\documentclass[12pt ]{article} \setlength{\textwidth}{14 true cm} \setlength{\textheight}{20 true cm} \usepackage{graphicx} \usepackage{hangul} \usepackage{amscd,amsmath} \usepackage{amsfonts} \usepackage{amssymb,theorem} \usepackage{longtable} \newcommand{\hs}{\hspace} \newcommand{\inv}{^{-1}} \newcommand{\vphi}{\varphi} \newcommand{\paxi}{\frac{\partial}{\partial x_i}} \newcommand{\payj}{\frac{\partial}{\partial y_j}} \newcommand{\paxj}{\frac{\partial}{\partial x_j}} \newcommand{\payi}{\frac{\partial}{\partial y_i}} \newcommand{\dis}{\displaystyle} \newcommand{\disi}{\displaystyle{\sum_{i=1}^n}} \newcommand{\disj}{\displaystyle{\sum_{j=1}^n}} \newcommand{\pa}{\partial} \newcommand{\Aff}{\mbox{\it Aff}} \newcommand{\aff}{\mbox{\it aff}} \newcommand{\cc}{\mathcal{C}} \newcommand{\dc}{\mathcal{D}} \newcommand{\ccn}{\mathcal{C}^{\infty}} \newcommand{\sbb}{\mathbb{S}} \newcommand{\rb}{\mathbb{R}} \newcommand{\rc}{\mathcal{R}} \newcommand{\alp}{\alpha} \newcommand{\bet}{\beta} \newcommand{\del}{\delta} \newcommand{\gam}{\gamma} \newcommand{\vep}{\varepsilon} \newcommand{\eps}{\epsilon} \newcommand{\lam}{\lambda} \newcommand{\kap}{\kappa} \newcommand{\sig}{\sigma} \newcommand{\ome}{\omega} \newcommand{\Gam}{\Gamma} \newcommand{\Ome}{\Omega} \newcommand{\Sig}{\Sigma} \newcommand{\Del}{\Delta} \newcommand{\Lam}{\Lambda} \newtheorem{thm}{정리} \newtheorem{cor}[thm]{따름정리} \newtheorem{lem}[thm]{보조정리} \newtheorem{prop}[thm]{명제} \newtheorem{cl}{Claim} {\theorembodyfont{\rm} \newtheorem{ex}{예} \newtheorem{que}{질문} \newtheorem{notation}{Notation}[section] \newtheorem{defn}{정의} \newtheorem{rem}{주} \newtheorem{note}{Note} } \renewcommand{\thenote}{} \renewcommand{\therem}{} \newenvironment{proof}{{\bf 증명}}{\hfill\framebox[2mm]{}} \newenvironment{proof1}{{\bf 정리증명}}{\hfill\framebox[2mm]{}} \begin{document} \parindent=0cm \section*{Global flow.} Let $X$ be a $\ccn$ vector field on $M$. For $p\in M,$ let $\alp_p:J_p=(a(p),b(p))\rightarrow M$ be the maximal integral curve through $p=\alp_p(0)$. Let $\mathcal{D}:=\{(t,p)\in\rb\times M\,|\,t\in J_p\,\,,\,\,p\in M\}$ , $\alp:\dc\rightarrow M$ defined by $\alp(t,p)=\alp_p(t)$ is called the (global) flow of $X$. \begin{thm} $\dc\subset\rb\times M$ and $\alp:\dc\rightarrow M$ as above. Then (1) $\dc$ is open. (2) $\alp$ is $\ccn$. (3) $\alp(t,p)=\alp_p(t)$ gives the maximal integral curve with $\alp_p(0)=p$. (4) $\alp(t,\alp(s,p))=\alp(t+s,p)$ for points in $\dc$ and $J_{\alp(t,p)}=J_p-t.$ \end{thm} \begin{proof} (3)은 정의로부터 당연하고, (4)를 증명하자. 