Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors

Ammann, Bernd; Kroencke, Klaus; Müller, Olaf

doi:10.48550/arxiv.1903.02064

Cited by 1 publication

(4 citation statements)

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“…The constraint equations. Using Lemma 2.6 together with the resolution of the initial value problem for a parallel null spinor presented in [5,33,35], see also [1,6], we obtain the following characterization of parallel spinors on globally hyperbolic Lorentzian four-manifolds. Proposition 2.9.…”

Section: ) and (25)mentioning

confidence: 98%

“…Existence of such S g is obstructed. The obstruction was shown in [31,32] to be equivalent to the existence of a spin structure Q g , in which case S g can be considered to be a vector bundle associated to Q g through the tautological representation induced by the natural embedding Spin + (3, 1) ⊂ Cl (3,1), where Spin + (3, 1) denotes the connected component of the identity of the spin group in signature (3, 1) = − + + + and Cl(3, 1) denotes the real Clifford algebra in signature (3,1). We will assume, without loss of generality, that (M, g) is spin and equipped with a fixed spin structure Q g .…”

Section: Parallel Real Spinors On Lorentzian Four-manifoldsmentioning

confidence: 99%

“…In all these cases (with µ = 1), (Σ, h e ) ֒→ (M, g) is a totally umbilical submanifold of (M, g). 1 Recall that −1 < µ ≤ 1 and µ = 0.…”

Section: 4mentioning

confidence: 99%

“…(Σ, h e ) = (R 2 × S 1 , dx 1 ⊗ dx 1 + dx 2 ⊗ dx 2 + e 2u dθ ⊗ dθ)/ ρ a (1), ρ b (1) , where ρ a (1), ρ b (1) ∈ Diff(S 1 ) are rotations of rationally independent constant irrational angle, respectively, and u ∈ C ∞ (R 2 ) is a function depending only on x 1 and x 2 .…”

Section: Introductionmentioning

confidence: 99%

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Parallel spinors on globally hyperbolic Lorentzian four-manifolds

Murcia¹,

Shahbazi²

2020

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We investigate the differential geometry and topology of globally hyperbolic fourmanifolds (M, g) admitting a parallel real spinor ε. Using the theory of parabolic pairs recently introduced in [22], we first formulate the parallelicity condition of ε on M as a system of partial differential equations, the parallel spinor flow equations, for a family of polyforms on any given Cauchy surface Σ ֒→ M . Existence of a parallel spinor on (M, g) induces a system of constraint partial differential equations on Σ, which we prove to be equivalent to an exterior differential system involving a cohomological condition on the shape operator of the embedding Σ ֒→ M . Solutions of this differential system are precisely the allowed initial data for the evolution problem of a parallel spinor and define the notion of parallel Cauchy pair (e, Θ), where e is a coframe and Θ is a symmetric two-tensor. We characterize all parallel Cauchy pairs on simply connected Cauchy surfaces, refining a result of Baum, Leistner and Lischewski. Furthermore, we classify all compact three-manifolds admitting parallel Cauchy pairs, proving that they are canonically equipped with a locally free action of R 2 and are isomorphic to certain torus bundles over S 1 , whose Riemannian structure we characterize in detail. Moreover, we classify all left-invariant parallel Cauchy pairs on simply connected Lie groups, specifying when they are allowed initial data for the Ricci flat equations and when the shape operator is Codazzi. Finally, we give a novel geometric interpretation of a class of parallel spinor flows and solve it in several examples, obtaining explicit families of four-dimensional Lorentzian manifolds carrying parallel spinors.

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