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\begin{frontmatter}

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\title{Construction of examples of b-completion}

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\author[Miguel]{A. M. Amores\thanksref{DGICYT}}
\ead{angel \underline{ }amores@mat.ucm.es}
\author[Manuel]{M. Guti\'{e}rrez\thanksref{DGES}}
\ead{mgl@agt.cie.uma.es}

\thanks[DGICYT]{Partially supported by DGICYT PB98-0758.}
\thanks[DGES]{Partially supported by DGES Grant PB97-1095.}

\address[Miguel]{Departamento de Geometr\'{\i }a y Topolog\'{\i }a, Facultad
de Matem\'{a}ticas, Universidad Complutense de Madrid, 28040 Madrid, Spain}
\address[Manuel]{Departamento de \'{A}lgebra, Geometr\'{\i}a y
Topolog\'{\i}a, Facultad de Ciencias, Universidad de M\'{a}laga, 29071 M\'{a}laga, Spain}

\begin{abstract}
% Text of abstract
We survey some techniques for constructing b-completions.
First, we give general results relating the b-completion of a product
to the b-completion of its factors, the b-completion of a quotient,
and the dependence of the b-completion on the bundle chosen to form it.
Second, we relate (when possible) some b-completions to those of Riemannian
metrics, and study invariant connections on homogenous spaces.  Some
particular examples are also considered.
\end{abstract}

\begin{keyword}
% keywords here, in the form: keyword \sep keyword
b-completion \sep homogeneus reductive spaces \sep singularity

% PACS codes here, in the form: \PACS code \sep code
%\PACS
\MSC 53C50 \sep 53C30 \sep 83C75
\end{keyword}
\end{frontmatter}

% main text

\section{Introduction}

The classical singularity theorems proved in the sixties by Hawking and
Penrose, being as they were timelike incompleteness theorems of Lorentz
manifolds, brought to the forefront the need to define those ``ideal
points'' forming the ``edge'' of the space-time, which were the ``limits''
of the incomplete geodesics. These ideal points are the singularities of the
space-time and its existence indicates the existence of particles with a
begining or end according to its proper time. A reasonable choice was to
define singularities as equivalence classes of incomplete timelike
geodesics. Nevertheless, Geroch [7] showed the shortcomings of this
definition, giving an example of a geodesically complete space-time
admitting endless timelike curves with finite length and bounded
acceleration. This example was singularity free with the mathematical
definition, but did not fit the intended physical condition that all
particles were ``eternal''.

A much more successful definition of singularity was given by Schmidt [14].
He defined the set of singularities and \emph{endowed it with a
topology.} This construction is very general in the sense that it depends
only on the existence of a connection on a manifold and not on a metric. The
example of Geroch has singularities with this definition. The Schmidt
definition has interesting properties, but some drawbacks too. It is very
hard to determine the singularity space in many examples, and when it has
been obtained, as in the case of closed Friedmann spaces, its properties
seem to contradict physical intuition (the Big Bang and the Big Crunch are
the same singularity). See [10], [4], and [3].
At any rate, the b-completion stands out as a reference for future
definitions, since endless finite length curves (measured via a parallel
basis along the curve) are very important for \emph{any }definition of
singularity. In this paper we study different examples of b-completion (with
the Schmidt definition) along with the tools required to construct them.
Some of them are well known; others not so much; therefore we put more
emphasis on them. A more complete version of this paper containing proofs are
available through the autors.

\section{The b-completion of some products and quotients}

Let $M$ be a connected $m$-dimensional manifold, $\pi :P\rightarrow M$ a
connected subbundle of the frame bundle with structural group $G\subset GL(m,%
\R)$. Let $\Phi $ be the fundamental form and $\lambda $ a connection
(form) on $P$. We define on $P$ the Riemann metric
\begin{equation}
g^{\lambda }(X,Y)=\left\langle \Phi (X),\Phi (Y)\right\rangle +\left\langle
\lambda (X),\lambda (Y)\right\rangle ,
\end{equation}
where $\left\langle \bullet ,\bullet \right\rangle $ is the standard
Euclidean product in $\R^{m}$ and $\left\langle A,B\right\rangle
=tr(AB^{t})$ ($\bullet ^{t}$ denoting transposition) the standard Euclidean
product in $\mathfrak{gl}(m,\R)$. All is clear since $\mathfrak {g}$, the Lie
algebra of $G$, is a subspace of $\mathfrak {gl}(m,\R)$. Let $d=d^{\lambda }$
be the distance in $P$ associated to $g^{\lambda }$ and $(\hat{P},\hat{d}%
)=\left( P,d\right) _{c}$ the Cauchy completion of the metric space $(P,d)$.
The right action of $G$ on $P$ can be extended to $\hat{P}$. The quotient
space $\widehat{M}=\hat{P}/G$ is now the \textbf{b-completion} of $%
(M,\lambda )$. The \textbf{singularity space}\emph{\ }or \textbf{b-boundary}
is $\partial M=\widehat{M}-M$. Since the distance $\hat{d}$ and the
projection $\hat{\pi}$ are extensions of $d$ and $\pi ,$ we simply write $d$
and $\pi $.