먼저 $\sigma(t)$가 integral curve(with $\sigma(0)=p$)라면 $\widetilde{\sigma}(t)=\sigma(t+a)$ 역시 integral curve(with $\widetilde{\sigma}(0)=\sigma(a))$ 이다. $\because $ $a$만큼 traslation시키는 $f$에 대해 $\,\,\widetilde{\sigma}=\sigma\circ f$ 로 보면, $\frac{d\widetilde{\sigma}}{dt}(t)=\widetilde{\sigma_*}(\frac{d}{dt}|_t)$ $\hs{2.3em}=\sigma_*f_*(\frac{d}{dt}|_t)$ $\hs{2.3em}=\sigma_*(\frac{d}{dt}|_{t+a})$ $\hs{2.3em}=\frac{d\sigma}{dt}(t+a)$ $\hs{2.3em}=X(\sigma(t+a))$ $\hs{2.3em}=X(\widetilde{\sigma}(t))$\\ $\alp(s,p)=q$라 두고 이 때 $\alp_q(t)=\alp(t,q)$ on $J_q$는 $q$에서의 maximal integral curve (with initial data $(0,q)$) 이다. 그런데 $\beta(t):=\alp(t+s,p)$ defined on $J_p-s$라 두면 이는 위의 내용에 의해 initial data를 $(0,q)$로 가지는 integral curve가 된다. 따라서 $\beta$는 $\alp$에 포함되고 즉 $J_q\supset J_p-s$, $\alp(t,q)=\alp_q(t)=\beta(t)=\alp(t+s,p)$ on $J_p-s$. 역으로 $\gamma(t):=\alp(t-s,q)$ defined on $J_q+s$ 라 두면 이는 initial data를 $(0,p)$로 가지는 integral curve가 된다. 따라서 위와 마찬가지로 $J_q+s\subset J_p$ 를 만족하고 그러므로 $J_p-s=J_q$ 이다.\\ (1) and (2) Let $A=\{t\in J_p\,|\,\exists U\,\,a\,\,neighborhood\,\,of\,\,(t,p)\,\,such\,\,that\,\,U\subset\dc\,and\,\,\alp\,\,is\,\,\ccn\,\,on\,\,U\}$. 먼저 local flow theorem으로부터 $0\in A$임을 알 수 있으므로 $A\neq\emptyset$이다. 또한 $A$의 정의로부터 $A$가 open임은 당연하다. 이제 $A$가 closed임을 보이기만 하면 connectedness로부터 (1),(2)를 증명할 수 있다. $\overline{A}\subset A$를 보이자. $s\in \overline{A}$ 이라 두면 $\alp(s,p)\in M$ 에 대해 local flow가 존재한다. i.e., $\exists I=(-\del,\del)\,\,and\,\,V$, a neighborhood of $\alp(s,p)$, such that $I\times V\subset\dc$ and $\alp_1=\alp|_{I\times V}$ is $\ccn$. 이 때 $\alp_p(t)$가 $t$에 대해 연속이므로 $\,\,\exists\, I_s$, a neighborhood of $s$, such that $\alp_p(I_s)\subset V$ and $I_s\subset I+s$. $s\in\overline{A}$이므로 $a\in I_s\cap A$를 잡을 수 있고, 이 $a$에 대해 $A$의 정의에 의해 $\exists (a-\eps,a+\eps)\times U_p=U$, a neighborhood of $(a,p)$, such that $U\subset\dc$ and $\alp_2=\alp|_U$ is $\ccn$ 이다. 이 때 $\alp(U)\subset V$라고 가정해도 무관하다. 이제 $I+a$를 생각해보자. 이것이 바로 $\dc$에 들어가는 $s$의 근방이 된다. 먼저 $a\in I_s\subset I+s$ 이므로 $a-s\in I,s-a\in I$가 되고 따라서 $ s\in I+a$이다. 이제 $W:=(I+a)\times U_p\subset\dc$ 임을 보이자. Let $\alp_3:=\alp|_{(I+a)\times U_p}$. Then for $t\in I+a,q\in U_p$ $\alp_3(t,q)=\alp(u+a,q)=\alp(u,\alp(a,q))=\alp_1(u,\alp_2(a,q)):\ccn.$ i.e., $\hs{2em}W\hs{1em}\rightarrow \hs{1em}I\times U\overset{id\times\alp_2}{\rightarrow} \hs{1em}I\times V\hs{1em}\overset{\alp_1}{\rightarrow}\hs{1.5em} M$ $\hs{2em}(u+a,q)\mapsto(u,(a,q))\mapsto(u,\alp_2(a,q))\mapsto\alp_1(u,\alp_2(a,q))$\\ 따라서 $s\in A$임을 보였으므로 $A$는 closed이다.\end{proof} \begin{figure}[htb] \centerline{\includegraphics*[scale=0.42,clip=true]{grp22.eps}} \end{figure} 그림 22\\ \begin{cor} Let $\dc_t=\{p\in M\,|\,(t,p)\in \dc\}$ and $\alp_t:\dc_t\rightarrow M$ be given by $\alp_t(p)=\alp(t,p).