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\dateposted{October 10, 2000}
\PII{S 1079-6762(00)00083-4}
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\title[The flow completion of a manifold with vector field]{The flow completion\\ of a manifold with vector field}
\author{Franz W. Kamber}
\address{Department of Mathematics, 
University of Illinois, 1409 West Green Street, 
Urbana, IL 61801}
\email{kamber@math.uiuc.edu }
\thanks{Supported by 
Erwin Schr\"{o}dinger International Institute of Mathematical Physics, 
Wien, Austria.
FWK was supported in part by The National Science Foundation under 
Grant No. DMS-9504084.
PWM was supported  
by `Fonds zur F\"{o}rderung der wissenschaftlichen  
Forschung, Projekt P~14195~MAT'}

\author{Peter W. Michor}
\address{Institut f\"{u}r Mathematik, Universit\"{a}t Wien,
Strudlhofgasse 4, A-1090 Wien, Austria; {\it and:} 
Erwin Schr\"{o}dinger Institut f\"{u}r Mathematische Physik,
Boltzmanngasse 9, A-1090 Wien, Austria}
\email{michor@pap.univie.ac.at }

\copyrightinfo{2000}{American Mathematical Society}
\date{July 27, 2000}
\commby{Alexandre Kirillov }
\keywords{Flow completion, non-Hausdorff manifolds}
\subjclass[2000]{Primary 37C10, 57R30}
\begin{abstract}For a vector field $X$ on a smooth manifold $M$ there 
exists a smooth but not necessarily Hausdorff manifold $M_{\mathbb{R}}$ 
a complete vector field $X_{\mathbb{R}}$ on it which is the universal 
completion of $(M,X)$. 

Let $X\in \X (M)$ be a smooth vector field on a 
smooth manifold $M$. 

Then there exists 
a universal flow completion $j:(M,X)\to 
(M_{\mathbb{R}},X_{\mathbb{R}})$ of $(M,X)$. 
Namely, there exists
a (connected) smooth not necessarily Hausdorff manifold 
a complete vector field $X_{\mathbb{R}}\in \X (M_{\mathbb{R}})$, and an 
$j:M\to M_{\mathbb{R}}$ onto an open submanifold such that $X$ and 
$X_{\mathbb{R}}$ are $j$-related: $Tj\myo X=X_{\mathbb{R}}\myo j$. 
Moreover, for any 
other equivariant morphism $f:(M,X)\to (N,Y)$ for a manifold $N$ and 
a complete vector field $Y\in X(N)$ there exists a unique equivariant 
morphism $f_{\mathbb{R}}:(M_{\mathbb{R}},x_{\mathbb{R}})\to (N,Y)$ with 
$f_{\mathbb{R}}\myo j=f$. The leaf spaces $M/X$ and 
$M_{\mathbb{R}}/X_{\mathbb{R}}$ are 

Consider the manifold $\mathbb{R}\x M$ with coordinate function $s$ on 
$\mathbb{R}$, the vector field $\bar X:=\partial _{s}\x X\in \X 
(\mathbb{R}\x M)$, 
and let $M_{\mathbb{R}}:= \mathbb{R}\x _{\bar X}M$ be the orbit space 
(or leaf 
space) of the vector field $\bar X$. 

Consider the flow mapping 
$\Fl ^{\bar X}:\mathcal{D}(\bar X)\to \mathbb{R}\x M$, given by 
$\Fl ^{\bar X}_{t}(s,x)=(s+t,\Fl ^{X}_{t}(x))$, where the domain of 
$\mathcal{D}(\bar X)\subset \mathbb{R}\x (\mathbb{R}\x M)$ is an open 
neighbourhood of $\{0\}\x (\mathbb{R}\x M)$ with the property that 
$\mathbb{R}\x \{x\}\cap \mathcal{D}(\bar X)$ is an open interval times 

For each $s\in \mathbb{R}$ we consider the mapping 
j_{s}:M @>{\myon {ins}_{t}}>> \{s\}\x M\subset \mathbb{R}\x M @>{\pi }>> 
\mathbb{R}\x _{\bar X}M= M_{\mathbb{R}}.
Each mapping $j_{s}$ is injective: A trajectory of $\bar X$ can 
meet $\{s\}\x M$ at most once since it projects onto the unit speed 
flow on $\mathbb{R}$. 

