The Cauchy Problems for Einstein Metrics and Parallel Spinors

Ammann, Bernd; Moroianu, Andrei; Moroianu, Sergiu

doi:10.1007/s00220-013-1714-1

Cited by 20 publications

(67 citation statements)

References 48 publications

(49 reference statements)

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“…Clearly, condition (1.6) arises as a generalisation of the equation for imaginary Killing spinors, for which W = i 2 Id, see [8]. Moreover, solutions to equation (1.6) are the counterpart to real generalised Killing spinors which have been in the focus of recent research, for example in [1,2]. Given a solution to (1.6), we denote by U ϕ ∈ X(M) the Dirac current of ϕ, given by 7) and assume that ϕ solves the algebraic constraint…”

Section: Background and Main Resultsmentioning

confidence: 99%

Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors

Leistner

Lischewski

2017

J Geom Anal

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We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups these flow equations have been investigated in the literature before in other contexts. As an application, the results provide a classification of Riemannian manifolds admitting imaginary generalised Killing spinors. We will also give new local normal forms for Lorentzian metrics with parallel null spinor in any dimension.

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Section: Background and Main Resultsmentioning

confidence: 99%

Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors

Leistner

Lischewski

2017

J Geom Anal

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“…In Section 5 we will study to the problem of finding parallel spinors on the Lorentzian manifold in (1). For Riemannian manifolds, the corresponding Cauchy problem was studied by Ammann, Moroianu and Moroianu [1] in relation to the Cauchy problem for Ricci-flat manifolds. But, in contrast to the Riemannian situation, Lorentzian manifolds with parallel spinors are not necessarily Ricci-flat.…”

Section: Background and Main Resultsmentioning

confidence: 99%

“…In fact, Lemma 3.3 reveals that the problematic evolution equation (39) for g t is equivalent to the curvature condition (26) from Lemma 3.2. In order to overcome the symmetry issue, we next show that for analytic Lorentzian manifolds of the form (5), one can alternatively characterize parallel null vector fields by relaxing (26) and using a method similar to the one in [1]. This approach turns out to yield evolution equations for symmetric bilinear forms g t .…”

Section: Solving the Evolution Equations For Analytic Datamentioning

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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“…where ϕ is a spinor field which is represented in a frames ∈Q by [ϕ] belonging to Spin (3,1). This is exactly the spinor representation of an immersion in H 3 as described in [4] Section 5, where it is moreover proved that it is equivalent to the…”

Section: 2mentioning

confidence: 98%

“…However the question whether in general a manifold of arbitrary dimension carrying a generalized Killing spinor can be immersed isometrically into some Euclidean space remained until now unanswered. Some of the few achievement in this direction were obtained in [1] for real analytic manifolds and in [3,19] when A is a Codazzi tensor, showing the existence of an immersion into a Ricci flat manifold admitting a parallel spinor which restricts to ϕ.…”

Section: Introductionmentioning

confidence: 99%

Spinorial representation of submanifolds in Riemannian space forms

2017

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In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Riemannian space forms in terms of the existence of so called generalized Killing spinors. We then discuss several applications, among them a new and concise proof of the fundamental theorem of submanifold theory. We also recover results of T. Friedrich, B. Morel and the authors in dimension 2 and 3.

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The Cauchy Problems for Einstein Metrics and Parallel Spinors

Cited by 20 publications

References 48 publications

Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors

Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors

Cauchy problems for Lorentzian manifolds with special holonomy

Spinorial representation of submanifolds in Riemannian space forms

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