194 research outputs found
Symmetries of Parabolic Geometries
We generalize the concept of affine locally symmetric spaces for parabolic
geometries. We discuss mainly --graded geometries and we show some
restrictions on their curvature coming from the existence of symmetries. We use
the theory of Weyl structures to discuss more interesting --graded
geometries which can carry a symmetry in a point with nonzero curvature. More
concretely, we discuss the number of different symmetries which can exist at
the point with nonzero curvature
The twistor spinors of generic 2- and 3-distributions
Generic distributions on 5- and 6-manifolds give rise to conformal structures
that were discovered by P. Nurowski resp. R. Bryant. We describe both as
Fefferman-type constructions and show that for orientable distributions one
obtains conformal spin structures. The resulting conformal spin geometries are
then characterized by their conformal holonomy and equivalently by the
existence of a twistor spinor which satisfies a genericity condition. Moreover,
we show that given such a twistor spinor we can decompose a conformal Killing
field of the structure. We obtain explicit formulas relating conformal Killing
fields, almost Einstein structures and twistor spinors.Comment: 26 page
Indices of quaternionic complexes
Methods of parabolic geometries have been recently used to construct a class
of elliptic complexes on quaternionic manifolds, the Salamon's complex being
the simplest case. The purpose of this paper is to describe an algorithm how to
compute their analytical indices in terms of characteristic classes. Using
this, we are able to derive some topological obstructions to existence of
quaternionic structures on manifolds.Comment: 14 page
A holonomy characterisation of Fefferman spaces
We prove that Fefferman spaces, associated to non--degenerate CR structures
of hypersurface type, are characterised, up to local conformal isometry, by the
existence of a parallel orthogonal complex structure on the standard tractor
bundle. This condition can be equivalently expressed in terms of conformal
holonomy. Extracting from this picture the essential consequences at the level
of tensor bundles yields an improved, conformally invariant analogue of
Sparling's characterisation of Fefferman spaces.Comment: AMSLaTeX, 15 page
Ricci-corrected derivatives and invariant differential operators
We introduce the notion of Ricci-corrected differentiation in parabolic
geometry, which is a modification of covariant differentiation with better
transformation properties. This enables us to simplify the explicit formulae
for standard invariant operators given in work of Cap, Slovak and Soucek, and
at the same time extend these formulae from the context of AHS structures
(which include conformal and projective structures) to the more general class
of all parabolic structures (including CR structures).Comment: Substantially revised, shortened and simplified, with new treatment
of Weyl structures; 24 page
On Nurowski's conformal structure associated to a generic rank two distribution in dimension five
For a generic distribution of rank two on a manifold of dimension five,
we introduce the notion of a generalized contact form. To such a form we
associate a generalized Reeb field and a partial connection. From these data,
we explicitly constructed a pseudo--Riemannian metric on of split
signature. We prove that a change of the generalized contact form only leads to
a conformal rescaling of this metric, so the corresponding conformal class is
intrinsic to the distribution. In the second part of the article, we relate
this conformal class to the canonical Cartan connection associated to the
distribution. This is used to prove that it coincides with the conformal class
constructed by Nurowski.Comment: AMSLaTeX, 23 page
Essential Killing fields of parabolic geometries
We study vector fields generating a local flow by automorphisms of a
parabolic geometry with higher order fixed points. We develop general tools
extending the techniques of [1], [2], and [3]. We apply these tools to almost
Grassmannian, almost quaternionic, and contact parabolic geometries, including
CR structures, to obtain descriptions of the possible dynamics of such flows
near the fixed point and strong restrictions on the curvature. In some cases,
we can show vanishing of the curvature on a nonempty open set. Deriving
consequences for a specific geometry entails evaluating purely algebraic and
representation-theoretic criteria in the model homogeneous space. Published in
Indiana University Mathematics Journal.Comment: 50 pages. Minor corrections, references update
Twisted Courant algebroids and coisotropic Cartan geometries
In this paper, we show that associated to any coisotropic Cartan geometry
there is a twisted Courant algebroid. This includes in particular parabolic
geometries. Using this twisted Courant structure, we give some new results
about the Cartan curvature and the Weyl structure of a parabolic geometry. As
more direct applications, we have Lie 2-algebra and 3D AKSZ sigma model with
background associated to any coisotropic Cartan geometry
C^1 Deformations of almost-Grassmannian structures with strongly essential symmetry
We construct a family of -almost Grassmannian structures of regularity
, each admitting a one-parameter group of strongly essential
automorphisms, and each not flat on any neighborhood of the higher-order fixed
point. This shows that Theorem 1.3 of [9] does not hold assuming only
regularity of the structure (see also [2, Prop 3.5]).Comment: 24 p
- …