\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{\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{\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*{Definitions.} \begin{defn}{\it Let $\phi:M\rightarrow N$ be a $\ccn$ map. (1)$\phi$ is an immersion if $d\phi_p$ is injective for $\forall p\in M$. (2)$(M,\phi)$ is a submanifold of $N$ if $\phi$ is an injective immersion. (3)$\phi$ is an embedding if $\phi$ is an injective immersion which is also a topological embedding. (4) $\phi$ is a submersion if $d\phi_p$ is surjective for $\forall p\in M$.} \end{defn} \begin{figure}[htb] \centerline{\includegraphics*[scale=0.5,clip=true]{grp15.eps}} \end{figure} $\hspace{15em}$ 그림 15\\ 예를 들어 projection map은 submersion이 된다.\\ {\bf Local picture} \begin{lem} $f:U(\subset\rb^m)\rightarrow\rb^n,\ccn$ and has constant rank $r$ on a neighborhood of $p\in U$. Then $\exists$a rectangular coordinate charts $x$ about $p$ and $y$ about $f(p)$ such that $\hspace{3em}(y\circ f\circ x^{-1})(a_1,\cdot\cdot\cdot,a_m)=(\overset{n}{\overbrace{a_1,\cdot\cdot\cdot,a_r,0,\cdot\cdot\cdot,0}}).$ \end{lem} \begin{proof} May assume $det(\frac{\pa f_i}{\pa u_j})\neq 0$, $i,j=1,\cdot\cdot,r$ on $U$ by rearranging coordinates $u_i$ on $\rb^m$ and $\rb^n$ and by restricting $U$. Define $x:U\rightarrow\rb^m$ by $x(u)=(f_1(u),\cdot\cdot,f_r(u),u_{r+1},\cdot\cdot,u_m)$, then \[ Dx= \left( \begin{array}{rr} (\frac{\pa f_i}{\pa u_j})_{r\times r} & * \\ 0 \,\,\,\,\,\,\,\, & I \end{array} \right)_{m\times m} :\,\,nonsingular \] By the inverse function theorem, $x$ is a coordinate chart on a neighborhood $"U"$ of $p$. Let $x(p)=(a,b)$ and $V_r,V_{m-r}$ as in the picture so that $V_r\times V_{m-r}\subset dom(x^{-1})$.\\ \vspace{3em} \begin{figure}[htb] \centerline{\includegraphics*[scale=0.4,clip=true]{grp16.eps}} \end{figure} $\hspace{15em}$ 그림 16\\ $(a,b)\overset{x^{-1}}{\mapsto}(h(a,b),b)\overset{x}{\mapsto}(f_{1,\cdot\cdot\cdot,r}(h(a,b),b),b)=(a,b)$ $\Rightarrow f_{1,\cdot\cdot\cdot,r}(h(a,b),b)=a$ (independent of $b$) $\Rightarrow (f\circ x^{-1})(a,b)=(a,f_{r+1,\cdot\cdot,n}(h(a,b),b))$ Consider \[ D(f\circ x^{-1})= \left[ \begin{array}{cc} I & 0 \\ * & \,\,\,\,\, (\frac{\pa (f_i\circ x^{-1} )}{\pa u_j}) \end{array} \right]_{i=r+1,\cdot\cdot,n\,\,and\,\,j=r+1,\cdot\cdot,m} \] $rank\,\,\,D(f\circ x^{-1})=rank\,\,Df=r$ at $\forall p$ of $U$. $\therefore (\frac{\pa(f_i\circ x^{-1})}{\pa u_j})=0$ for $i=r+1,\cdot\cdot,n$ and $j=r+1,\cdot\cdot,m$.(i.e., $\frac{\pa(f_{r+1\cdot\cdot, n}\circ x^{-1})}{\pa b}=0$) $\therefore f_{r+1,\cdot\cdot,n}\circ x^{-1}(a,b)=f_{r+1,\cdot\cdot,n}(h(a,b),b)=g(a)$ (i.e., independent of $b$) for some $\ccn$ function $g$. Define coordinate chart $y$ on $V_r\times \rb^{n-r}$ by $y(a,c)=(a,c-g(a))$, then $(y\circ f\circ x^{-1})(a,b)=y(a,g(a))=(a,0)\hspace{2em}$ and hence $(y\circ f\circ x^{-1})(a_1,\cdot\cdot\cdot,a_r,a_{r+1},\cdot\cdot,a_m)=(a_1,\cdot\cdot,a_r,0,\cdot\cdot,0)$ on $V_r\times V_{m-r}$. \end{proof}\\ \begin{cor} $\phi:M^m\rightarrow N^n\,\,is\,\,\ccn,$ $d\phi$ has constant rank $\,r$ on neighborhood of $p\in M.$ Then $\exists$ rectangular coordinate charts $x$ about $p$, y about $f(p)$ such that $(y\circ \phi\circ x^{-1})(a_1,\cdot\cdot\cdot,a_m)=(a_1,\cdot\cdot,a_r,0,\cdot\cdot,0)$. \end{cor} \begin{cor} $\phi:M^m\rightarrow N^n$ , $\ccn$, is an immersion($m\leq n$). $\Rightarrow \forall p\in M,\exists\,\,$rectangular coordinate charts $x$ about $p$, y about $f(p)$ such that $(y\circ\phi\circ x^{-1})(a_1,\cdot\cdot\cdot,a_m)=(a_1,\cdot\cdot,a_m,0,\cdot\cdot,0)$ \end{cor} \begin{proof} $d\phi$가 injective하므로 rank가 m이고 따라서 바로 위의 정리에 따르면 된다. \end{proof}\\ \begin{cor} $\phi:M^m\rightarrow N^n$,$\ccn$, is a submersion ($m\geq n) $ then $\forall p\in M$, $\exists\,$ rectangular coordinate charts $x$ about $p$, y about $f(p)$ such that $(y\circ\phi\circ x^{-1})(a_1,\cdot\cdot,a_n,\cdot\cdot,a_m)=(a_1,\cdot\cdot\cdot,a_n)$.