Lorentzian manifolds with causal Killing vector field: causality and geodesic connectedness
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Lorentzian manifolds with causal Killing vector field: causality and geodesic connectedness C. Atindogbe1 · M. Gutiérrez2 · R. Hounnonkpe1 Received: 22 October 2019 / Accepted: 23 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove that a compact Lorentzian manifold (M, 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. Keywords Causality · Killing horizons · Causal Killing vector field Mathematics Subject Classification 53C50 · 53C40
1 Introduction Hopf–Rinow theorem is an important tool in Riemannian geometry. It gives an equivalence between Cauchy completeness, geodesic completeness and finite compactness (i.e., bounded sets have compact closure). It also guarantees the existence of at least one geodesic joining any two distinct points of a complete Riemannian manifold. This last property is known as geodesic connectedness.
* M. Gutiérrez [email protected] C. Atindogbe [email protected] R. Hounnonkpe [email protected] 1
Université d’Abomey-Calavi, Abomey‑Calavi, Benin
2
Dto. Álgebra, Geometría y Topología, Universidad de Málaga, Málaga, Spain
13
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C. Atindogbe et al.
For semi-Riemannian manifolds the situation is quite different. Compact Riemannian manifolds are always complete and geodesically connected. However, there are examples showing compact spacetimes not geodesically complete and not geodesically connected, [4], even for globally hyperbolic spacetimes. The problem of geodesic connectedness in semi-Riemannian manifolds has been widely studied from very different viewpoints. This topic has applications in Physics, and it is challenging from both an analytical and a geometrical point of view. In [6] it is used pseudoconvexity and disprisonment to study geodesic connectedness. The variational approach is used in [8]. For stationary spacetime (with complete stationary vector field), it is proved that the spacetime is geodesically connected if it admits a complete (smooth, spacelike) Cauchy hypersurface [7] and a similar result was obtained in the case the Killing vector field is assumed lightlike [3]. It is proved that compact static spacetimes are totally vicious and geodesically connected. In fact the first claim holds even for compact conformally stationary spacetime; nevertheless, it is no longer true if we consider a ca
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