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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
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