\documentclass[11pt ]{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{\we}{\wedge} \newcommand{\vts}{v_1\otimes\cdot\cdot\cdot\ot v_k} \newcommand{\vs}{V_{1}\times\cdot\cdot\cdot\times V_k} \newcommand{\ot}{\otimes} \newcommand{\pauo}{\frac{\partial}{\partial u_1}} \newcommand{\paui}{\frac{\partial}{\partial u_i}} \newcommand{\pauj}{\frac{\partial}{\partial u_j}} \newcommand{\wid}{\widetilde} \newcommand{\ml}{\mathcal{L}} \newcommand{\hs}{\hspace} \newcommand{\inv}{^{-1}} \newcommand{\vphi}{\varphi} \newcommand{\pax}{\frac{\partial}{\partial x}} \newcommand{\pay}{\frac{\partial}{\partial y}} \newcommand{\paz}{\frac{\partial}{\partial z}} \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{\li}{[X,Y]} \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{\fc}{\mathcal{F}} \newcommand{\h}{\hspace{1em}} \newcommand{\hh}{\hspace{2em}} \newcommand{\hhh}{\hspace{3em}} \newcommand{\hhhh}{\hspace{4em}} \newcommand{\ccn}{\mathcal{C}^{\infty}} \newcommand{\sbb}{\mathbb{S}} \newcommand{\rb}{\mathbb{R}} \newcommand{\cb}{\mathbb{C}} \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} \newcommand{\pr}{(\,\,,\,\,)} \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*{ 1. Integration of n-forms on oriented M.} {\bf Recall} : Change of variable formula.\\ $\int_{\vphi(D)}fdx_1\cdot\cdot\cdot dx_n=\int_D f\circ\vphi|det\vphi_*|du_1\cdot\cdot\cdot du_n$.\\ If $\vphi$ is orientation-preserving, $\int_{\vphi(D)}fdx_1\cdot\cdot\cdot dx_n=\int_Df\circ\vphi \,\,(det\,\vphi_*)du_1 \cdot\cdot\cdot du_n$.\\ $\rb^n$에서의 n-form $\ome=fdx_1\we\cdot\cdot\cdot\we dx_n$를 생각해보자. $\rb^n$에서 $n$-form에 대한 적분은 $\int\ome=\int fdx_1\we\cdot\cdot\cdot \we dx_n:=\int fdx_1\cdot\cdot\cdot dx_n$ 으로 정의한다. 그러면\\ $\vphi^*\ome=\vphi^*(f)\vphi^*(dx_1)\we\cdot\cdot\cdot\we\vphi^*(dx_n)$ $\hs{2em}=(f\circ\vphi)d(\vphi^*x_1)\we\cdot\cdot\cdot\we d(\vphi^*x_n)$ $\hs{2em}=(f\circ\vphi)d(x_1\circ\vphi)\we\cdot\cdot\cdot d(x_n\circ\vphi)$ $\hs{2em}=(f\circ\vphi)det(\frac{\pa x}{\pa u})du_1\we\cdot\cdot\cdot\we du_n\hs{2em}$ (note that $d(x_i\circ\vphi)=\sum_j\frac{\pa x_i}{\pa u_j} du_j )$ $\hs{2em}= f\circ\vphi\,\,(det\vphi_*)du_1\we\cdot\cdot\cdot\we du_n$.\\ 따라서 위 변수변환 공식은 다음과 같이 간단히 표시할 수 있다.\\ $(*)\hs{4em}\dis{\int_{\vphi(D)}w=\int_D\vphi^*w}$.\\ If $\vphi$ is orientation-reversing, then $\hs{1em}\dis{\int_{\vphi(D)}w=-\int_D\vphi^*w}$.\\ 이제 manifold 위에서의 n-form의 적분을 정의하자. 먼저 $\ome$를 $supp\,\,\ome\subset(U,x)$, a positive coordinate chart(i.e., $x$는 orientation preserving )를 만족하는 $n$-form이라고 두자. (즉 single chart에 포함된다.) 이 $\ome$에 대해 적분을 다음과 같이 정의한다. $\dis{\int_M\ome:=\int_{\widetilde{U}}\vphi^*\ome}$, $\hs{2em}\vphi=x\inv,\widetilde{U}=x(U)$. $\dis{(\int_M\ome:=-\int_{\widetilde{U}}\vphi^*\ome},\hs{1.8em}$ if $\vphi$ is orientation-reversing.)\\ 이 정의가 잘 정의됨을 보이자. coordinate chart의 선택에 무관함을 보이기 위해 $(U,y)$를 다른 positive coordinate chart 라고 두자.\\ $\widetilde{\widetilde{U}}=y(U),\,\,\psi=y\inv$라고 두면, transition map $x\circ\psi$는 $\rb^n$간의 함수가 되고 여기에 변수변환 공식 $(*)$ 을 쓸 수 있다. 그러면 $\dis{\int_{\widetilde{U}}\vphi^*\ome=\int_{\widetilde{\widetilde{U}}}(x\circ\psi)^*(\vphi^*\ome)}= \int_{\widetilde{\widetilde{U}}}\psi^*\ome$, $\hs{1em}(x\circ\vphi=x\circ x\inv=id$이므로) $\therefore\,\,\,\dis{\int_{\widetilde{U}}\vphi^*\ome=\int_{\widetilde{\widetilde{U}}}\psi^*\ome}$.\\ 이제 일반적인 $\ome\,\,(n$-form with compact support on orientable $M$) 에 대해서 적분을 정의하자. 먼저 finite cover $\mathcal{U}=\{U_i\}$ of positive coordinate charts를 $\mathcal{U}=\bigcup U_i\supset $supp$ \,\,\ome$ 를 만족하게끔 잡는다. $\{\rho_i\}$ 를 partition of unity(subordinate to $\mathcal{U}=\{U_i\}$ )로 두고 다음과 같이 적분을 정의한다.\\ $\hs{2em}\dis{\int_M\ome:=\sum_i\int_M\rho_i\ome}$. 위의 정의가 cover $\mathcal{U}$와 partition of unity 의 선택에 무관하게 잘 정의됨을 보이자. $\mathcal{V}=\{V_j\},\{\tau_j\}$를 또다른 cover,partition of unity라고 두자. 그러면 $\dis{\sum_j\int_M\tau_j\ome=\sum_j\sum_i\int_M\rho_i(\tau_j\ome)\hs{1em}}$(적분의 정의에 의해)\\ 각 $\rho_i\ome$들은 single coordinate $ U_i$안에 있으므로 additive 이다. 따라서 위 식은 $\dis{\sum_j\int_M\tau_j\ome=\sum_i\int_M(\sum_j\tau_j)\rho_i\ome=\sum_i\int_M\rho_i M.}$\\ {\bf Exercise.} (1) $\mathcal{E}_c^k(M)$=the space of differentiable $k$-forms on $M$ with compact support. Then $\hs{1em}\dis{\int_M:\mathcal{E}_c^k(M)\rightarrow\rb}\hs{1em} $ is linear.\\ (2) $\vphi:M\rightarrow N$, a diffeomorphism. Then $\dis{\int_M\vphi^*\ome=\pm\int_N\ome}\hs{1em}$($+$: if $\vphi$ is orientation-preserving, $-:$ if $\vphi$ is orientation-reversing.)\\ (3) $\vphi:\widetilde{M}\rightarrow M$, an n-fold covering with induced orientation. Then $\hs{3em}\dis{\int_{\widetilde{M}}\vphi^*\ome=n\,\int_M\ome}$. \end{document}