contact manifoldContactManifold2003-06-27 08:19:022003-06-27 08:19:02Definitioncontact structurecontact formcontactomorphism% 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 hereLet $M$ be a smooth manifold and $\alpha$ a one form on $M$. Then $\alpha$ is a
\emph{contact form} on $M$ if
\begin{enumerate}
\item for each point $m \in M$, $\alpha_{m} \neq 0$ and
\item the restriction $d\alpha_{m}|_{\ker{\alpha_{m}}}$ of the differential of $\alpha$
is nondegenerate.
\end{enumerate}
Condition 1 ensures that $\xi=\ker{\alpha}$ is a subbundle of the vector bundle $TM$. Condition 2 equivalently says $d\alpha$ is a symplectic structure on the vector bundle $\xi \to M$. A \emph{contact structure} $\xi$ on a manifold $M$ is a subbundle of $TM$ so that for each $m \in M$, there is a contact form $\alpha$ defined on some neighborhood of $m$ so that $\xi = \ker{\alpha}$. A co-oriented contact structure is a subbundle of $TM$ of the form $\xi=\ker{\alpha}$ for some globally defined contact form $\alpha$.
A (co-oriented) \emph{contact manifold} is a pair $(M,\xi)$ where $M$ is a manifold and $\xi$ is a (co-oriented) contact structure. Note, symplectic linear algebra implies that $\dim{M}$ is odd. If $\dim{M} = 2n+1$ for some positive integer $n$, then a one form $\alpha$ is a contact form if and only if $\alpha \wedge (d\alpha)^{n}$ is everywhere nonzero.
Suppose now that $(M_1,\xi_1=\ker{\alpha_1})$ and $(M_2,\xi_2=\ker{\alpha_2})$ are co-oriented contact manifolds. A diffeomorphism $\phi:M_1 \to M_2$ is called a \emph{contactomorphism} if the pullback along $\phi$ of $\alpha_2$ differs from $\alpha_1$ by some positive smooth function $f:M_1 \to \mathbb{R}$, that is, $\phi^{*}\alpha_{2} = f\alpha_{1}$.
\textbf{Examples}:
\begin{enumerate}
\item $\mathbb{R}^3$ is a contact manifold with the contact structure induced by the one form $\alpha = dz + xdy$.
\item Denote by $\mathbb{T}^{2}$ the two-torus $\mathbb{T}^{2} = S^1 \times S^1$. Then, $\mathbb{R} \times \mathbb{T}^2$ (with coordinates $t,\theta_1,\theta_2$) is a contact manifold with the contact structure induced by
$\alpha = \cos{t\theta_1}+\sin{t\theta_2}$.
\end{enumerate}