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\begin{document}
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  \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 ¼÷Á¦ 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}