Obviously, the image $j_{s}(M)$ is open in $M_{\mathbb{R}}$ in the 
quotient topology:
If a trajectory hits $\{s\}\x M$ in a point $(s,x)$, let $U$ be an open 
neighborhood of $x$ in $M$ such that 
$(-\ep ,\ep )\x (s-\ep ,s+\ep )\x U\subset \mathcal{D}(\bar X)$. Then 
trajectories hitting $(s-\ep ,s+\ep )\x U$ fill a flow invariant open 
neighborhood which projects on an open neighborhood of $j_{s}(x)$ in 
$M_{\mathbb{R}}$ which lies in $j_{s}(M)$. This argument also shows that 
$j_{s}$ is a homeomorphism onto its image in $M_{\mathbb{R}}$.

Let us use the mappings $j_{s}:M\to M_{\mathbb{R}}$ as charts. The chart 
change then looks as follows: For $r0$. The charts $j_{r}(M)$ and $j_{s}(M)$ are glued 
together by the shift $x\mapsto x+s-r$. 
In this example $M_{\mathbb{R}}$ is not Hausdorff, but its Hausdorff 
quotient (given by the equivalence relation generated 
by identifying non-separable points) is again a smooth manifold and 
has the universal property described in {1}. 

\subsection*{{3}. Example }Let $(M,X)=(\mathbb{R}
^{2}\setminus \{0\}\x [-1,1],\partial _{x})$.
The trajectories of $\bar X$ on $\mathbb{R}\x M$ in the 
slices $y=\text{constant}$ for $|y|\le 1$ and $|y|\ge 1$ then look as 
in the second and third illustration above. The flow completion 
$M_{\mathbb{R}}$ then becomes $\mathbb{R}^{2}$ with the part 
$\mathbb{R}\x [-1,1]$ 
doubled and the topology such that the points $(x,-1)_{-}$ and 
$(x,-1)_{+}$ cannot be separated as well as the points $(x,1)_{-}$ and 
$(x,1)_{+}$. The flow is just $(x,y)\to (x+t,y)$:
\noindent In this example $M_{\mathbb{R}}$ is not Hausdorff, and its Hausdorff 
quotient is not a smooth manifold any more. There are two obvious 
quotient manifolds which are Hausdorff, the cylinder and the plane. 
Thus none of these two has the universal property of {1}.

\subsection*{{4}. Non-Hausdorff smooth manifolds }We met second 
countable smooth manifolds which need not be 
Hausdorff. Let us discuss a little their properties. They are $T_{1}$, 
since all points are closed; they are closed in a chart. The 
construction of the tangent bundle is by glueing the local tangent 
Smooth mappings and vector fields are defined as usual: non-separable 
pairs of points are mapped to non-separable pairs. Vector fields 
admit flows as usual: these are given locally in the charts and are 
then glued together. If $x$ and $y$ are non-separable points and
$X$ is a vector field on the manifold, then for each $t$ the points 
$\Fl ^{X}_{t}(x)$ and $\Fl ^{X}_{t}(y)$ are non-separable. 
Theorem {1} can be extended to the category of not necessarily 
Hausdorff smooth manifolds and vector fields, without any change in 
the proof. 

\subsection*{{5}. Remark }The ideas in this paper generalize to the 
setting of 
$\mathfrak{g}$-manifolds, where $\mathfrak{g}$ is a finite dimensional 
Lie group. 
Let $G$ be the simply connected Lie group with Lie algebra 
Then one may construct the $G$-completion of a non-complete 
$\mathfrak{g}$-manifold. There are difficulties with the property 
not only with Hausdorff.
This was our original road which was inspired by \cite{1}. 
We treat the full theory in \cite{2}.
We thought that the special case of a vector field is 
interesting in its own.   


D. V. Alekseevsky and Peter W. Michor, {\em Differential geometry of 
$\mathfrak{g}$-manifolds.}, Differ. Geom. Appl. {\bf 5} (1995), 
371--403, math.DG/9309214.
F. W. Kamber and P. W. Michor, {\em Completing Lie algebra actions to 
Lie group actions}, in preparation.