\\ \end{cor} {\bf Remark.} Hence an immersion is locally an inclusion and so embedding.(the local topology is the same as that of a slice)\\ \begin{cor} (1) $i:M^m\hookrightarrow N^n$ is a submanifold $\Leftrightarrow \forall p\in M,\exists$ rectangular coordinate chart $(U,x)$ about $p$ such that $x(p)=0$, and $V=\{p\in U\,\,|\,\,x_{m+1}(p)=\cdot\cdot\cdot=x_n(p)=0\}$ is a neighborhood of $p$ in M and $(x_1|_{V},\cdot\cdot\cdot,x_m|_{V})$ is a coordinate chart of $M$.\\ (2) $i:M^m\hookrightarrow N^n$ is an embedding $\Leftrightarrow \forall p\in M,\exists$ rectangular coordinate chart $(U,x)$ about $p$ such that $x(p)=0$, $V=\{p\in U\,\,|\,\,x_{m+1}(p)=\cdot\cdot\cdot=x_n(p)=0\}$ is a neighborhood of $p$ in M , $(x_1|_{V},\cdot\cdot\cdot,x_m|_{V})$ is a coordinate chart of $M$ and $M\cap U=V$ .\\ \end{cor} \begin{figure}[htb] \centerline{\includegraphics*[scale=0.5,clip=true]{grp17.eps}} \end{figure} $\hspace{15em}$ 그림 17\\ \begin{proof} (1)의 $\Rightarrow$ 는 따름정리 3에 의해 잡을 수 있는 coordinate chart들과 local inclusion map을 이용하면 되고, $\Leftarrow$는 $i$가 locally immersion이라는 사실로부터 당연하다. (2)의 $\Rightarrow$를 보이자. (1)의 내용을 만족하면서 $M\cap U=V$ 를 만족하는 $U$가 있음을 보이면 된다. embedding은 immersion이기도 하므로 (1)의 내용을 만족하는 $U$가 존재하고, 이에 대해 $i(V)$를 생각해보자. $i$는 embedding이므로 $i(V)$는 $i(M)$에서 open이다. 따라서 $i(V)=i(M)\cap U'$ ( $U'$ is open in $N$) 을 알 수 있고, 우리가 원하는 $"U"$를 $"U"=U\cap U'$으로 두면 된다. (rectangular coordinate를 잡으려면 다시 $"U"$ 안에서 더 작은 $p$의 rectangular 근방을 잡으면 된다.) (2)의 $\Leftarrow$을 보이기 위해서는 $i(open)=open$ in $i(M)=M$ 임을 보이면 된다. 그런데 $i(V)=V=M\cap U$ 에서 $U$가 N의 open set이므로 $M\cap U $는 $i(M)=M$에서 open이 된다. basic open set에 대해 $i(basic \,\,open)=open$을 만족하므로 $i$가 open map(onto $i(M)$ )임을 알 수 있다. \end{proof}\\ {\bf Remark.} Corollary 3 shows {\it the uniqueness of smooth structure on a submanifold }i.e., if a subset $M\subset N$ has a topology, then there is at most one smooth structure on $M$ such that $i:M\hookrightarrow N$ is a submanifold.\\ \begin{proof} 두 개의 $\mathcal{F}_1,\mathcal{F}_2$가 $i:M\hookrightarrow N$ 를 submanifold로 만드는 smooth structure라고 하자. 두 structure 에 대해 $i$는 submanifold이므로 따름정리5의 (1)에 따라 chart $(U_1,\phi_1),V_1,(U_2,\phi_2),V_2$를 잡을 수 있다. $(U_1,\phi_1)\in\mathcal{F}_1,(U_2,\phi_2)\in\mathcal{F}_2$ 일 때 $U_1\cap U_2$를 생각해보자. 원래 $\phi_1,\phi_2\in \mathcal{F}(N)$이므로 $\ccn$인 transition map $\phi_2\circ \phi_1^{-1}$이 존재한다. 이것을 $\phi_1(V_1)$에 restriction시키면 이것 역시 $\ccn$이다. 왜냐하면 우리가 구하는 $\phi_1(V_1)$에서 $\phi_2(V_2)$로 가는 transition map $\phi_2|_{V_2}\circ\phi_1|_{V_1}^{-1}$은 $\pi\circ\phi_2\circ\phi_1^{-1}\circ i$ 로 표현되기 때문이다. 따라서 $\phi_1|_{V_1}$, $\phi_2|_{V_2}$는 $\ccn$-related되어 있고 $\mathcal{F}_1=\mathcal{F}_2$이다. \end{proof} \begin{prop} Let $\varphi:M\rightarrow N$ be $\ccn$ , $P$ is a submanifold of $N$ and $\vphi(M)\subset P$. If $\vphi:M\rightarrow P$ is continuous, then $\vphi$ is $\ccn$. \end{prop} {\bf 숙제 5.} 위 명제를 증명하라.\\ 위 명제에서 연속성이 필요한 이유를 다음과 같은 예에서도 볼 수 있다.\\ \begin{figure}[htb] \centerline{\includegraphics*[scale=0.5,clip=true]{grp18.eps}} \end{figure} $\hspace{15em}$ 그림 18\\ $N=\rb^2$,$M=(-1,1)$이고 $P$를 figure eight으로 보면 이 때 $i:M\rightarrow N$은 $\ccn$ 이나 $M\rightarrow P$로 가는 map은 불연속이다. \end{document}