If $(M,g)$ is a space-time, $P$ is usually the bundle of all positive(ly
oriented) frames $(u_{1},u_{2},\ldots ,u_{m})$ such that $u_{1}$ is timelike
future, $G$ is the connected component of the Lorentz group and $\lambda $
the connection form of the Levi-Civita connection of $g$. If $(M,h)$ is a
Riemann manifold and similar choices are made, $\widehat{M}$ is homeomorphic
to the Cauchy completion of $\left( M,\delta \right) ,$ where $\delta $ is
the distance in $M$ attached to $g$ [14]. In any case we refer to
the b-completion of $g$ or $h.$ Although we will not use it, we cite an
equivalent definition of the b-completion due to Sachs [13], \emph{%
only for space-times }$\left( M,g\right) .$ In the unit bundle $UM=\{X\in
TM\;|\;X$ unit timelike future$\}$ a Riemann metric $g^{D},$ depending on $%
g, $ is given and the $g^{D}$-distance $d$ is used to get the Cauchy
completion of $\left( UM,d\right) .$ Then an equivalence relation $R$ in $%
\left( UM,d\right) _{c}$ is defined and the quotient space is shown to be
homeomorphic to $\widehat{M}.$

It is not hard to prove that $\pi :\hat{P}\rightarrow \widehat{M}$ is open, $%
\widehat{M}$ is arc connected, $M$ is dense in $\widehat{M}$, and the
manifold topology of $M$ is the one inherited as a subspace of $\widehat{M}$%
. Nevertheless $\widehat{M}$ fails to be locally compact or Hausdorff. This
happens in important examples of space-times and adds to the difficulties of
mathematical handling those of physical interpretation.

Quite often we require $M$ to be inextendible. If not, there are simple
examples of b-completion as we may see.

\begin{proposition}
Let $M$ be a connected manifold with $m>1$ with a connection. Given $%
x_{1},\ldots ,x_{n}\in M$, let $N=M-\{x_{1},\ldots ,x_{n}\}$ with the
induced connection. Then $\widehat{N}$ is homeomorphic to $\widehat{M}$ and $%
\partial N=\partial M\cup \{x_{1},\ldots ,x_{n}\}$. $\square$
\end{proposition}

Singularities are closely related to b-incomplete curves [9].
Let $c:[a,b)\rightarrow M$ be a smooth curve and $C:[a,b)\rightarrow P$ a
(non unique) horizontal lift of $c$. It happens that the finiteness of the
length of $C$, measured with $g^{\lambda },$ depends only on $c.$
Accordingly we say that $c$ has \textbf{finite or infinite fibre length}.%

\begin{definition}
A smooth curve $c:[a,b)\rightarrow M$ is \textbf{extendible} if $%
\lim_{t\rightarrow b}c(t)$ exists in $M$, and is \textbf{inextendible}
otherwise. We say that $c$ is \textbf{b-incomplete} if it is inextendible
and has finite fibre length.
\end{definition}

Incomplete geodesics are b-incomplete if $b<\infty ,$ since it is well known
that the existence of $\lim_{t\rightarrow b}c(t)\in M$ implies that $c$ is
extendible as a geodesic to $[a,b+\varepsilon )$ for some $\varepsilon >0.$

\begin{theorem}
\label{CurvaHorizontal}For any Cauchy sequence $\left( u_{n}\right) $ in $P$
with limit $\widehat{u}\in \widehat{P},$ there is a finite length horizontal
curve $C:[0,1)\rightarrow P$ such that $\lim_{t\rightarrow 1}C(t)=\widehat{u}$. $\square$

\end{theorem}

If $c:[0,1)\rightarrow M$ is b-incomplete, any horizontal lift $C$ has
finite length. This implies the existence of $\lim_{t\rightarrow 1}C(t)=%
\widehat{u},$ necessarily unique and in $\hat{P}-P.$ Therefore, $\pi (%
\widehat{u})$ is a singularity and $c$ converges to it. Nevertheless, since $%
\widehat{M}$ may be non-Haussdorf, $c$ may converge to other points in $%
\widehat{M}$. To sum up, b-incomplete curves converge to singularities, but not
uniquely (there is a limit set rather than a limit point). In any case,
theorem \ref{CurvaHorizontal} shows that any singularity is in the limit set
of a curve: If $\widehat{x}=\pi (\widehat{u}),$ and $c=\pi \circ C,$ then $c$
converges to $\widehat{x}.$ However, we must stress that we have no
information about a possible limit of $c$ in $M$ (recall $\widehat{M}$ is
not Haussdorf). If all limits of $c$ were in $\partial M,$ $c$ would be
b-incomplete. Schmidt proved (see [15], [9]) that in
space-times all singularities are limits of b-incomplete curves.

