Geodesical connectedness of compact Lorentzian manifolds (Q2785856)

In this paper, using global variational methods, the authors prove the geodesic connectedness (i.e., the existence of a geodesic joining two arbitrary points) of compact Lorentzian manifolds of the form \({\mathcal M}= {\mathcal M}_0 \times S^1\), where \({\mathcal M}_0\) is a connected, compact Riemannian manifold, \(S^1\) is the circle, immersed as a timelike submanifold of \({\mathcal M}\), and the metric \(g\) is of the form NEWLINE\[NEWLINEg(z) [\zeta,\zeta] =g(z) \bigl[(\xi, \tau), (\xi,\tau) \bigr] =\bigl\langle \alpha(x, t) \xi,\xi\bigr \rangle -\beta(x,t)\tau^2,NEWLINE\]NEWLINE where \(z=(x,t) \in{\mathcal M}\), \(\zeta= (\xi,\tau) \in T_z {\mathcal M} =T_x{\mathcal M}_0 \times T_tS^1\), \(\langle \cdot, \cdot \rangle\) is the Riemannian metric on \({\mathcal M}_0\), \(\alpha(x,t): T_x{\mathcal M}_0 \to T_x {\mathcal M}_0\) is a positive linear operator on \(T_x{\mathcal M}_0\), smoothly depending on \(z\), and \(\beta (z)\) is a smooth positive scalar field.NEWLINENEWLINENEWLINEThe authors also prove the existence of geodesics of arbitrarily large energy between two fixed points and they give conditions on the existence of timelike geodesics (i.e., with negative energy) joining the points.