Abstract
We prove that a compact Lorentzian manifold \(({\overline{M}},{\overline{g}})\) admitting a causal Killing vector field is totally vicious or it contains a compact achronal Killing horizon. In particular a compact spacetime which satisfies the null generic condition and admits a causal Killing vector field is totally vicious. If in addition, its universal Lorentzian covering is globally hyperbolic then it is geodesically connected. In the non-compact case, we prove that a chronological spacetime admitting a complete causal Killing vector field, a smooth spacelike partial Cauchy hypersurface S and satisfying the null generic condition is stably causal. If additionally S is compact then the spacetime is globally hyperbolic. We also determine the geodesic connectedness in this case.
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Acknowledgements
The second and third authors has been partially supported by the Ministerio de Economía y Competitividad, [Grant Number FEDER-MTM2016-78647-P]. The third author has been partially supported by the African Centre of Excellence in Mathematical Sciences and Applications ACE-SMA.
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Atindogbe, C., Gutiérrez, M. & Hounnonkpe, R. Lorentzian manifolds with causal Killing vector field: causality and geodesic connectedness. Annali di Matematica 199, 1895–1908 (2020). https://doi.org/10.1007/s10231-020-00948-9
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DOI: https://doi.org/10.1007/s10231-020-00948-9