\begin{definition}
A b-incomplete curve $c$ \textbf{induces a singularity} $p\in \partial M$ if
there is a horizontal lift $C$ of $c$ with limit in $\pi ^{-1}(p)$. (Then
all horizontal lifts have limit in $\pi ^{-1}(p).)$
\end{definition}

\begin{lemma}
\label{Comparación de longitudes en P}Let $C,D:[a,b)\rightarrow P$ be curves
such that $C$ is horizontal and $D(t)=C(t)s(t)$ for some $s:[a,b)\rightarrow
G$. Then

\begin{enumerate}
\item  $Length(s)\leq Length(D)$, where lengths in $G$ are measured with the
left invariant metric induced by the Euclidean product $\left\langle \bullet
,\bullet \right\rangle $ already chosen in $\mathfrak {g}$.

\item  For some $k\in \R$ we have $Length(C)\leq k\,Length(D)$. $\square$

\end{enumerate}
\end{lemma}

This lemma is used to prove the following theorem.

\begin{theorem}
\label{EquivPuntFinalMismaFfibr}Let $C$, $D:[a,b)\rightarrow P$ be finite
length curves having limit in $\hat{P}$ as $t\rightarrow b.$ Then the
following statements are equivalent

\begin{enumerate}
\item  The limits of $C$ and $D$ belong to the same fibre $\pi
^{-1}(p)\subset \hat{P}$, $p\in \widehat{M}$.

\item  \label{EquivPuntFinSuces}There is an increasing sequence $%
t_{n}\rightarrow b$ and a family of horizontal curves $\Delta
_{n}:[0,1]\rightarrow P,$ joining $C(t_{n})$ with the fibre $\pi
^{-1}(d(t_{n})),$ such that $Length(\Delta _{n})\rightarrow 0$ and the
sequence $\left( s_{n}\right) $ in $G$ defined by $\Delta
_{n}(1)s_{n}=D(t_{n})$ converges to some $s\in G$.

Conditions \textbf{1 }and \textbf{2} are a consequence of

\item  There is an increasing sequence $t_{n}\rightarrow b$, a family of
curves $\Gamma _{n}:[0,1]\rightarrow P$ joining $C(t_{n})$ to $D(t_{n})$ and
a constant $k\in $ $\R$ such that
\begin{equation}
\int_{0}^{1}\left| \lambda (\dot{\Gamma}_{n}(t))\right| dt\leq k \quad \mbox{for all }n \in \N \quad and \quad \int_{0}^{1}\left| \Phi (\dot{%
\Gamma}_{n}(t))\right| dt\rightarrow 0.
\end{equation}
\end{enumerate}

If $G$ is connected, conditions \textbf{1},\textbf{2} and \textbf{3} are
equivalent. $\square$

\end{theorem}

We remark that if $C$ and $D$ are horizontal lifts of b-incomplete curves $c$
and $d$, condition \textbf{1} can be substituted by the condition that $c$
and $d$ induce the same singularity. As an application we give necessary
conditions to hold if two b-incomplete curves induce the same singularity.

\begin{theorem}
\label{IdentifSingularidades}Let $\alpha _{i}:[0,1)\rightarrow M,$ $i=1,2$
be b-incomplete curves inducing the same singularity. If $J_{i}=$\quad $Im \thinspace \left( \alpha _{i}\right) $, suppose the closures in $M$ of $J_{1}$ and $J_{2}$ are
disjoint. Then, there is a b-incomplete curve $\beta :[0,1)\rightarrow M$
inducing the same singularity as $\alpha _{1}$ and $\alpha _{2}$ and an
increasing sequence $\left( t_{n}\right) $ in $[0,1)$ with limit $1$ such
that $\beta (t_{2n})\in J_{1}$ and $\beta (t_{2n+1})\in J_{2}.$
\end{theorem}

\begin{theorem}
\label{bComplHomeo}Let $\pi :P\rightarrow M$ and $\chi :Q\rightarrow M$ be
connected subbundles of the frame bundle $L(M)$ with structural groups $G$
and $H$ and connections $\lambda $ and $\sigma$. Let $F:P\rightarrow Q$ be
a morphism of principal bundles such that

\begin{enumerate}
\item  $F$ is an injective immersion, inducing a difeomorphism $f$ on $M$
and the inclusion of $G$ in $H$.

\item  \label{SubgrupoCerrado}$G$ is a closed subgroup of $H$.

