Abstract
We investigate the differential geometry and topology of globally hyperbolic four-manifolds (M, g) admitting a parallel real spinor \(\varepsilon \). Using the theory of parabolic pairs recently introduced in [22], we first formulate the parallelicity condition of \(\varepsilon \) on M as a system of partial differential equations, the parallel spinor flow equations, for a family of polyforms on an appropriate Cauchy surface \(\Sigma \hookrightarrow M\). The existence of a parallel spinor on (M, g) induces a system of constraint partial differential equations on \(\Sigma \), which we prove to be equivalent to an exterior differential system involving a cohomological condition on the shape operator of the embedding \(\Sigma \hookrightarrow M\). 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 \(({\mathfrak {e}},\Theta )\), where \({\mathfrak {e}}\) is a coframe and \(\Theta \) is a symmetric two-tensor. We characterize all parallel Cauchy pairs on simply connected Cauchy surfaces, refining a result of 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 \({\mathbb {R}}^2\) 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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Notes
Recall that \(-1 < \mu \le 1\) and \(\mu \ne 0\).
Note however that there is a typo in Exercise 11, the correct condition being, using the notation of the exercise, \(\vert X(p)\vert < c\) rather than \(\vert X(p)\vert > c\).
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We would like to thank V. Cortés, T. Leistner, and A. Moroianu for very interesting discussions and comments. Part of this work was undertaken during a visit of C.S.S. to the University Paris-Saclay under the Deutsch–Französische Procope Mobilität program. C.S.S. would like to thank this very welcoming institution for providing a nice and stimulating working environment. The work of Á.M. is funded by the Spanish FPU Grant No. FPU17/04964, with additional support from the MCIU/AEI/FEDER UE Grant PGC2018-095205-B-I00 and the Centro de Excelencia Severo Ochoa Program Grant SEV-2016-0597. The work of C.S.S. is supported by the Germany Excellence Strategy Quantum Universe - 390833306.
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Murcia, Á., Shahbazi, C.S. Parallel spinors on globally hyperbolic Lorentzian four-manifolds. Ann Glob Anal Geom 61, 253–292 (2022). https://doi.org/10.1007/s10455-021-09808-y
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DOI: https://doi.org/10.1007/s10455-021-09808-y