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

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.

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