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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package %% Translation via Omnimark script a2l, August 28, 2000 (all in one day!) \controldates{6-OCT-2000,6-OCT-2000,6-OCT-2000,6-OCT-2000} \documentclass{era-l} \issueinfo{6}{12}{}{2000} \dateposted{October 10, 2000} \pagespan{95}{97} \PII{S 1079-6762(00)00083-4} \usepackage{amscd} \usepackage{graphicx} %% Declarations: \theoremstyle{plain} \newtheorem*{theorem1}{{1}. Theorem} %% User definitions: \newcommand{\myo}{\circ } \newcommand{\X}{\mathfrak X} \newcommand{\ep}{\varepsilon } \newcommand{\myi}{^{-1}} \newcommand{\x}{\times } \newcommand{\Fl}{\operatorname {Fl}} \newcommand{\myon}{\operatorname } \begin{document} \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 $M_{\mathbb{R}}$ and a complete vector field $X_{\mathbb{R}}$ on it which is the universal completion of $(M,X)$. \end{abstract} \maketitle \begin{theorem1} Let $X\in \X (M)$ be a smooth vector field on a (connected) 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 $M_{\mathbb{R}}$, a complete vector field $X_{\mathbb{R}}\in \X (M_{\mathbb{R}})$, and an embedding $j:M\to M_{\mathbb{R}}$ 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 homeomorphic. \end{theorem1} \begin{proof} 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 definition $\mathcal{D}(\bar X)\subset \mathbb{R}\x (\mathbb{R}\x M)$ 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 $\{x\}$. For each $s\in \mathbb{R}$ we consider the mapping \begin{equation*}\begin{CD} j_{s}:M @>{\myon {ins}_{t}}>> \{s\}\x M\subset \mathbb{R}\x M @>{\pi }>> \mathbb{R}\x _{\bar X}M= M_{\mathbb{R}}. \end{CD}\end{equation*} 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 the 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 $r~~0$. 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 $(x,y)\to (x+t,y)$: \begin{center} \includegraphics[scale=.52]{era83e-fig-4} \end{center} \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 bundles. 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 $\mathfrak{g}$. Then one may construct the $G$-completion of a non-complete $\mathfrak{g}$-manifold. There are difficulties with the property $T_{1}$, 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. \bibliographystyle{amsalpha} \begin{thebibliography}{99} \bibitem[1]{1} 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. \MR{96k:53035} \bibitem[2]{2} F. W. Kamber and P. W. Michor, {\em Completing Lie algebra actions to Lie group actions}, in preparation. \end{thebibliography} \end{document}~~