\item  \label{PullbackFormasConexion}$F^{*}\sigma =\lambda $.
\end{enumerate}

Then the b-completions $\widehat{M}_{P}$ and $\widehat{M}_{Q}$ are
homeomorphic.
\end{theorem}


\begin{corollary}
Let $(M,g)$ be a semi-riemannian manifold, $P$ a connected component of the
orthonormal bundle $OM$ and $Q$ a connected component of the frame bundle $%
LM.$ If they are endowed with the Levi-Civita connections, the completions
of $M$ are homeomorphic. $\square$

\end{corollary}

\begin{corollary}
Let $M$ be a manifold with a connection, $P$ a connected component in the
holonomy bundle and $Q$ a connected component in the frame bundle $L(M).$ If
the holonomy group is closed, both b-completions of $M$ are homeomorphic. $\square$
\end{corollary}

Let $(M,\lambda )$, $(N,\sigma )$ be manifolds of dimension $m$, $n$ with
connections $\lambda $, $\sigma $. Let $L^{\prime }(M\times N)$, $L^{\prime
}M$, and $L^{\prime }N$ be connected components of the frame bundles and $%
\varphi $, $\pi $, $\chi $ the projections. Depending on manifold
orientability, the structural groups will be the full $GL(k,\R),$ $%
k=m,n,mn$ or connected components of them. They will be denoted by $G$, $H$,
$K.$ Let $\pi \times \chi :L^{\prime }M\times L^{\prime }N\rightarrow M$ be
the product bundle, which is a principal bundle with structural group $%
G\times H.$ In a natural way, $G\times H$ is a subgroup of $K$ and $%
L^{\prime }M\times L^{\prime }N$ a subbundle of $L^{\prime }\left( M\times
N\right) .$ If $p:L^{\prime }M\times L^{\prime }N\rightarrow L^{\prime }M$,$%
\;q:L^{\prime }M\times L^{\prime }N\rightarrow L^{\prime }N$ are the
projections there is on $L^{\prime }M\times L^{\prime }N$ a connection $%
\lambda \times \sigma $ defined by $\lambda \times \sigma =p^{*}\lambda
+q^{*}\sigma .$ (Since $G$ and $H$ are subgroups of $K,$ its Lie algebras
are subalgebras of $\mathfrak {k}$, and $p^{*}\lambda ,q^{*}\sigma $ are seen as
$\mathfrak {k}$-valued forms). Finally $\lambda \times \sigma $ induces a
connection $\omega $ in $L^{\prime }(M\times N)$ with formula $\Psi
^{*}\omega =\lambda \times \sigma $.

\begin{theorem}
\label{bComplecionProducto}With the above notation the completion of $\left(
M\times N,\omega \right) $ is homeomorphic to the product of the
b-completions of $\left( M,\lambda \right) $ and $\left( N,\sigma \right)$.
\end{theorem}


\begin{corollary}
Let $(M,g)$ and $(N,h)$ be semi-riemannian manifolds, then the b-completion of $%
\left( M\times N,g\times h\right) $ is the product of the b-completions of $%
(M,g)$ and $(N,h)$. $\square$
\end{corollary}

Some particular cases of the results derived from theorem \ref{bComplHomeo}
are in [6], and others appear in [2].

Let $\psi :M\rightarrow M$ be an affine map for a connection $\lambda $ and $%
\Psi $ its lift to $LM$; it is an isometry of $(LM,d^{\lambda })$ which can
be extended to an isometry of $(LM,d^{\lambda })_{c}.$ In the quotient we
get a homeomorphism $\widehat{\psi }:\widehat{M}\rightarrow \widehat{M}$;
hence a group $A$ of affine transformations of $M$ acts also on $\widehat{M}$
and the orbit space $\widehat{M}/A$ can be considered.

If $A$ acts discretely on $M$, then $N=M/A$ is a manifold and the projection
$p:M\rightarrow N$ is smooth. Let $L^{\prime }N$ be a connected component of
the frame bundle $LN$. Then $p$ induces a fibre space morphism $%
q:LM\rightarrow LN$ and there is a unique connection $\lambda _{0}$ in $LN$
determined by $q^{*}\lambda _{0}=\lambda $. We have [1]

\begin{theorem}
\label{bComplecionCociente}If the action of $A$ on $\left( LM\right) _{c}$
has closed orbits, the b-completion of $(N,\lambda _{0})$ is homeomorphic to
$\widehat{M}/A$ with the quotient topology. $\square$
\end{theorem}

\section{Other examples}

In the preceding section we developed theorems which allow us to get new
examples from old ones. We give now concrete examples.

\subsection{Riemann metrics and related cases}

We said at the begining that if $\left( M,h\right) $ is a Riemann manifold,
the b-completion of the Levi-Civita connection is just the Cauchy completion
of $\left( M,\delta \right) ,$ where $\delta $ is the distance associated to
the metric. These cases are considered trivial in the sense that there is a
general simple answer (although $\left( M,\delta \right) _{c}$ may be hard
to determine). If a Lorentz manifold $\left( M,g\right) $ had the same
Levi-Civita connection $\nabla $ as that of the Riemann metric $h,$ we would
know that the b-completion of $g$ is the b-completion of $h.$ This is clear,
since the b-completion can be computed in the full frame bundle (not
necessarily in the specific orthonormal bundle of $g$ or $h$, see theorem
\ref{bComplHomeo}) and the b-completion depends only on the connection.

