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Geodesics in static Lorentzian manifolds with critical quadratic behavior. (English) Zbl 1049.53048

A Lorentzian manifold \(M\) is static if \(M = N\times R\) and the Lorentzian metric is of the form: \(g-f(x)dt^2\) for a Riemannian metric \(g\) on \(N\). Among other things, the authors prove that \(M\) is geodesically connected under the assumption that \(N\) is complete and \(f\) grows at most quadratically at infinity. The problem is reduced to a critical variational problem for a certain functional in a suitable set of curves on \(N\). Multiplicity of connecting geodesics and extensions to the noncomplete case are also studied and discussed.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C22 Geodesics in global differential geometry
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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