Normal geodesics in Space-Times

Let $\m = \mo \times \R$ be a Lorentzian manifold equipped with a static metric $\g_L = \langle\alpha(x)\cdot,\cdot\rangle - \beta(x) dt^2$ where $\beta$ has a subquadratic growth. Then, fixed $P_0$, $P_1$ submanifolds of $\mo$, a suitable version of the Fermat principle and the classical Ljusternik-Schnirelman theory allow to prove that the existence of normal lightlike geodesics joining $P_0 \times \{0\}$ to $P_1 \times \R$ is influenced by the topology of $\mo$, $P_0$ and $P_1$