Let $g$ and $h$ be a Lorentz and Riemann metric on $M$ and $g^{\flat
},h^{\flat }:TM\rightarrow T^{*}M$ the induced isomorphisms. We have a field
of automorphisms $J$ defined by $h(X,Y)=g(JX,Y)$ for $X,Y\in \mathfrak {X}(M)$.
The symmetry of $g$ and $h$ imply that $J$ is self adjoint for both metrics.

\begin{proposition}
If $J$ is parallel for either metric, the b-completions of them coincide,
and $\widehat{M}_{g}$ is homemorphic to the Cauchy completion of $\left(
M,\delta \right) .$ This is the case if the metrics are related by $%
h(Y,Z)=g(Y,Z)+2g(X,Y)g(X,Z),$ where $X$ is a unit timelike parallel field.
\end{proposition}


We may then show the equivalence of another definition of singularity set
(called the \textit{Cauchy boundary} [8]) and the
b-completion. Using the fact that $X$ is parallel if and only if the
connection form of $g$ is reducible to the $O(m-1)$-structure of frames
having $X$ as the first vector$,$ it can be proved that the Riemann metric
associated to the space time given in [8] is just the metric $%
h$ given above.

On the other hand, the b-completion of the example in the last section of
[13], can be computed via a Riemann metric $h$ constructed with the
help of a parallel field $X.$ It can also be computed with the corollary of
theorem \ref{bComplecionProducto}.

\subsection{Reductive homogeneous spaces}

We fix a connected Lie group $P$ and a closed Lie subgroup $G$. The quotient
space $P/G$ is denoted by $M$, and the projection of $P$ onto $M$ by $\pi $.
We have the corresponding Lie algebras $\mathfrak {p}$ and $\mathfrak {g}$. It is well
known that $P$ is a principal fibre bundle with base $M$, group $G$, and
such that the right action of $G$ onto $P$ is the multiplication. Therefore,
if $A\in \mathfrak {g}\subset \mathfrak {p}$, the fundamental field $A_{\#}$ in
$\mathfrak {X}(P)$ associated to $A$ is just $A$. This bundle has the important property
that $P$ acts on the left on $P$ and (more important) on $M$ by left
multiplication: For $v\in P$ we have $\tau _{v}:M\rightarrow M,$ $\tau
_{v}(uG)=vuG.$ We assume our homogeneus space to be \textbf{reductive.} This
means there is a supplementary $\mathfrak {m}$ of $\mathfrak {g}$ in $\mathfrak {p}$ such
that $Ad_{s}(\mathfrak {m})\subset \mathfrak {m}$ for all $s\in G$. We have a
homomorphism ${\mathcal A}:G\rightarrow GL(\mathfrak {m})$ by defining ${\mathcal A}%
(s)={\mathcal A}_{s}=Ad(s)|_\mathfrak {m}.$ The associated Lie algebra
homomorphism is $X\rightarrow {\mathbf a}_{X}$ , obtained by restriction of
$ad(X)\in \mathfrak {gl(p)} \mbox{ to } \mathfrak {m}$.

It is convenient to consider frames in $M$ as isomorphisms $r:\mathfrak {m}
\rightarrow T_{x}M$. Define
$j:P\rightarrow LM\quad ,\quad j(u)(Z)=\pi _{*u}(Z_{u})\quad u\in P,\,Z\in
\mathfrak {m}$.

\begin{theorem}
The map $j:P\rightarrow LM$ is a principal fibre bundle homomorphism over
the identity of $M$, with ${\mathcal A}:G\rightarrow GL(\mathfrak {m})$ as
associated group homomorphism; i.e., $j(us)=j(u)\circ {\mathcal A}_{s}$ for $
u\in P$, $s\in G$. For $v\in P$, $j\circ \tau _{v}=T_{v}\circ j,$ where $%
T_{v}$ is the lift of $\tau _{v}$ to $LM$. $\square$
\end{theorem}

Let $L^{\prime }M$ be a connected component of $LM.$ In a reductive
homogeneous space $M=P/G$ an \textbf{invariant connection} is considered.
This means by definition that the translations $\tau _{v}:M\rightarrow M$
are affine maps. Due to high symmetry, it is easier to get information about
the b-completion. The study of these examples was started by the authors
[2]. We outline the main results.

