Codazzi spinors and globally hyperbolic manifolds with special holonomy

Baum, Helga; Müller, Olaf

doi:10.1007/s00209-007-0169-5

Cited by 22 publications

(39 citation statements)

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“…On the other hand, all complete Riemannian spin manifolds (M, g) with imaginary W-Killing spinor for an invertible Codazzi tensor W arise in this way. (For a proof see [6]). For a proof of all these statements see [16].…”

Section: Riemannian Manifolds Satisfying the Constraint Equationsmentioning

confidence: 99%

Cauchy problems for Lorentzian manifolds with special holonomy

Baum¹,

Leistner²,

Lischewski³

2016

Differential Geometry and its Applications

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Abstract. On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions can be extended to a Lorentzian manifold with a parallel null vector field. Similarly, every parallel null spinor on a Lorentzian manifold induces an imaginary generalised Killing spinor on a space-like hypersurface. Then, based on the fact that a parallel spinor field induces a parallel vector field, we can apply the first result to prove: every real analytic Riemannian manifold carrying a real analytic, imaginary generalised Killing spinor can be extended to a Lorentzian manifold with a parallel null spinor. Finally, we give examples of geodesically complete Riemannian manifolds satisfying the constraint conditions. Background and main resultsThis paper is a contribution to the research programme of studying global and causal properties of Lorentzian manifolds with special holonomy. A Lorentzian manifold has special holonomy if the connected component of its holonomy group is reduced from the full group SO 0 (1, n), but still acts indecomposably, i.e., without non-degenerate invariant subspaces. In this situation the Lorentzian manifold admits a bundle of tangent null lines that is invariant under parallel transport. The possible special Lorentzian holonomy groups were classified in [7] and [14], all of them can be realised by local metrics [12], but many questions about the consequences of special holonomy for global and causal properties of the manifold are still open. A special case of this situation is when the parallel null line bundle is spanned by a parallel null vector field. This is the case we will study in this paper. It is motivated by the question which Lorentzian manifolds admit a parallel spinor field, which in turn draws its motivation from mathematical physics. Since a parallel spinor is invariant under the spin representation of the holonomy group, indecomposable Lorentzian manifolds with parallel spinors have special holonomy. However, since SO 0 (1, n) has no proper irreducible subgroups, the situation is very different from the Riemannian case, where we have several irreducible holonomy groups that admit an invariant spinor. In fact, a spinor field φ on any Lorentzian manifold (M, g) induces a causal vector field V φ , its Dirac current, which is defined by g(X, V φ ) = − X · φ, φ , 2010 Mathematics Subject Classification. Primary 53C50, 53C27; Secondary 53C44, 35A10, 83C05.

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Section: Riemannian Manifolds Satisfying the Constraint Equationsmentioning

confidence: 99%

Cauchy problems for Lorentzian manifolds with special holonomy

Baum¹,

Leistner²,

Lischewski³

2016

Differential Geometry and its Applications

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“…We can extend Σ with little modifications to oriented isometric codimension-one immersions instead of isometric diffeomorphisms. Here, df : SO g1 M 1 → SO g2 M 2 is replaced by (df, ν) which completes the pushed-forward basis by the right choice of normal vector ν to an oriented orthonormal basis, for details and applications see [BGM05] or [BM08]. If the dimension of the hypersurface is even and we want to interpret Σ in a contravariant manner, we need two spinor bundles on the hypersurface, which can be identified with the spinor bundle in the ambient space.…”

Section: Naturality Questionsmentioning

confidence: 99%

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Müller

Nowaczyk

2016

Lett Math Phys

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Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.where ρ := ρ r,s : Spin r,s → GL(Σ r,s ) is a complex spinor representation, Θ g : Spin g M → SO g M is a metric spin structure and Spin r,s ⊂ GL + m is the spin group. Main Theorem 1 (universal spinor bundle, cf. Theorem 2.25). There exists a finite dimensional vector bundleπ Σ SM :ΣM → J 1 π r,s such that for each metric g ∈ S r,s (M ), there existsῑ g such thatcommutes. Here, J 1 π r,s denotes the first jet bundle of π r,s . Moreover,π Σ SM carries a connection, a metric and a Clifford multiplication such thatῑ g is a morphism of (generalized) Dirac bundles (see Definition 2.27). In addition,π Σ SM is natural with respect to spin diffeomorphisms. ♦The claims of the last assertions mean that not only the vector bundle structure of any metric spinor bundle π g M can be recovered fromπ Σ SM , but also its spinorial connection, its metric and its Clifford multiplication, see Theorem 2.25 for the precise meaning and a proof. We would like to emphasize that the morphism j 1 (g) in (1.2) is the 1-jet of the metric g and that the naturality assertion is formulated with respect to diffeomorphisms and not just isometries, see Section 2.6 for details. arXiv:1504.01034v3 [math.DG] 6 Oct 2015 ∼ = 9 9 We setκ V := κ V • θ. ♦ Diagram (2.1) easily extends to spin manifolds as follows: Let {ρ r,s : Spin r,s → GL(Σ r,s )} r,s∈N be a fixed choice of Spin r,s -representations. The construction of κ V andκ V induces bundle maps κ M : GL + M → S r,s M,κ M : GL + M → S r,s M, (2.2) by setting κ M | GL + x M := κ TxM andκ M | GL + x M :=κ TxM for any x ∈ M . Definition 2.3 (universal spinor bundle). The map π Σ SM : ΣM → S r,s M [b, σ] →κ M (b), where ΣM := GL + M × ρr,s Σ r,s is called universal spinor (vector) bundle. We obtain the following diagram ΣM π Σ SM / / π Σ M 9 9 S r,s M π r,s / / M, (2.3)where π Σ M := π r,s • π Σ SM . The map π Σ M is called universal spinor (fiber) bundle of M . Its sections are called universal spinor fields.♦

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“…It would be interesting to obtain examples of conformally flat Lorentzian manifolds satisfying some global geometric properties, e.g., important are globally hyperbolic Lorentzian manifolds with special holonomy groups [17], [19].…”

Section: )mentioning

confidence: 99%

“…Similar problem for Lorentzian manifolds was considered in [40], [52], and it was solved in [98], [99]. The relation of the holonomy groups of Lorentzian manifolds with the solutions of some other spinor equations is discussed in [12], [13], [17] and in physical literature that is cited below.…”

mentioning

confidence: 99%

“…The problem to construct examples of Lorentzian manifolds with various holonomy groups and additional global geometric properties springs up. In [17], [19] constructions of globally hyperbolic Lorentzian manifolds with some classes of the holonomy groups are given. The global hyperbolicity is a strong casuality condition in Lorentzian geometry that generalizes the general notion of completeness in Riemannian geometry.…”

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Holonomy groups of Lorentzian manifolds

Galaev¹

2015

Russ. Math. Surv.

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In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian manifolds with parallel and recurrent spinor fields, conformally flat Walker metrics and the classification of 2-symmetric Lorentzian manifolds are considered.Comment: 49 pages; this is a survey on Lorentzian holonomy groups and their application

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Codazzi spinors and globally hyperbolic manifolds with special holonomy

Cited by 22 publications

References 16 publications

Cauchy problems for Lorentzian manifolds with special holonomy

Cauchy problems for Lorentzian manifolds with special holonomy

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Holonomy groups of Lorentzian manifolds

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