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

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

2010-11-04

arxiv.org/abs/1011.1154v3

Mathematics

Differential Geometry

Final version with minor modifications, to appear in Memoirs of the American Mathematical Society. 80 pages, 10 figures, latex

Scientific paper

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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