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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

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.