# Weyl structures for parabolic geometries

@article{ap2000WeylSF,
title={Weyl structures for parabolic geometries},
author={Andreas {\vC}ap and Jan Slov{\'a}k},
journal={Mathematica Scandinavica},
year={2000},
volume={93},
pages={53-90}
}
• Published 28 January 2000
• Mathematics
• Mathematica Scandinavica
Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces $G/P$ with $G$ semisimple and $P$ parabolic, Weyl structures and preferred connections are introduced in this general framework. In particular, we extend the notions of scales, closed and exact Weyl connections, and Rho-tensors, we characterize the classes of such objects, and we use the results to give a new description of the Cartan bundles and connections…

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## References

SHOWING 1-10 OF 45 REFERENCES
Tractor calculi for parabolic geometries
• Mathematics
• 2001
Parabolic geometries may be considered as curved analogues of the homogeneous spaces G/P where G is a semisimple Lie group and P C G a parabolic subgroup. Conformal geometries and CR geometries are
Invariant operators on manifolds with almost Hermitian symmetric structures, III. Standard operators
• Mathematics
• 1998
This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g.
Parabolic geometries and canonical Cartan connections
• Mathematics
• 2000
Let G be a (real or complex) semisimple Lie group, whose Lie algebra g is endowed with a so called |k|–grading, i.e. a grading of the form g = g−k ⊕ · · · ⊕ gk, such that no simple factor of G is of
On Conformal Geometry.
• T. Y. Thomas
• Computer Science
Proceedings of the National Academy of Sciences of the United States of America
• 1926
Four functionals on the space of normalized almost Hermitian metrics on almost complex manifolds are discussed and the Euler-Lagrange equations for all these functionals are computed – as a tool for characterizing these metrics.
Complex Paraconformal Manifolds – their Differential Geometry and Twistor Theory
• Mathematics
• 1991
A complex paraconformal manifold is a/^-dimensional complex manifold (/?, q > 2) whose tangent bündle factors äs a tensor product of two bundles of ranks p and q. We also assume that we are given a
Invariants and calculus for projective geometries
An understanding of the local invariants is essential in any study of a differential geometry with local structure. In his ground-breaking paper "Parabolic invariant theory in complex analysis" [F2]
Conformal tensors
• P. Szekeres
• Mathematics
Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
• 1968
The existence of conformally invariant geometrical objects and tensors which are functions of the metric tensor is investigated. It is shown that every such conformal object is a differential
Natural operations in differential geometry
• Mathematics
• 1993
I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII.
The geometry of hyperbolic and elliptic CR-manifolds of codimension two
• Mathematics
• 1999
The general theory of parabolic manifolds is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two.