\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}{����}
\newtheorem{cor}[thm]{��������}
\newtheorem{lem}[thm]{��������}
\newtheorem{prop}[thm]{����}
\newtheorem{cl}{Claim}

{\theorembodyfont{\rm}
\newtheorem{ex}{��}
\newtheorem{que}{����}
\newtheorem{notation}{Notation}[section]
\newtheorem{defn}{����}
\newtheorem{rem}{��}
\newtheorem{note}{Note}
}
\renewcommand{\thenote}{}
\renewcommand{\therem}{}

\newenvironment{proof}{{\bf ����}}{\hfill\framebox[2mm]{}}
\newenvironment{proof1}{{\bf ��������}}{\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)�� ���Ƿκ��� �翬�ϰ�, (4)�� ��������.

���� $\sigma(t)$�� integral curve(with $\sigma(0)=p$)���
$\widetilde{\sigma}(t)=\sigma(t+a)$ ���� integral curve(with
$\widetilde{\sigma}(0)=\sigma(a))$ �̴�.

$\because $ $a$��ŭ traslation��Ű�� $f$�� ����
$\,\,\widetilde{\sigma}=\sigma\circ f$ �� ����,

$\frac{d\widetilde{\sigma}}{dt}(t)=\widetilde{\sigma_*}(\frac{d}{dt}|_t)$

$\hs{2.3em}=\sigma_*f_*(\frac{d}{dt}|_t)$


$\hs{2.3em}=\sigma_*(\frac{d}{dt}|_{t+a})$

$\hs{2.3em}=\frac{d\sigma}{dt}(t+a)$

$\hs{2.3em}=X(\sigma(t+a))$

$\hs{2.3em}=X(\widetilde{\sigma}(t))$\\

$\alp(s,p)=q$�� �ΰ�  �� ��  $\alp_q(t)=\alp(t,q)$ on $J_q$��
$q$������ maximal integral curve (with initial data $(0,q)$) �̴�.
�׷��� $\beta(t):=\alp(t+s,p)$ defined on $J_p-s$�� �θ� �̴� ����
���뿡 ���� initial data�� $(0,q)$�� ������ integral curve�� �ȴ�.
���� $\beta$�� $\alp$�� ���Եǰ�  �� $J_q\supset J_p-s$,
$\alp(t,q)=\alp_q(t)=\beta(t)=\alp(t+s,p)$ on $J_p-s$.

������ $\gamma(t):=\alp(t-s,q)$ defined on $J_q+s$ �� �θ� �̴�
initial data�� $(0,p)$�� ������ integral curve�� �ȴ�. ���� ����
���������� $J_q+s\subset J_p$ �� �����ϰ� �׷��Ƿ� $J_p-s=J_q$
�̴�.\\

(1) and (2)

 Let $A=\{t\in J_p\,|\,\exists
 U\,\,a\,\,neighborhood\,\,of\,\,(t,p)\,\,such\,\,that\,\,U\subset\dc\,and\,\,\alp\,\,is\,\,\ccn\,\,on\,\,U\}$.

 ����  local flow theorem���κ��� $0\in A$����
 �� �� �����Ƿ� $A\neq\emptyset$�̴�. ���� $A$�� ���Ƿκ��� $A$��
 open���� �翬�ϴ�. ���� $A$�� closed���� ���̱⸸ �ϸ� connectedness�κ���
 (1),(2)�� ������ �� �ִ�.

 $\overline{A}\subset A$�� ������. $s\in \overline{A}$ �̶� �θ�  $\alp(s,p)\in M$
�� ���� local flow�� �����Ѵ�. 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
$\ccn$.

 �� �� $\alp_p(t)$�� $t$�� ���� �����̹Ƿ�  $\,\,\exists\, I_s$, a
 neighborhood of $s$, such that

 $\alp_p(I_s)\subset V$ and $I_s\subset
 I+s$.

 $s\in\overline{A}$�̹Ƿ�  $a\in I_s\cap A$�� ���� �� �ְ�, �� $a$�� ���� $A$��
  ���ǿ� ���� $\exists (a-\eps,a+\eps)\times U_p=U$, a neighborhood of $(a,p)$, such that $U\subset\dc$ and
 $\alp_2=\alp|_U$ is $\ccn$ �̴�. �� �� $\alp(U)\subset V$��� �����ص�
 �����ϴ�.

