Correction to: ”Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains” (Ann. of Math. (2) 103 (1976), no. 2, 395 – 416), Ann. of Math. (2) 104 (1976), no. 2, 393 – 394. · Zbl 0332.32018
[15] Charles Fefferman and C. Robin Graham, Conformal invariants, Astérisque Numéro Hors Série (1985), 95 – 116. The mathematical heritage of Élie Cartan (Lyon, 1984). · Zbl 0602.53007
[16] A. B. Goncharov, Generalized conformal structures on manifolds, Selecta Math. Soviet. 6 (1987), no. 4, 307 – 340. Selected translations. · Zbl 0632.53038
[17] A. Rod Gover, Invariants and calculus for projective geometries, Math. Ann. 306 (1996), no. 3, 513 – 538. · Zbl 0904.53014
[18] A.R. Gover, Invariants and calculus for conformal geometry, to appear in Adv. Math. · Zbl 1004.53010
[19] A. Rod Gover, Aspects of parabolic invariant theory, Rend. Circ. Mat. Palermo (2) Suppl. 59 (1999), 25 – 47. The 18th Winter School ”Geometry and Physics” (Srní, 1998). · Zbl 0967.53033
[20] A.R. Gover, K. Hirachi, In progress.
[21] A. Rod Gover and Jan Slovák, Invariant local twistor calculus for quaternionic structures and related geometries, J. Geom. Phys. 32 (1999), no. 1, 14 – 56. · Zbl 0981.53031
[22] A.R. Gover, C.R. Graham, CR calculus and invariant powers of the sub-Laplacian, In progress. · Zbl 1076.53048
[23] C. Robin Graham, Invariant theory of parabolic geometries, Complex geometry (Osaka, 1990) Lecture Notes in Pure and Appl. Math., vol. 143, Dekker, New York, 1993, pp. 53 – 66. · Zbl 0794.53028
[24] Kengo Hirachi, Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. (2) 151 (2000), no. 1, 151 – 191. · Zbl 0954.32002
[25] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. · Zbl 0254.17004
[26] Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329 – 387. · Zbl 0134.03501
[27] Tohru Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), no. 3, 263 – 347. · Zbl 0801.53019
[28] Takushiro Ochiai, Geometry associated with semisimple flat homogeneous spaces, Trans. Amer. Math. Soc. 152 (1970), 159 – 193. · Zbl 0205.26004
[29] J. Slovák, Conformal differential geometry lecture notes, University of Vienna, 1992, electronically available at http://www.math.muni.cz/\(\sim\)slovak.
[30] Noboru Tanaka, On the equivalence problems associated with a certain class of homogeneous spaces, J. Math. Soc. Japan 17 (1965), 103 – 139. · Zbl 0132.16303
[31] Noboru Tanaka, On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), no. 1, 23 – 84. · Zbl 0409.17013
[32] T.Y. Thomas, On conformal geometry, Proc. N.A.S. 12 (1926), 352-359; Conformal tensors, Proc. N.A.S. 18 (1931), 103-189. · JFM 52.0736.01
[33] Keizo Yamaguchi, Differential systems associated with simple graded Lie algebras, Progress in differential geometry, Adv. Stud. Pure Math., vol. 22, Math. Soc. Japan, Tokyo, 1993, pp. 413 – 494. · Zbl 0812.17018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.