\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{\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{\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*{Foliation.}
  \begin{defn}{\it Let $M$ be a n-dimensional $\ccn$ manifold. A k-dimensional submanifold
  $(N,incl)$ of $M$ is called a ${\bf foliation}$ if}

{\it  (1) $M=N$ as a set.}

{\it  (2) $\forall p\in N, \,\,\exists(U,x)$, a coordinate chart
of
  $M$ such that $x(U)=(-\eps,\eps)^n$ and the components of $N\cap
  U$ are the slices}

  $S_{a}=\{q\in
  U|(x^{k+1}(q),\cdot\cdot\cdot,x^n(q))=(a_{k+1},\cdot\cdot\cdot,a_n)=a\},
|a_j|<\eps,\,\,j=k+1,\cdot\cdot,n.$\\


  \end{defn}

{\bf Remark.} $N\cap U$ is open in $N$ : $N\cap U=i\inv(U)$ for
$i:N\hookrightarrow M$ ,  continuous.

And each component of $N\cap U$ is open in $N$. Therefore $S_{a}$
is a coordinate open neighborhood for $N$ automatically and hence
the topology of $N$ is the coherent topology coming from
$\{S_{a}\}$.\\

A connected component of a foliation $N$ is called a leaf and $N$
is partitioned by leaves.\\

\begin{cor}
Let $\dc$ be a k-dimensional $\ccn$ involutive distribution on
$M$. An integral manifold of $\dc$ gives a foliation on $M$ and
each leaf is a maximal connected integral manifold of $\dc$.
Conversely, a foliation $N$ on $M$ gives rise to an involutive
distribution given by $\dc(p)=T_pN\subset T_pM$ whose integral
manifold is the given foliation $N$.
\end{cor}

{\bf ���� 13.} $N$ is a foliation of $M\Rightarrow \exists$ an
atlas of $M$ such that each coordinate transition is given in the
form

$\hs{6em}h:\rb^k\times \rb^{n-k}\rightarrow \hs{0.8em}\rb^k\times
\rb^{n-k}$

$\hs{9em}(x,y)\hs{0.8em}\mapsto (f(x,y),g(y))$

Conversely such atlas gives a foliation structure on $M$.

  \end{document}
