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Recent Trends in Lorentzian Geometry pp 231–242Cite as

Can We Make a Finsler Metric Complete by a Trivial Projective Change?

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 26))

Abstract

A trivial projective change of a Finsler metric F is the Finsler metric F + d f. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change.Though the problem is purely Finslerian, it was inspired by Lorentz geometry and mathematical relativity: it was observed that it is possible to understand the light-like geodesics of a (normalized, standard) stationary 4-dimensional space time as geodesics of a certain Finsler Randers metric on a 3-dimensional manifold. The trivial projective change of the Finsler metric corresponds to the choice of another 3-dimensional slice, and the existence of a trivial projective change that is forward and backward complete is equivalent to the global hyperbolicity of the space time.

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Acknowledgements

The work was started during the VI International Meeting on LorentzianGeometry (Granada, September 6–9, 2011) and was initiated by the questions by E. Caponio, M.A. Javaloyes, and M. Sánchez; I thank them for this andfor the stimulating discussions at the final stage of the preparation ofthis chapter, and Mike Scherfner for pointing out a misprint. I thankthe anonymous referees for correcting Remark 1 and for many usefulsuggestions.

The standard proof of the existence of a smooth Lipschitz approximation of a Lipschitz function on \({\mathbb{R}}^{n}\) whose generalization for Finsler metrics is the main result of the appendix was explained to me by Yu. Burago, S. Ivanov, and A. Petrunin.

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Authors and Affiliations

  1. Institute of Mathematics, Friedrich-Schiller-Universität Jena, 07737, Jena, Germany

    Vladimir S. Matveev

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  1. Vladimir S. Matveev
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Correspondence to Vladimir S. Matveev .

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Editors and Affiliations

  1. , Dept of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E18071, Granada, Spain

    Miguel Sánchez

  2. Departamento de Mátematica Aplicada, Universidad de Granada, Campus de Fuentenueva, Granada, 18071, Spain

    MIguel Ortega

  3. , Dept. of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E-18071, Granada, Spain

    Alfonso Romero

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Matveev, V.S. (2012). Can We Make a Finsler Metric Complete by a Trivial Projective Change?. In: Sánchez, M., Ortega, M., Romero, A. (eds) Recent Trends in Lorentzian Geometry. Springer Proceedings in Mathematics & Statistics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4897-6_10

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