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

Recently, the old notion of causal boundary for a spacetime V has been redefined consistently. The computation of this boundary partial derivative V on any 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 derivative M-B is constructed in terms of Busemann-type functions. Roughly, partial derivative M-B represents the set of all the directions in M including both, asymptotic and "finite" (or "incomplete") directions. This Busemann boundary partial derivative M-B is related to two classical boundaries: the Cauchy boundary partial derivative M-C and the Gromov boundary partial derivative M-G. In a natural way partial derivative M-C subset of partial derivative M-B subset of partial derivative M-G, but the topology in partial derivative M-B is coarser than the others. Strict coarseness reveals some remarkable possibilities - in the Riemannian case, either partial derivative M-C is not locally compact or partial derivative M-G 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, partial derivative(+)(B) M(equivalent to partial derivative M-B), partial derivative M--(B), 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 partial derivative V of any standard conformally stationary spacetime.