For each connection $\lambda $ in $LM$ define $\mu =j^{*}\lambda \in \Lambda
^{1}(P,\mathfrak {gl(m)})$. If $X\in \mathfrak {g}\subset \mathfrak {p}\subset
 \mathfrak {X}(P)$, then $\mu(X) $ is the \emph{constant} map ${\mathbf a}_{X}.$ This shows that the information that $\mu $ may carry about $\lambda $ is in the maps $\mu (Z)$,
$Z\in \mathfrak {m}$. We also denote by $\mu :\mathfrak {m}\rightarrow C^{\infty
}(P,\mathfrak {gl(m)})$ the induced linear map.

\begin{theorem}
\label{FuncionDeP1a1Conexiones}The connections $\lambda $ on $LM$ are in
bijective correspondence with the linear maps $\mu :\mathfrak {m}\rightarrow
C^{\infty }(P,\mathfrak {gl(m)})$ such that
$$\mu ({\mathcal A}_{s}(Z))(us^{-1})={\mathcal A}_{s}\circ (\mu (Z)(u))\circ
{\mathcal A}_{s}^{-1}\quad ,\quad \mbox{for all}\;u\in P,\;s\in G,\;Z\in
\mathfrak {m}.$$
The relation between $\lambda $ and $\mu $ is $\mu (Z)(u)=j^{*}\lambda (Z(u))
$. $\square$
\end{theorem}

If $\lambda $ is invariant, $\mu (Z)\in C^{\infty }(P,\mathfrak {gl(m)})$ is
constant. We identify $\mu (Z)$ with its constant value in $\mathfrak {gl(m)}$.

\begin{corollary}
\label{Aplicacionbilineal1a1ConexionInvariante}The invariant connections $%
\lambda $ on $LM$ are in bijective correspondence with the linear maps $\mu :
\mathfrak {m}\rightarrow \mathfrak {gl(m)}$ such that
$$\mu ({\mathcal A}_{s}(Z))={\mathcal A}_{s}\circ \mu (Z)\circ {\mathcal A}
_{s}^{-1}\quad \mbox{for all}\;s\in G,\;Z\in \mathfrak {m}$$
or the bilinear maps $\mu :\mathfrak {m}\times \mathfrak {m}\rightarrow \mathfrak {m}$ such that
$$\mu ({\mathcal A}_{s}(X),{\mathcal A}_{s}(Y))={\mathcal A}_{s}\left( \mu
(X,Y)\right) \quad \mbox{for all }s\in G,\;X,Y\in \mathfrak {m}.$$ $\square$
\end{corollary}

\begin{theorem}
Let $\lambda $ be an invariant connection $\lambda $ on $LM$ such that there
is a connection $\omega $ in the $G$-principal bundle $\pi :P\rightarrow M$
with the property that
$$\mu (X)=j^{*}\lambda (X)={\mathbf a}_{\omega (X)}\quad \mbox{for all}\;X\in
\mathfrak {X}(P).$$
Then $\lambda $ has no singularities. $\square$
\end{theorem}

However, many invariant connections have singularities. Let $C_{0}:\R%
\rightarrow P$ be given by $C_{0}(t)=u\exp (\varepsilon tZ)$, for some $Z\in
\mathfrak {m}$, $u\in P$ and $\varepsilon =\pm 1,$ and let $c:\R\rightarrow
M $ be its projection on $M;$ i.e., $c=\pi \circ C_{0}$. The horizontal lift
$C $ of $c$ to $L^{\prime }M$ such that $C(0)=j(u)$ is
\begin{equation}
C(t)=j(C_{0}(t))EXP(-\varepsilon t\mu (Z))  \label{CurvaHorizontalFormula}
\end{equation}
and its length measured by $g^{\lambda }$ for $a\leq t\leq b$ is
\begin{equation}
\int_{a}^{b}\left| EXP(\varepsilon t\mu (Z))(Z)\right| dt,
\label{LongCurvaHorizontal}
\end{equation}
where $EXP$ is the exponential of an endomorphism.

\begin{theorem}
Consider an invariant connection $\lambda $ on $LM$ given by $\mu :\mathfrak {m}%
\rightarrow \mathfrak {gl(m)}$, and $Z\in \mathfrak {m}$ such that $\mu (Z)(Z)=kZ$
with $k\neq 0$. If $\varepsilon =\pm 1$ has opposite sign to $k$, the curve $%
c(t)=\pi (u\exp (\varepsilon tZ))$, $t\geq 0$, is b-incomplete and thus
converges to a singularity.
\end{theorem}


Invariant connections on Lie groups and homogeneous spaces are classified in
[11] and [12].

\subsection{An example of b-completion for a homogeneous space.}

Let $P=(\R^{m},+)$ and $G=0$. We identify $P=M=\R^{m}\mbox{, } \mathfrak {p}=\mathfrak {m}=\R^{m}$,  $\pi =id$, $u\exp (tZ)=u+tZ$, and $L^{\prime }M=\R^{m}\times GL^{+}(m;\R)$.
The connection $\lambda $ is associated to
$\mu :\mathfrak {m}=\R^{m}\rightarrow \mathfrak {gl(m)}$, $\mu (Z)=f(Z)I,$ where $f:
\R^{m}\rightarrow \R$ is $f(x^{1},...,x^{m})=x^{m}$.

