Abstract. Let M be a compact spacetime which admits a regular globally hyperbolic covering, and C a nontrivial free timelike homotopy class of closed timelike curves in M. We prove that C contains a longest curve (which must be a closed timelike geodesic) if and only if the timelike injectivity radius of C is finite; i.e., C has a bounded length. As a consequence among others, we deduce that for a compact static spacetime there exists a closed timelike geodesic within every nontrivial free timelike homotopy class having a finite timelike injectivity radius.

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Abstract. The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold N/Γ, where N is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and Γ a discrete subgroup of N acting on N by left translations. For this purpose, we shall first show that if N is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric g, then the restriction of g to the center Z of N is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the threedimensional Heisenberg Lie group H 3 . If (N, g) is one such group, we prove that no timelike geodesic in (N, g) can be translated by an element of N. By the way, we rediscover that the Heisenberg Lie group H 2k+1 admits a flat left-invariant Lorentz metric if and only if k = 1.

We study geodesic completeness for left-invariant Lorentz metrics on solvable Lie groups.

Abstract. Let M be a compact spacetime which admits a regular globally hyperbolic covering, and C a nontrivial free timelike homotopy class of closed timelike curves in M. We prove that C contains a longest curve (which must be a closed timelike geodesic) if and only if the timelike injectivity radius of C is finite; i.e., C has a bounded length. As a consequence among others, we deduce that for a compact static spacetime there exists a closed timelike geodesic within every nontrivial free timelike homotopy class having a finite timelike injectivity radius.

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We construct the first examples of geodesically complete compact spacetimes admitting a regular globally hyperbolic covering, but which do not contain closed causal geodesics.