 ���� $I+a$�� �����غ���. �̰��� �ٷ� $\dc$�� ���� $s$�� �ٹ���
 �ȴ�. ���� $a\in I_s\subset I+s$ �̹Ƿ� $a-s\in I,s-a\in I$�� �ǰ� ���� $ s\in
 I+a$�̴�.  ���� $W:=(I+a)\times U_p\subset\dc$ ���� ������.

 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., $\hs{2em}W\hs{1em}\rightarrow \hs{1em}I\times U\overset{id\times\alp_2}{\rightarrow} \hs{1em}I\times
  V\hs{1em}\overset{\alp_1}{\rightarrow}\hs{1.5em} M$

 $\hs{2em}(u+a,q)\mapsto(u,(a,q))\mapsto(u,\alp_2(a,q))\mapsto\alp_1(u,\alp_2(a,q))$\\

���� $s\in A$���� �������Ƿ� $A$�� closed�̴�.\end{proof}





\begin{figure}[htb]
 \centerline{\includegraphics*[scale=0.42,clip=true]{grp22.eps}}

\end{figure}

�׸� 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 $p\in\dc_t,\,\,\alp_t(p)\in\dc_{-t}$���� ������.

$\alp(-t,\alp_t(p))=\alp(-t,\alp(t,p))=\alp(0,p)$(recall that
$J_{\alp(t,p)}=J_p-t$)

 �̹Ƿ� �� ���ǵȴ�.

���� $(-t,\alp_t(p))\in\dc$�̴�. �׸���


$\alp_{-t}\circ\alp_t(p)=\alp_{-t}(\alp_{t}(p))=\alp(-t,\alp(t,p))=\alp(0,p)=p$
�̰�

���� ���������� $\alp_{t}\circ\alp_{-t}=id$�̴�.\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.} �Ϲ������� $a,b$�� continuous���� �ʴ�.

 �� : $\rb^2\setminus\{0\}$���� $X=\frac{\pa}{\pa u_1}$ �� �����غ���.

\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 $t=b(p)-\frac{\eps}{2}$.

$\alp(t,p)$�� ���� $J_{\alp(t,p)}=J_p-t$ �̰� ����
$b(\alp(p,t))=b(p)-t$ �̴�.

$\therefore\,\,\frac{\eps}{2}+t=b(p)=b(\alp(p,t))+t>\eps+t$,
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
$t_o<t<b(p)\Rightarrow \alp_p(t)\notin K$.

\end{prop}

{\bf ���� 9.} �� ������ �����϶�.


\begin{cor}
$supp \,\,X=\overline{\{p\in M\,|\,X(p)\neq 0\}}$ is compact
$\Rightarrow\,X$ is complete.

In particular, $M$ is compact $\Rightarrow$ $\forall\,X$ on $M$ is
complete.

\end{cor}

\begin{proof}
$\forall p\in K= supp\,X,\,$ there is a local flow of $p$ , i.e.,
$\exists (-\eps_p,\eps_p),U_p$ such that

 $(-\eps_p,\eps_p)\times
U_p\subset\dc$. $K$�� compact�̹Ƿ� finite subcover
$\{U_1,\cdot\cdot\cdot,U_k\}$�� �����Ѵ�. ���� �׿� �ش��ϴ�
$(-\eps_1,\eps_1),\cdot\cdot\cdot,(-\eps_k,\eps_k)$�� ����
$\eps=min\{\eps_1,\cdot\cdot\cdot,\eps_k\}$ �̶� �θ�
$(-\eps,\eps)\times M\subset\dc$�̴�. ���⼭ $p\notin supp\,X$ ��
��쿡�� constant map�� integral curve�� �ǹǷ� �翬��
$J_p=(-\infty,\infty)$�̴�.
\end{proof}\\

{\bf ���� 10.} $X$ is a vector field on $M$ and $f\in\ccn(M,\rb)$
is a positive function. Let $\widetilde{X}=fX$,
$\widetilde{J_p}=(\widetilde{a}(p),\widetilde{b}(p))$. Then

(1) The integral curve for $\widetilde{X}$ is a reparametrization
of the integral curve for $X $.

(2) $\widetilde{a}(p)=\int_{0}^{a(p)}\frac{1}{f(\alp(t,p))}dt$ ,
$\widetilde{b}(p)=\int_{0}^{b(p)}\frac{1}{f(\alp(t,p))}dt$,
$\forall\,p\in M.$

  \end{document}
