Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Tractor calculi for parabolic geometries

Authors: Andreas Cap and A. Rod Gover
Journal: Trans. Amer. Math. Soc. 354 (2002), 1511-1548
MSC (2000): Primary 53B15, 53C05, 53C07, 53C15; Secondary 32V05, 53A20, 53A30, 53A40, 53A55
Published electronically: November 20, 2001
MathSciNet review: 1873017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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$ a 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.

References [Enhancements On Off] (What's this?)

  • 1. T.N. Bailey and M.G. Eastwood, Complex paraconformal manifolds: their differential geometry and twistor theory, Forum Math. 3 (1991), 61-103. MR 92a:32038
  • 2. T.N. Bailey, M.G. Eastwood, A.R. Gover, Thomas's structure bundle for conformal, projective and related structures, Rocky Mountain J. 24 (1994), 1191-1217. MR 96e:53016
  • 3. T.N. Bailey, M.G. Eastwood and C. R. Graham, Invariant theory for conformal and CR geometry. Ann. of Math. 139 (1994), 491-552. MR 95h:53016
  • 4. R. J. Baston, Almost Hermitian symmetric manifolds, I: Local twistor theory, II: Differential Invariants, Duke Math. J. 63 (1991), 81-111, 113-138. MR 93d:53064
  • 5. T. Branson, A.R. Gover, Conformally Invariant Non-Local Operators, to appear in Pacific J. Math.
  • 6. A. Cap, H. Schichl, Parabolic Geometries and Canonical Cartan Connections, Hokkaido Math. J. 29 No.3 (2000), 453-505. CMP 2001:04
  • 7. A. Cap, J. Slovák, On local flatness of manifolds with AHS-structures in Proc. of the Winter School Geometry and Physics, Srni 1995 Supp ai Rend. Circ. Mat. Palermo 43 (1996), 95-101. MR 98j:53059
  • 8. A. Cap, J. Slovák, V. Soucek, Invariant operators on manifolds with almost Hermitian symmetric structures, I. invariant differentiation, Acta Math. Univ. Commenianae, 66 No. 1 (1997), 33-69, electronically available at; MR 98m:53036 II. normal Cartan connections, Acta Math. Univ. Commenianae, 66 No. 2 (1997), 203-220, electronically available at; MR 2000a:53045 III. Standard operators, Diff. Geom. Appl. 12 No. 1 (2000) 51-84. MR 2001h:53038
  • 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
  • 10. E. Cartan, Les espaces à connexion conforme, Ann. Soc. Pol. Math. 2 (1923), 171-202.
  • 11. K. Dighton, An introduction to the theory of local twistors, Int. J. Theor. Phys. 11 (1974), 31-43. MR 54:9475
  • 12. M.G. Eastwood, Notes on Conformal Differential Geometry, Supp. Rend. Circ. Matem. Palermo 43 (1996), 57-76. MR 98g:53021
  • 13. M.G. Eastwood and J.W. Rice, Conformally invariant differential operators on Minkowski space and their curved analogues, Commun. Math. Phys. 109 (1987), 207-228. Erratum:144 (1992), 213. MR 89d:22012; MR 92k:22026
  • 14. C. Fefferman, Monge-Ampère equations, the Bergman kernel and geometry of pseudoconvex domains, Ann. of Math. 103 (1976), 395-416; Erratum 104 (1976), 393-394. MR 53:11097a; MR 53:11097b
  • 15. C. Fefferman and C.R. Graham, Conformal invariants, in Élie Cartan et les Mathématiques d'Adjourd'hui, (Astérisque, hors serie), (1985), 95-116. MR 87g:53060
  • 16. A.B. Goncharov, Generalized conformal structures on manifolds, Selecta Math. Soviet. 6 (1987), 308-340. MR 89e:53050
  • 17. A.R. Gover, Invariants and Calculus for Projective Geometries, Math. Ann. 306 (1996) 513-538. MR 97k:53019
  • 18. A.R. Gover, Invariants and calculus for conformal geometry, to appear in Adv. Math.
  • 19. A.R. Gover, Aspects of parabolic invariant theory, in Proc. of the Winter School Geometry and Physics, Srni 1998 Supp. Rend. Circ. Matem. Palermo, Ser. II. Suppl. 59 (1999), 25-47. MR 2001a:58047
  • 20. A.R. Gover, K. Hirachi, In progress.
  • 21. A.R. Gover, J. Slovák, Invariant Local Twistor Calculus for Quaternionic Structures and Related Geometries, J. Geom. Phys. 32 (1999) 14-56. MR 2001b:53051
  • 22. A.R. Gover, C.R. Graham, CR calculus and invariant powers of the sub-Laplacian, In progress.
  • 23. Graham C.R.: Invariant theory of parabolic geometries in Komatsu, Sakane, (eds.) Complex Geometry. Proceedings Osaka 1990 (Marcel Dekker Lecture Notes in Pure and Applied Mathematics) New York, Basel, Hong Kong: Marcel Dekker, 1993. MR 94a:32013
  • 24. K. Hirachi, Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. 151 no. 1 (2000), 151-191. MR 2001f:32003
  • 25. J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York (1972). MR 48:2197
  • 26. B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 no. 2 (1961), 329-387. MR 26:265
  • 27. T. Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), 263-347. MR 94m:58243
  • 28. T. Ochiai, Geometry associated with semisimple flat homogeneous spaces, Trans. Amer. Math. Soc. 152 (1970), 159-193. MR 44:2160
  • 29. J. Slovák, Conformal differential geometry lecture notes, University of Vienna, 1992, electronically available at$\sim$slovak.
  • 30. N. Tanaka, On the equivalence problem associated with a certain class of homogeneous spaces, J. Math. Soc. Japan 17 (1965), 103-139. MR 32:6358
  • 31. N. Tanaka, On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 23-84. MR 80h:53034
  • 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.
  • 33. K. Yamaguchi, Differential Systems Associated with Simple Graded Lie Algebras, Advanced Studies in Pure Math. 22 (1993), 413-494. MR 95g:58263

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53B15, 53C05, 53C07, 53C15, 32V05, 53A20, 53A30, 53A40, 53A55

Retrieve articles in all journals with MSC (2000): 53B15, 53C05, 53C07, 53C15, 32V05, 53A20, 53A30, 53A40, 53A55

Additional Information

Andreas Cap
Affiliation: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A–1090 Wien, Austria and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A–1090 Wien, Austria

A. Rod Gover
Affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand

Keywords: Parabolic geometry, Cartan connection, tractor bundle, tractor calculus, invariant differential operator, invariant calculus
Received by editor(s): July 17, 2000
Published electronically: November 20, 2001
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society