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Gromov, Cauchy and Causal Boundaries for Riemannian, Finslerian and Lorentzian Manifolds
J. L. Flores and J. Herrera, University of Malaga, Spain, and M. Sánchez, University of Granada, Spain
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Memoirs of the American Mathematical Society
2013; 76 pp; softcover
Volume: 226
ISBN-10: 0-8218-8775-0
ISBN-13: 978-0-8218-8775-2
List Price: US$69
Individual Members: US$41.40
Institutional Members: US$55.20
Order Code: MEMO/226/1064
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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 \(\partial_GM\).

The authors' 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.

Table of Contents

  • Introduction
  • Preliminaries
  • Cauchy completion of a generalized metric space
  • Riemannian Gromov and Busemann completions
  • Finslerian completions
  • C-boundary of standard stationary spacetimes
  • Bibliography
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