Tractor calculi for parabolic geometries
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- by Andreas Čap and A. Rod Gover
- Trans. Amer. Math. Soc. 354 (2002), 1511-1548
- DOI: https://doi.org/10.1090/S0002-9947-01-02909-9
- Published electronically: November 20, 2001
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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
- Toby N. Bailey and Michael G. Eastwood, Complex paraconformal manifolds—their differential geometry and twistor theory, Forum Math. 3 (1991), no. 1, 61–103. MR 1085595, DOI 10.1515/form.1991.3.61
- 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. MR 1322223, DOI 10.1216/rmjm/1181072333
- 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. MR 1283869, DOI 10.2307/2118571
- R. J. Baston, Almost Hermitian symmetric manifolds. I. Local twistor theory, Duke Math. J. 63 (1991), no. 1, 81–112. MR 1106939, DOI 10.1215/S0012-7094-91-06305-2
- T. Branson, A.R. Gover, Conformally Invariant Non-Local Operators, to appear in Pacific J. Math.
- A. Čap, H. Schichl, Parabolic Geometries and Canonical Cartan Connections, Hokkaido Math. J. 29 No.3 (2000), 453–505.
- 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. MR 1463512
- Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
- A. Čap, J. Slovák, V. Souček, Bernstein–Gelfand–Gelfand Sequences to appear in Ann. of Math., extended version electronically available as Preprint ESI 722 at http://www.esi.ac.at
- E. Cartan, Les espaces à connexion conforme, Ann. Soc. Pol. Math. 2 (1923), 171–202.
- Kenza Dighton, An introduction to the theory of local twistors, Internat. J. Theoret. Phys. 11 (1974), 31–43. MR 421473, DOI 10.1007/BF01807935
- Michael Eastwood, Notes on conformal differential geometry, The Proceedings of the 15th Winter School “Geometry and Physics” (Srní, 1995), 1996, pp. 57–76. MR 1463509
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
- Max Zorn, Continuous groups and Schwarz’ lemma, Trans. Amer. Math. Soc. 46 (1939), 1–22. MR 53, DOI 10.1090/S0002-9947-1939-0000053-7
- Charles Fefferman and C. Robin Graham, Conformal invariants, Astérisque Numéro Hors Série (1985), 95–116. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837196
- A. B. Goncharov, Generalized conformal structures on manifolds, Selecta Math. Soviet. 6 (1987), no. 4, 307–340. Selected translations. MR 925263
- A. Rod Gover, Invariants and calculus for projective geometries, Math. Ann. 306 (1996), no. 3, 513–538. MR 1415076, DOI 10.1007/BF01445263
- A.R. Gover, Invariants and calculus for conformal geometry, to appear in Adv. Math.
- 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). MR 1692257
- A.R. Gover, K. Hirachi, In progress.
- 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. MR 1723137, DOI 10.1016/S0393-0440(99)00018-2
- A.R. Gover, C.R. Graham, CR calculus and invariant powers of the sub-Laplacian, In progress.
- 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. MR 1201601
- Kengo Hirachi, Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. (2) 151 (2000), no. 1, 151–191. MR 1745015, DOI 10.2307/121115
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842, DOI 10.1007/978-1-4612-6398-2
- Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
- Tohru Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), no. 3, 263–347. MR 1245130, DOI 10.14492/hokmj/1381413178
- Takushiro Ochiai, Geometry associated with semisimple flat homogeneous spaces, Trans. Amer. Math. Soc. 152 (1970), 159–193. MR 284936, DOI 10.1090/S0002-9947-1970-0284936-6
- J. Slovák, Conformal differential geometry lecture notes, University of Vienna, 1992, electronically available at http://www.math.muni.cz/$\sim$slovak.
- Noboru Tanaka, On the equivalence problems associated with a certain class of homogeneous spaces, J. Math. Soc. Japan 17 (1965), 103–139. MR 188930, DOI 10.2969/jmsj/01720103
- Noboru Tanaka, On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), no. 1, 23–84. MR 533089, DOI 10.14492/hokmj/1381758416
- T.Y. Thomas, On conformal geometry, Proc. N.A.S. 12 (1926), 352–359; Conformal tensors, Proc. N.A.S. 18 (1931), 103–189.
- 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. MR 1274961, DOI 10.2969/aspm/02210413
Bibliographic Information
- Andreas Čap
- 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
- Email: Andreas.Cap@esi.ac.at
- A. Rod Gover
- Affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
- MR Author ID: 335695
- Email: gover@math.auckland.ac.nz
- Received by editor(s): July 17, 2000
- Published electronically: November 20, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 1511-1548
- MSC (2000): Primary 53B15, 53C05, 53C07, 53C15; Secondary 32V05, 53A20, 53A30, 53A40, 53A55
- DOI: https://doi.org/10.1090/S0002-9947-01-02909-9
- MathSciNet review: 1873017