On the Distribution of Conjugate Points along semi-Riemannian Geodesics

Helfer in [Pacific J. Math. 164/2 (1994), p. 321--350] was the first to produce an example of a spacelike Lorentzian geodesic with a continuum of conjugate points. In this paper we show the following result: given an interval $[a,b]$ of $IR$ and any closed subset $F$ of $IR$ contained in $]a,b]$, then there exists a Lorentzian manifold $(M,g)$ and a spacelike geodesic $\gamma:[a,b]\to M$ such that $\gamma(t)$ is conjugate to $\gamma(a)$ along $\gamma$ iff $t\in F$.