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Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds


About this Title

J. L. Flores, Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Teatinos, E-29071 Málaga, Spain, J. Herrera, Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Teatinos, E-29071 Málaga, Spain and M. Sánchez, Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, E-18071 Granada, Spain

Publication: Memoirs of the American Mathematical Society
Publication Year 2013: Volume 226, Number 1064
ISBNs: 978-0-8218-8775-2 (print); 978-1-4704-1064-3 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2013-00680-6
Published electronically: May 23, 2013
Keywords: Causal boundary, Gromov compactification, Busemann function, Busemann boundary, Eberlein and O'Neill compactification of Hadamard manifolds, Finsler manifold, Randers metric, non-symmetric and generalized distances, Cauchy completion, quasi-distance, conformally stationary spacetime.
MSC (2010): Primary 53C23, 53C50, 53C60, 83C75

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. Cauchy completion of a generalized metric space
  • Chapter 4. Riemannian Gromov and Busemann completions
  • Chapter 5. Finslerian completions
  • Chapter 6. C-boundary of standard stationary spacetimes

Abstract


Recently, the old notion of causal boundary for a spacetime has been redefined consistently. The computation of this boundary on any standard conformally stationary spacetime , suggests a natural compactification associated to any Riemannian metric on or, more generally, to any Finslerian one. The corresponding boundary is constructed in terms of Busemann-type functions. Roughly, represents the set of all the directions in including both, asymptotic and “finite” (or “incomplete”) directions. This Busemann boundary is related to two classical boundaries: the Cauchy boundary and the Gromov boundary . In a natural way , but the topology in is coarser than the others. Strict coarseness reveals some remarkable possibilities—in the Riemannian case, either is not locally compact or contains points which cannot be reached as limits of ray-like curves in . In the non-reversible Finslerian case, there exists always a second boundary associated to the reverse metric, and many additional subtleties appear. The spacetime viewpoint interprets the asymmetries between the two Busemann boundaries, , , and this yields natural relations between some of their points. Our aims are: (1) to study the subtleties of both, the Cauchy boundary for any generalized (possibly non-symmetric) distance and the Gromov compactification for any (possibly incomplete) Finsler manifold, (2) to introduce the new Busemann compactification , relating it with the previous two completions, and (3) to give a full description of the causal boundary of any standard conformally stationary spacetime.

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