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
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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.

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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