4,222 research outputs found
Two constructions with parabolic geometries
This is an expanded version of a series of lectures delivered at the 25th
Winter School ``Geometry and Physics'' in Srni.
After a short introduction to Cartan geometries and parabolic geometries, we
give a detailed description of the equivalence between parabolic geometries and
underlying geometric structures.
The second part of the paper is devoted to constructions which relate
parabolic geometries of different type. First we discuss the construction of
correspondence spaces and twistor spaces, which is related to nested parabolic
subgroups in the same semisimple Lie group. An example related to twistor
theory for Grassmannian structures and the geometry of second order ODE's is
discussed in detail.
In the last part, we discuss analogs of the Fefferman construction, which
relate geometries corresponding different semisimple Lie groups
AHS-structures and affine holonomies
We show that a large class of non-metric, non-symplectic affine holonomies
can be realized, uniformly and without case by case considerations, by Weyl
connections associated to the natural AHS-structures on certain generalized
flag manifolds.Comment: AMS-LaTeX, 8 pages, v2: changes in exposition; final version; to
appear in Proc. Amer. Math. So
On left invariant CR structures on SU(2)
There is a well known one--parameter family of left invariant CR structures
on . We show how purely algebraic methods can be used to
explicitly compute the canonical Cartan connections associated to these
structures and their curvatures. We also obtain explicit descriptions of
tractor bundles and tractor connections.Comment: 10 page
Curved Casimir Operators and the BGG Machinery
We prove that the Casimir operator acting on sections of a homogeneous vector
bundle over a generalized flag manifold naturally extends to an invariant
differential operator on arbitrary parabolic geometries. We study some
properties of the resulting invariant operators and compute their action on
various special types of natural bundles. As a first application, we give a
very general construction of splitting operators for parabolic geometries. Then
we discuss the curved Casimir operators on differential forms with values in a
tractor bundle, which nicely relates to the machinery of BGG sequences. This
also gives a nice interpretation of the resolution of a finite dimensional
representation by (spaces of smooth vectors in) principal series
representations provided by a BGG sequence.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
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