Projective holonomy II: cones and complete classifications

Armstrong, Stuart

doi:10.1007/s10455-007-9075-7

Cited by 22 publications

(103 citation statements)

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“…Of course a parallel tractor may be reinterpreted as a reduction of the (projective) tractor holonomy. By definition this is additional geometric structure, and as mentioned in the introduction, local and generic aspects of this have been explored in, for example, [3,4]. In the examples of the next section we shall see, from our current point of view, how such classical structures arise.…”

Section: 12mentioning

confidence: 99%

“…By their definition, normal solutions are related to holonomy reductions of the Cartan/tractor connection and from this perspective certain local aspects have been investigated in [3,4]. There one sees that, on the one hand, the available holonomy groups restrict the range of curved cases (although the treatment there classifies irreducible holonomy algebras, and so is not exhaustive), but on the other hand within the allowed groups interesting geometric structures arise including various pseudo-Riemannian Einstein (mentioned above) and contact adapted projective structures in the sense of [24].…”

Section: Introductionmentioning

confidence: 99%

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Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

Čap

Gover

Hammer

2012

Journal of the London Mathematical Society

109

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Abstract. For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geometric contact with the model to a level that is optimal, in a suitable sense. In terms of the trivialisations arising from the special frames, normal solutions of classes of natural linear PDE (so-called first BGG equations) are shown to be necessarily polynomial in the generalised homogeneous coordinates; the polynomial system is the pull back of a polynomial system that solves the corresponding problem on the model. Thus questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of the corresponding polynomial systems and algebraic sets. We show that a normal solution determines a canonical manifold stratification that reflects an orbit decomposition of the model. Applications include the construction of structures that are analogues of Poincaré-Einstein manifolds.

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Section: 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

Čap

Gover

Hammer

2012

Journal of the London Mathematical Society

109

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“…This cone result will allow us, using [3] and the original papers [23,24], to construct all examples of possible irreducible projective Tractor holonomy, and demonstrate that there are p + q ≥ 5 g 2 R (4,3) so(n, C) C n n ≥ 5 g 2 (C) C 7 su( p, q)…”

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confidence: 95%

Projective holonomy I: principles and properties

Armstrong

2007

Ann Glob Anal Geom

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The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection. After a brief historical background, this paper presents and analyses the projective Cartan and Tractor connections, the various structures they can preserve, and their geometric interpretations. Preserved subbundles of the Tractor bundle generate foliations with Ricci-flat leaves. Contact-and Einstein-structures arise from other reductions of the Tractor holonomy, as do U (1) and Sp(1, H) bundles over a manifold of smaller dimension.

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“…For though we have excluded many holonomy algebras from being Ricci-flat, paper [Arm2] has constructed Ricci-flat cones for most of the others, we have not settle the existence of general Ricci-flat connections in some cases. These are the algebras concerned (those that can be Ricci-flat have been marked with a * ):…”

Section: Low Dimensional Casesmentioning

confidence: 99%

“…This is done mainly in papers [Arm1] and [Arm2], as an adjunct to proving the existence of various projective Tractor holonomies. In fact, projective Tractor holonomies correspond to Ricci-flat Torsion-free affine holonomies on a cone one dimension higher, and [Arm2] constructs these Ricci-flat cones for all higher dimensional holonomy algebras that are not ruled out by this paper.…”

Section: Introductionmentioning

confidence: 99%

Ricci-flat holonomy: A classification

Armstrong¹

2007

Journal of Geometry and Physics

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The reductive holonomy algebras for a torsion-free affine connection are analysed, with the goal of establishing which ones can correspond to a Ricci-flat connection with the same properties. Various families of holonomies are eliminated through different algebraic means, and examples are constructed (in this paper and in 'Projective Geometry II: Holonomy Classification', by the same author) in the remaining cases, thus solving this problem completely, for reductive holonomy.

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Projective holonomy II: cones and complete classifications

Cited by 22 publications

References 19 publications

Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

Projective holonomy I: principles and properties

Ricci-flat holonomy: A classification

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