distributionDistribution52004-11-30 12:10:142005-03-07 04:54:20Definitioninvolutiveinvolutive distributionlocal basis% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{theorem}
\newtheorem*{thm}{Theorem}
\newtheorem*{lemma}{Lemma}
\newtheorem*{conj}{Conjecture}
\newtheorem*{cor}{Corollary}
\newtheorem*{example}{Example}
\theoremstyle{definition}
\newtheorem*{defn}{Definition}In the following we will \PMlinkescapetext{mean} $C^\infty$ when we say smooth.
\begin{defn}
Let $M$ be a smooth manifold of dimension $m$. Let $n \leq m$ and for each $x \in M$, we assign an $n$-dimensional subspace
$\Delta_x \subset T_x(M)$ of the tangent space in such a way that for a
neighbourhood $N_x \subset M$ of $x$ there exist $n$ linearly independent
smooth vector fields $X_1,\ldots,X_n$ such that for any point $y \in N_x$,
$X_1(y),\ldots,X_n(y)$ span $\Delta_y$. We let $\Delta$ refer to the
collection of all the $\Delta_x$ for all $x \in M$ and we then call $\Delta$ a
{\em distribution} of dimension $n$ on $M$, or sometimes a
{\em $C^\infty$ $n$-plane distribution} on $M$. The set of smooth
vector fields $\{ X_1,\ldots,X_n \}$ is called a {\em local basis} of $\Delta$.
\end{defn}
Note: The naming is unfortunate here as these distributions have nothing
to do with \PMlinkname{distributions in the sense of analysis}{Distribution}.
However the naming is in wide use.
\begin{defn}
We say that a distribution $\Delta$ on $M$
is {\em involutive} if for every point $x \in M$ there exists a local basis
$\{ X_1,\ldots,X_n \}$
in a neighbourhood of $x$ such that for all $1 \leq i, j \leq n$, $[X_i,X_j]$
(the commutator of two vector fields) is in the span of
$\{ X_1,\ldots,X_n \}$. That is, if
$[X_i,X_j]$ is a linear combination of $\{ X_1,\ldots,X_n \}$.
Normally this is written as $[ \Delta , \Delta ] \subset \Delta$.
\end{defn}
\begin{thebibliography}{9}
\bibitem{boothby}
William M.\@ Boothby.
{\em \PMlinkescapetext{An Introduction to Differentiable Manifolds and
Riemannian Geometry}},
Academic Press, San Diego, California, 2003.
\end{thebibliography}