An intrinsic approach to Ljusternik-Schnirelman theory for light rays on Lorentzian manifolds

Abstract

In this paper it is proven the existence of light--like geodesics joining an event $p$ and a time--like vertical curve $\gamma$ of a Lorentzian manifold $\mathcal{M}$ endowed with a Universal Time Function $T$, under a certain compactness condition. Moreover, it is developed a Ljusternik--Schnirelman theory for light rays, using which it is shown that, if the topology of $\mathcal{M}$ satisfies a non--triviality condition, then there are multiple light rays joining $p$ with $\gamma$. The results are obtained under intrinsic assumptions on the manifold $\mathcal{M}$, that do not involve the coefficients of the Lorentzian metric