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58
Obstructions to conformally Einstein metrics in n dimensions
 JOURNAL OF GEOMETRY AND PHYSICS
, 2005
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Infinitesimal Automorphisms and Deformations of Parabolic Geometries
, 2005
"... We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de–Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to ..."
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Cited by 33 (8 self)
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We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de–Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to the underlying geometric structure using the machinery of BGG sequences. In the locally flat case, this leads to a deformation complex, which generalizes the is well know complex for locally conformally flat manifolds. Recently, a theory of subcomplexes in BGG sequences has been developed. This applies to certain types of torsion free parabolic geometries including, quaternionic structures, quaternionic contact structures and CR structures. We show that for these structures one of the subcomplexes in the adjoint BGG sequence leads (even in the curved case) to a complex governing deformations in the subcategory of torsion free geometries. For quaternionic structures, this deformation complex is elliptic.
Conformal holonomy of Cspaces, Ricciflat, and Lorentzian manifolds
, 2008
"... The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2dimensional totally isotropic invariant subspace. Furthermore, ..."
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Cited by 24 (11 self)
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The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2dimensional totally isotropic invariant subspace. Furthermore, for semiRiemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a Cspace is a Berger algebra. For Ricciflat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and obtain results for
Nonlinear Realizations of Conformal Symmetry and Effective Field Theory for the PseudoConformal Universe
 JCAP
, 2012
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Contact Projective Structures
 Indiana Univ. Math. J
"... Abstract. A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact distribution and such that given a point and a onedimensional subspace of the contact distribution at that point there is a unique path of the family passing through the g ..."
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Cited by 15 (2 self)
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Abstract. A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact distribution and such that given a point and a onedimensional subspace of the contact distribution at that point there is a unique path of the family passing through the given point and tangent to the given subspace. A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with onedimensional fibers over the contact manifold and using this the local equivalence problem for contact projective structures is solved by the construction of a canonical regular Cartan connection. This Cartan connection is normal if and only if an invariant contact torsion vanishes. Every contact projective structure determines canonical paths transverse to the contact structure which fill out the contact projective structure to give a full projective structure, and the vanishing of the contact torsion implies the contact
CR tractors and the Fefferman space
"... ABSTRACT. We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman’s construction of a canonical conformal class on the total space of a circle bundle over a nondegenerate CR manifold of hypersurface type. In particular, we co ..."
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Cited by 14 (5 self)
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ABSTRACT. We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman’s construction of a canonical conformal class on the total space of a circle bundle over a nondegenerate CR manifold of hypersurface type. In particular, we construct and treat the basic objects that relate the natural bundles and natural operators on the two spaces. This is illustrated with several applications: We prove that a number of conformally invariant overdetermined systems, including Killing form equations and the equations for twistor spinors, admit nontrivial solutions on any Fefferman space. We show that, on a Fefferman space, the space of infinitesimal conformal isometries naturally decomposes into a direct sum of subspaces, which admit an interpretation as solutions of certain CR invariant PDE’s. Finally we explicitly analyse the relation between tractor calculus on a CR manifold and the complexified conformal tractor calculus on its Fefferman space, thus obtaining a powerful computational tool for working with the Fefferman construction. 1.
Parabolic geometries, CR–tractors, and the Fefferman construction
 DIFFERENTIAL GEOM. APPL
, 2002
"... This is a survey on recent joint work with A.R. Gover on the geometry of non–degenerate CR manifolds of hypersurface type. Specifically we discuss the relation between standard tractors on one side and the canonical Cartan connection, the construction of the Fefferman space and the ambient metric ..."
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Cited by 14 (2 self)
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This is a survey on recent joint work with A.R. Gover on the geometry of non–degenerate CR manifolds of hypersurface type. Specifically we discuss the relation between standard tractors on one side and the canonical Cartan connection, the construction of the Fefferman space and the ambient metric construction on the other side. To put these results into perspective, some parts of the general theory of parabolic geometries are discussed.