Let $I=\left[ a,b\right] ,$ $c:I\rightarrow M$ a curve and $C:I\rightarrow
LM=\R^{m}\times GL^{+}(m;\R)$ its horizontal lift with initial
condition $C(0)=(x,1).$ Then
\begin{equation}
C(t)=\left( c(t),e^{-\left( f(c(t))-f(x)\right) }\right) \in \R
^{m}\times GL^{+}(m;\R),  \label{CurvaHorizontalRm}
\end{equation}
$$Length(C)=\int_{a}^{b}e^{\left( f(c(t))-f(x)\right) }\left| \dot{c}%
(t)\right| dt\geq \left| \left[ e^{\left( f(c(t))-f(x)\right) }\right]
_{a}^{b}\right| ,$$
since $\left| f(\dot{c}(t))\right| \leq \left| \dot{c}(t)\right| .$

We show that \emph{every singularity is induced by a b-incomplete curve}.
Let $p\in \partial M$. By theorem \ref{CurvaHorizontal}, there is a
horizontal $C:I\rightarrow LM$ of finite length such that $%
\lim_{t\rightarrow b}C(t)\in \pi ^{-1}(p)\subset \pi ^{-1}\left( \partial
M\right) $. If $c=\pi \circ C,$ then $C(t)=\left( c(t),e^{-\left(
f(c(t))-f(x)\right) }\right) $. If $c$ converged to $y\in M$, $%
\lim_{t\rightarrow b}C(t)=\left( y,e^{-\left( f(y)-f(x)\right) }\right) \in
LM.$ Therefore $c$ is b-incomplete and induces $p$.

Define $\gamma :[0,\infty )\rightarrow M=\R^{m}$, $\gamma (t)=\left(
0,...,0,-t\right) $. Then $\Gamma (t)=\left( \gamma (t),e^{t}\right) $ is
the horizontal lift of $\gamma $ such that $\Gamma (0)=(0,1).$ Clearly, $%
Length(\Gamma )=\int_{0}^{\infty }e^{-t}dt<\infty $; therefore $\gamma $ is
b-incomplete and induces $p^{*}\in \partial M$. We say that \emph{there is a
unique singularity}; i.e., $\partial M=\{p^{*}\}$. We should prove that any
b-incomplete curve $c$ and $\gamma $ induce the same singularity. This is
shown with theorem \ref{EquivPuntFinalMismaFfibr} using as curves $\Delta
_{n}$ the horizontal lifts of curves joining $c(t)$ and $\left(
0,...,0,c^{m}(t)\right) =\gamma (c^{m}(t)).$

Let $K\subset \R^{m}$ be a closed subset such that $\inf \left\{
f(x)\;|\;x\in K\right\} >-\infty .$ It can be proved that $\pi ^{-1}(K)$ is
closed in \emph{the completion }$\left( L\R^{m}\right) _{c}.$ Therefore
$\widehat{M}-K=\widehat{\R^{m}}-K$ is an open neighbourhood of $p^{*}$
and $\widehat{M}$ is Hausdorff. Moreover, $M$ is open in $\widehat{M},$ but
this information is not enough to know exactly the topology of $\widehat{M}$
except in the case $m=1$.

Consider a \emph{discrete} subgroup $A$ of $\R^{m}$. Every $a\in A$ can
be seen as a translation of $\R^{m}=M$. Translations are affine maps
for $\lambda $; therefore theorem \ref{bComplecionCociente} is applicable to
get the b-completion of the manifold $N=M/A$ with the connection $\lambda
_{0}$ induced by $\lambda .$ We get $\widehat{N}=\left( \R^{m}\cup
\{p^{*}\}\right) /A$ under the action of $A$ on $\widehat{M}=\R^{m}\cup
\{p^{*}\}$ given by
$$a\cdot x=a+x\;\mbox{if}\;x\in \R^{m}\quad \mbox{and}\quad a\cdot
p^{*}=p^{*}.$$
The topology of $\widehat{N}$ is the quotient topology. As we said, the
topology of $\widehat{M}$ is not fully determined, but the sets $%
\{p^{*}\}\cup f^{-1}\left( (-\infty ,r)\right) $ are open in $\widehat{M}.$
It follows that if some $a\in A$ is not orthogonal to $\left(
0,...,0,1\right) ,$ then $\widehat{N}$ has a unique singularity $q^{*}$, and
that $q^{*}$ has a unique neighbourhood $\widehat{N}$. On the other hand, if
$A\subset Ker(f)$ and $Ker(f)/A$ is compact, any neighbourhood $V$ of the
unique singularity $q^{*}$ of $\hat{N}=\widehat{\R^{m}}/A$ contains one
of the form
$$\left( (\S^{1})^{m-1}\times (-\infty ,r)\right) \cup \{q^{*}\}.$$
The homogeneous spaces in this section generalize those given in [5] where $\R$ and $\R /\Z = \S^{1}$ with a ''constant''
connection were considered. The b-completions obtained were either $\R
\cup \{-\infty \}$ with intervals $[$-$\infty ,r)$ as neighbourhoods of $
p^{*}=-\infty $, or $\S^{1}\cup \{q^{*}\}$ having $q^{*}$ only one
neighbourhood. The techniques of proof were different.

