On a class of geodesically connected Lorentzian manifolds (Q1365220)

The classical Hopf-Rinow theorem for Riemannian (i.e., positive definite) manifolds does not hold in Lorentzian geometry, and a Lorentzian manifold \((M,g)\) may fail to be geodesically connected even under the assumptions of compactness. In this paper, the authors study the problem of the existence of one geodesic, of any causal type, between two fixed points of a Lorentzian manifold of splitting type. These are product manifolds of the form \(M=M_0\times \mathbb{R}\), with the metric given by \(ds^2=\alpha_{ij}dx^idx^j-\beta dt^2\), where \(dx^2=\delta_{ij}dx^idx^j\) is a positive definite metric on \(M_0\), \(\alpha(x,t)=(\alpha_{ij}(x,t))\) is a positive operator on \(T_xM_0\) and \(\beta\) is a smooth positive scalar field on \(M\). The main result of existence, that improves previous results by F.\ Giannoni and A.\ Masiello, is given in terms of suitable conditions on the metric coefficients \(\alpha\) and \(\beta\). The authors use techniques from critical point theory, and in particular the Rabinowitz saddle point theory is used to get the existence of a critical point for a penalized action functional \(f_\varepsilon\), \(\varepsilon>0\). A critical point for the action functional \(f\) is then obtained by a limit process as \(\varepsilon\downarrow0\) based on a priori estimates on the critical points of \(f_\varepsilon\).