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1
Content available remote

Subtleties concerning conformal tractor bundles

100%
Graham C., Willse T.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1721-1732
EN
The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures.
2
Content available remote

Geometry of some twistor spaces of algebraic dimension one

100%
Honda N.
Complex Manifolds
|
2015
|
tom 2
|
nr 1
EN
It is shown that there exists a twistor space on the n-fold connected sum of complex projective planes nCP2, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over nCP2 for any n ≥ 5, while the latter kind of example is constructed over 5CP2. Both of these seem to be the first such example on nCP2. The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces. It is also shown that the former kind of twistor spaces contain a pair of non-normal Hopf surfaces.
3
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Projective relatedness and conformal flatness

81%
Hall G.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1763-1770
EN
This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensional manifolds admitting a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds is conformally flat and not of the most general holonomy type for that signature then, in general, the connections of the manifolds involved are the same and the second manifold is also conformally flat. Counterexamples are provided which place limitations on the potential strengthening of the results.
4
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Transitive conformal holonomy groups

62%
Alt J.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1710-1720
EN
For (M, [g]) a conformal manifold of signature (p, q) and dimension at least three, the conformal holonomy group Hol(M, [g]) ⊂ O(p + 1, q + 1) is an invariant induced by the canonical Cartan geometry of (M, [g]). We give a description of all possible connected conformal holonomy groups which act transitively on the Möbius sphere S p,q, the homogeneous model space for conformal structures of signature (p, q). The main part of this description is a list of all such groups which also act irreducibly on ℝp+1,q+1. For the rest, we show that they must be compact and act decomposably on ℝp+1,q+1, in particular, by known facts about conformal holonomy the conformal class [g] must contain a metric which is either Einstein (if p = 0 or q = 0) or locally isometric to a so-called special Einstein product.
5
Content available remote

Smooth metric measure spaces, quasi-Einstein metrics, and tractors

62%
Case J.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1733-1762
EN
We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.
6
Content available remote

Erratum to: “Smooth metric measure spaces, quasi-Einstein metrics, and tractors”

62%
Case J.
Open Mathematics
|
2013
|
tom 11
|
nr 1
196-196
EN
The original version of the article was published in Central European Journal of Mathematics, 2012, 10(5), 1733–1762, DOI: 10.2478/s11533-012-0091-x. Unfortunately, it contains a typographical error in display of (27). Here we display the correct version of the equations.
7
Content available remote

Essential Killing fields of parabolic geometries: projective and conformal structures

52%
Čap A., Melnick K.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2053-2061
EN
We use the general theory developed in our article [Čap A., Melnick K., Essential Killing fields of parabolic geometries, Indiana Univ. Math. J. (in press)], in the setting of parabolic geometries to reprove known results on special infinitesimal automorphisms of projective and conformal geometries.
8
Content available remote

On conformally flat Lorentz parabolic manifolds

52%
Kamishima Y.
Open Mathematics
|
2014
|
tom 12
|
nr 6
861-878
EN
We introduce conformally flat Fefferman-Lorentz manifold of parabolic type as a special class of Lorentz parabolic manifolds. It is a smooth (2n+2)-manifold locally modeled on (Û(n+1, 1), S 2n+1,1). As the terminology suggests, when a Fefferman-Lorentz manifold M is conformally flat, M is a Fefferman-Lorentz manifold of parabolic type. We shall discuss which compact manifolds occur as a conformally flat Fefferman-Lorentz manifold of parabolic type.
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