Abstract
Recently, the authors have obtained criteria for the integral curves of a nonsingular smooth vector field X on a smooth manifold M to be timelike, null or spacelike geodesics for some Lorentzian metric g for M. In this paper, we show that for smoothly contractible subsets S of ℝ2 null geodesibility of a vector field X is equivalent to X being preHamiltonian on S and timelike, spacelike or Riemannian pregeodesibility of X are all equivalent to X being gradient-like. It turns out that null geodesibility is quite rare as we prove that even among real analytic vector fields on S there are many open sets of vector fields which fail to be preHamiltonian.
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Partially supported by a grant from the Weldon Springs endowment of the University of Missouri-Columbia
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Chicone, C., Ehrlich, P. Gradient-like and integrable vector fields on ℝ2 . Manuscripta Math 49, 141–164 (1984). https://doi.org/10.1007/BF01168748
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DOI: https://doi.org/10.1007/BF01168748