CHAPTER 1
Introduction
In Differential Geometry there are quite a few boundaries which can be attached
to a space, depending on the problem one would like to study. For a Riemann-
ian manifold (M,g), when g is incomplete the Cauchy completion yields a simple
boundary. When g is complete, Gromov introduced a general compactification by
using quotients of Lipschitz functions [17]. Such a compactification coincides with
Eberlein and O’Neill’s one for a Hadamard manifold, which defines the boundary
points as Busemann functions associated to rays, up to additive constants, and
uses the cone topology [9]. This construction can be extended to more general
spaces (as the CAT(0) ones), but it was not conceived for an arbitrary Riemann-
ian manifold. Among the different boundaries in Lorentzian Geometry (Schmidt’s
bundle boundary, Geroch’s geodesic boundary, Penrose’s conformal boundary...), the
so-called causal boundary (or c-boundary for short) becomes especially interesting.
The c-boundary was introduced by Geroch, Kronheimer and Penrose [14] by using
a conformally invariant construction, which is explicitly intrinsic and systematic (in
advantage with the conformal boundary, widely used in Mathematical Relativity).
But some problems about its consistency originated a long sequence of redefinitions
of this boundary (see [30] for a critical review). Recently, with the additional stim-
ulus of finding a general boundary for the AdS-CFT correspondence, the notions
of c-boundary and c-completion have been widely developed [18, 19, 26, 10, 30],
and the recent detailed study in [12] justifies that a satisfactory definition is avail-
able now. Moreover, several new ideas have been introduced for the computation
of the c-boundary (or several non-problematic elements of it) in relevant cases
[1, 11, 13, 20].
In this article we carry out a systematic study of the c-boundary of a natural
class of spacetimes, the (conformally) stationary ones. However, our motivation is
not only to compute this boundary for a remarkable class of spacetimes, but also the
revision of other classical boundaries in Differential Geometry —which turn out to
be related with the causal one. More precisely, in order to describe the c-boundary
of a standard stationary spacetime V = R × M, the space of, say, “diverging
directions of curves on M” appears in a natural way. These directions must be
computed with a Riemannian metric g in the particular case that the spacetime is
static, and with a Finsler metric F of Randers type (and its reverse metric F
rev)
in the general stationary case. Such directions are computed from Busemann-type
functions constructed for arbitrary curves of bounded velocity, and its topology is
naturally defined from the chronological topology for any c-completion. Of course, in
the particular case that (M,g) is a Hadamard manifold, the boundary agrees with
Eberlein and O’Neill’s one. But, in general, one obtains a new compactification
of both, any Riemannian manifold and any Randers manifold, the latter trivially
extensible to any Finslerian manifold. Then, it is natural to compare this new
1
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