The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 33–82 | Cite as

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

  • Thomas LeistnerEmail author
  • Andree Lischewski


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.


Lorentzian geometry Holonomy groups Parallel null vector field Cauchy problem Killing spinors Parallel spinors Symmetric hyperbolic system 

Mathematics Subject Classification

Primary 53C50 53C27 53C29 Secondary 53C26 53C44 35L02 35L10 83C05 



We would like to thank Helga Baum and Vicente Cortés for inspiring discussions. TL would like to thank Nick Buchdahl, Mike Eastwood, Jason Lotay and Spiro Karigiannis for helpful discussions in regards to the open problem in Sect. 7.3. AL would like to thank Todd Oliynyk for discussions about symmetric hyperbolic systems during his visit to Monash University. This work was supported by the Australian Research Council via the Grants FT110100429 and DP120104582


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Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  2. 2.Insitut für MathematikHumboldt Universität zu BerlinBerlinGermany

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