\subsection{Other invariant connection in $\R^{m}$.}

For each $x\in \R^{m}$, $m \geq 2$, let $\mu \left( x\right) \in \frak {gl}(\R^{m})$
denote the diagonal matrix $A$ such that $A_{i}^{i}=x^{i}.$ We consider the
Lie group $P=\left( \R^{m},+\right) .$ The invariant connection on $P$
is given by $\mu :\R^{m}\longrightarrow \mathfrak {gl}(\R^{m})$ just defined.
The curve $c:[0,a)\rightarrow \R^{m}$ has
$$C(t)=\left( c(t),\exp \left( -\mu \left( c\left( t\right) -c(0)\right)
\right) S\right)$$
as the horizontal lift such that $C(0)=\left( c(0),S\right)$ for $S\in GL(m,\R)$.
The length of $C$ is
$$Length(C)=\int_{0}^{a}\left| S^{-1}\cdot \exp \left(\mu \left(
c(t)-c(0)\right)\right) \left( \dot{c}(t)\right) \right| dt.$$
Suppose $S=1.$ Then
$$Length(C)\geq \int_{0}^{a}\left| \dot{c}^{i}(t)\exp \left( c^{i}(t)-c^{i}(0)\right) \right| dt\geq \int_{0}^{a}\left| \frac{d}{dt}\exp \left( c^{i}(t)-c^{i}(0)\right) \right| dt.$$
This implies the existence of the limits $\lim_{t\rightarrow a}\exp \left(
c^{i}(t)\right) $, if $Length(C)<\infty .$ Consider the curve $\alpha
:[0,\infty )\rightarrow \R^{m},$ $\alpha (t)=tu,$ where $u$ is a \emph{%
unit} vector with all $u^{i}\leq 0.$ The length of the horizontal lift $A$
such that $A(0)=\left( 0,1\right) $ is
$$Length(A)=\int_{0}^{\infty }\left| \left( e^{tu^{1}},...,e^{tu^{m}}\right)
\left( u^{1},...,u^{m}\right) \right| dt\leq \sum_{u^{i}\neq
0}\int_{0}^{\infty }\left| u^{i}\right| e^{tu^{i}}dt<\infty .$$
The curve $\alpha $ is b-incomplete and induces a singularity $z=z(u).$

\begin{theorem}
The singularities $z$ are different.
\end{theorem}


\subsection{The Friedmann spaces.}

It is proved in [4] and [10] that the closed Friedmann
space in dimension two has exactly one essential past singularity. This
singularity has a single neighbourhood (the whole b-completion). Using this
as stepping stone, these authors show that in dimension four there is at
least one essential singularity with a single neighbourhood and at least one
essential past and one essential future singularity are in fact the same
singularity. It was conjectured in [10] that there was a unique
essential past singularity. Other results on Schwarzschild spaces are also
obtained. The definition given in [10] of \textbf{essential
singularity} is specific for the cases considered and of a technical nature.
Its purpose is to rule out singularities which are induced by ``physically
meaningless'' curves.

The present authors [3] prove that 4-dimensional Friedmann
closed space has only one singularity (not restricted to being essential)
and this singularity has a unique neighbourhood. Briefly $\partial M=\{p\}$
and $\widehat{M}$ is the only neighbourhood of $p.$ They also prove that the
non-closed Friedmann four dimensional space has a unique essential past
singularity. As for future essential singularities, it is shown that maximal
non-spacelike future geodesics are not b-incomplete; hence, if these
singularities existed, they would be of little physical importance. To sum
up in a very simplified way: There is a Big Bang, only one Big Bang, and if
there is a Big Crunch it will be the same as the Big Bang.

\section{Final comments}

The techniques shown to get the b-completion of a space, related to some
other whose b-completion is known gives an important collection of examples,
at least from the mathematical viewpoint. Unfortunately, the collection is
very short if we look for physically interesting cases. The closed Friedmann
space is the only case, and there is interesting information for non-closed
Friedmann spaces, but Schwarzschild space has not been tackled. At present,
everything seems to indicate that to compute the b-completion, a specific
way has to be found for every case. The knowledge of the b-completion of the
most important space-times and its physical interpretation (if any) are the
more important steps to be taken.


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\end{document}
