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

Recently, the old notion of causal boundary for a spacetime V V has been redefined consistently. The computation of this boundary ∂ V ∂ V on any standard conformally stationary spacetime V = R × M V=R× 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 ∂ _BM is constructed in terms of Busemann-type functions. Roughly, ∂ B M ∂ _BM represents the set of all the directions in M M including both, asymptotic and “finite” (or “incomplete”) directions. This Busemann boundary ∂ B M ∂ _BM is related to two classical boundaries: the Cauchy boundary ∂ C M ∂ CM and the Gromov boundary ∂ G M ∂ _GM . In a natural way ∂ C M ⊂