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Gromov, Cauchy and Causal Boundaries for Riemannian, Finslerian and Lorentzian Manifolds

出版	American Mathematical Soc., 2013-10-23
主題	Mathematics / Geometry / GeneralMathematics / Geometry / DifferentialMathematics / Topology
ISBN	08218877509780821887752
URL	http://books.google.com.hk/books?id=CZ8CAQAAQBAJ&hl=&source=gbs_api
EBook	SAMPLE

註釋Recently, the old notion of causal boundary for a spacetime V has been redefined consistently. The computation of this boundary ∂V on any standard conformally stationary spacetime V=R×M, suggests a natural compactification MB associated to any Riemannian metric on M or, more generally, to any Finslerian one. The corresponding boundary ∂BM is constructed in terms of Busemann-type functions. Roughly, ∂BM represents the set of all the directions in M including both, asymptotic and "finite" (or "incomplete") directions. This Busemann boundary ∂BM is related to two classical boundaries: the Cauchy boundary ∂CM and the Gromov boundary ∂GM. In a natural way ∂CM⊂∂BM⊂∂GM, but the topology in ∂ B M ∂BM is coarser than the others. Strict coarseness reveals some remarkable possibilities --in the Riemannian case, either ∂CM is not locally compact or ∂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, ∂B+M(≡∂BM), ∂B−M, 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 MB, relating it with the previous two completions, and (3) to give a full description of the causal boundary ∂V of any standard conformally stationary spacetime.