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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

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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