Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds

J. L. Flores; J. Herrera; M. Sanchez

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arXiv · 1011.1154

Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds

J. L. Flores·J. Herrera·M. Sanchez

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Abstract

Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary $\partial V$ for a standard conformally stationary spacetime V = R x M, suggests a natural compactification $M_B$ associated to any Riemannian metric on M or, more generally, to any Finslerian one. The corresponding boundary $\partial_B M$ is constructed in terms of Busemann-type functions. Roughly, $\partial_B M$ represents the set of all the directions in M including both, asymptotic and "finite" (or "incomplete") directions. This Busemann boundary $\partial_B M$ is related to two classical boundaries: the Cauchy boundary and the Gromov boundary. 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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BibTeX RIS

J. L. Flores, J. Herrera, M. Sanchez. 2012-03-05. Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds. https://arxiv.org/abs/1011.1154

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