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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 $V$ has been redefined consistently. The computation of this boundary $\partial V$ on any standard conformally stationary spacetime $V=\mathbb {R}\times M$, suggests a natural compactification $M_B$ associated to any Riemannian metric on $M$ or, more generally, to any Finslerian one. The corresponding boundary $\partial _BM$ is constructed in terms of Busemann-type functions. Roughly, $\partial _BM$ represents the set of all the directions in $M$ including both, asymptotic and “finite” (or “incomplete”) directions. This Busemann boundary $\partial _BM$ is related to two classical boundaries: the Cauchy boundary $\partial _{C}M$ and the Gromov boundary $\partial _GM$. In a natural way $\partial _CM\subset \partial _BM\subset \partial _GM$, but the topology in $\partial _BM$ is coarser than the others. Strict coarseness reveals some remarkable possibilities—in the Riemannian case, either $\partial _CM$ is not locally compact or $\partial _GM$ contains points which cannot be reached as limits of ray-like curves in $M$. 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, $\partial ^+_BM (\equiv \partial _BM)$, $\partial ^-_BM$, 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 $M_B$, relating it with the previous two completions, and (3) to give a full description of the causal boundary $\partial V$ of any standard conformally stationary spacetime.

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