$ Then (1) $\dc_t$ is open $\forall t$ and $\dis{\bigcup_{t>0}\dc_t=M}$. (2) $\alp_s\circ\alp_t=\alp_{s+t}$ wherever defined. (3) $\alp_t:\dc_t\rightarrow\dc_{-t}$ is a diffeomorphism with inverse $\alp_{-t}$. \end{cor} \begin{proof} (1) Consider the slice map $i_t:\dc_t\rightarrow \dc$, $i_t(p)=(t,p)$ , which is continuous. $\Rightarrow i_t\inv(\dc)=\dc_t$ is open.\\ (2) $(\alp_s\circ\alp_t)(p)=\alp_s(\alp_t(p))=\alp_s(\alp(t,p))=\alp(s,\alp(t,p))=\alp(s+t,p)$ $\hs{7em}=\alp_{s+t}(p),\,\,\,\forall p\in M $.\\ (3) For $p\in\dc_t,\,\,\alp_t(p)\in\dc_{-t}$임을 보이자. $\alp(-t,\alp_t(p))=\alp(-t,\alp(t,p))=\alp(0,p)$(recall that $J_{\alp(t,p)}=J_p-t$) 이므로 잘 정의된다. 따라서 $(-t,\alp_t(p))\in\dc$이다. 그리고 $\alp_{-t}\circ\alp_t(p)=\alp_{-t}(\alp_{t}(p))=\alp(-t,\alp(t,p))=\alp(0,p)=p$ 이고 역시 마찬가지로 $\alp_{t}\circ\alp_{-t}=id$이다.\end{proof}\\ {\bf Remark.} $\{ \alp_t\}$ is called {\it the local 1-parameter family of lacal diffeomorphisms }given by a vector field $X$ or simply {\it the flow} generated by $X$. \\ \begin{cor} Let $J(p)=(a(p),b(p))$ be the maximal domain for integral curve at $p$. Then $b(a, respectively):M\rightarrow\rb\cup\{\pm\infty\}$ is lower(upper, resp.) semi-continuous. i.e., $b\inv(\lam,\infty]$($a\inv[-\infty,\lam)$, resp.) is open , $\forall \lam\in\rb$. \end{cor} \begin{proof} $b\inv(\lam,\infty]=\{p\in M\,|\,b(p)>\lam\}=\{p\in M\,|\,(t,p)\in \dc$ for some $t>\lam\}$ $\hs{6.8em}=\dis{\bigcup_{t>\lam}\{p\in M\,|\,(t,p)\in\dc\}=\bigcup_{t>\lam}\dc_t}$. $a\inv[-\infty,\lam)=\{p\in M\,|\,a(p)<\lam\}=\{p\in M\,|\,(t,p)\in \dc$ for some $t<\lam\}$ $\hs{6.8em}=\dis{\bigcup_{t<\lam}\{p\in M\,|\,(t,p)\in\dc\}=\bigcup_{t<\lam}\dc_t}$. \end{proof}\\ {\bf Remark.} 일반적으로 $a,b$는 continuous하지 않다. 예 : $\rb^2\setminus\{0\}$에서 $X=\frac{\pa}{\pa u_1}$ 을 생각해보라. \begin{defn}{\it A vector field $X$ is {\bf complete} if $J_p=(-\infty,\infty)$ for all $p\in M$. In this case $\{\alp_t\}$ is a global 1-parameter family of (global) flows of M.} \end{defn} \begin{prop} If $(-\eps,\eps)\times M\subset \dc$ for some $\eps>0, $ then $X$ is complete. \end{prop} \begin{proof} Suppose $b(p)<\infty,$ then take $t=b(p)-\frac{\eps}{2}$. $\alp(t,p)$에 대해 $J_{\alp(t,p)}=J_p-t$ 이고 따라서 $b(\alp(p,t))=b(p)-t$ 이다. $\therefore\,\,\frac{\eps}{2}+t=b(p)=b(\alp(p,t))+t>\eps+t$, contradiction. \end{proof} \begin{prop} $p\in M$, If $b(p)<\infty,$ then $"\lim_{t\rightarrow b(p)}\alp_p(t)=\infty"$, i.e., $\forall K^{cpt}\subset M,\exists t_0 $ such that $t_o