An intrinsic approach to the geodesical connectedness of stationary Lorentzian manifolds
It is shown that any two points in a stationary Lorentzian manifold can be joined by a geodesic, provided that a certain coercivity condition is satisfied. As a matter of fact, this coercivity condition is very restrictive. In particular, it requires the stationary Lorentzian manifold to be globally hyperbolic. The authors prove their result with the help of techniques from global nonlinear analysis, characterizing geodesics joining two points as the solution curves of a certain variational problem in an infinite dimensional Hilbert manifold setting. V.Perlik (Berlin) A Lorentzian manifold \(M\) is said to be geodesically connected if, for any two points \(p,q\in M\) there exists at least one geodesic in \(M\) with endpoints \(p\) and \(q\). On the other hand, \(M\) is said to be stationary if it admits a time orientation given by a timelike Killing vector field. The main purpose of the paper is to prove the geodesical connectedness for stationary Lorentzian manifolds satisfying a coercivity condition. The authors show that the coercivity condition implies global hyperbolicity of \(M\). Also, they give an example to show that global hyperbolicity, in general, does not imply the geodesical connectedness of stationary manifolds. Finally, whenever \(M\) is non-contractible the authors obtain a multiplicity result for geodesics in \(M\) joining two fixed points. Aurel Bejancu (Iaşi)
scientific article; zbMATH DE number 4219088