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

Source abstract

Recently, the old notion of causal boundary for a spacetime V V has been redefined consistently. The computation of this boundary ∂ V \partial V on any standard conformally stationary spacetime V = R × M V=\mathbb {R}\times M , suggests a natural compactification M B M_B associated to any Riemannian metric on M M or, more generally, to any Finslerian one. The corresponding boundary ∂ B M \partial _BM is constructed in terms of Busemann-type functions. Roughly, ∂ B M \partial _BM represents the set of all the directions in M M including both, asymptotic and “finite” (or “incomplete”) directions. This Busemann boundary ∂ B M \partial _BM is related to two classical boundaries: the Cauchy boundary ∂ C M \partial _{C}M and the Gromov boundary ∂ G M \partial _GM . In a natural way ∂ C M ⊂ ∂ B M ⊂ ∂ G M \partial _CM\subset \partial _BM\subset \partial _GM , but the topology in ∂ B M \partial _BM is coarser than the others. Strict coarseness reveals some remarkable possibilities —in the Riemannian case, either ∂ C M \partial _CM is not locally compact or ∂ G M \partial _GM contains points which cannot be reached as limits of ray-like curves in M 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 ( ≡ ∂ B M ) \partial ^+_BM (\equiv \partial _BM) , ∂ B − M \partial ^-_BM , 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 M B M_B , relating it with the previous two completions, and (3) to give a full description of the causal boundary ∂ V \partial V of any standard conformally stationary spacetime.