The purpose of this paper is on the one hand to extend and generalize, in terms of Clifford translations, some results in a previous paper (Math. Z. 239 (2002), 277-291) concerning the existence of closed timelike geodesics in compact spacetimes, and on the other hand to prove that a compact flat spacetime (M, g) contains a closed timelike geodesic if and only if the fundamental group pi(1)(M) contains a non-trivial timelike translation. [References: 23]
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