Weyl structures for parabolic geometries. (English) Zbl 1076.53029

Čap, Andreas; Slovák, Jan

Math. Scand. 93, No. 1, 53-90 (2003).

The Weyl structures and preferred connections are introduced here in the framework of geometries modeled on homogeneous spaces \(G/P\), with \(G\) semisimple and \(P\) parabolic [É. Cartan, Ann. Soc. Pol. Math. 2, 171–221 (1923; JFM 50.0493.01)]. The conformal Riemannian, projective, almost quaternionic and CR-structures are examples of parabolic geometries. As in the conformal Riemannian case, the class of Weyl structures underlying a parabolic geometry on a manifold \(M\) is an affine space modeled on one forms on \(M\) and each of them determines a linear connection on \(M\). The difference between the linear connection induced by a Weyl structure and the canonical Cartan connection is encoded in the so called Rho-tensor. The bundles of scales are defined here as certain affine line bundles generalizing the distinguished bundles of conformal metrics. The closed and exact Weyl geometries are given by scales. All objects related to a choice of a Weyl structure are characterized and the results are used to give a new description of the Cartan bundles and connections for all parabolic geometries.

Reviewer: Jürgen Berndt (Hull)

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
53A30	Conformal differential geometry (MSC2010)
32V05	CR structures, CR operators, and generalizations
53C05	Connections (general theory)

Citations:

JFM 50.0493.01