Poincare1Form 2004-10-23 12:28:46 2007-02-18 16:33:43 Definition cotangent bundle % 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} \usepackage{amsthm} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions % % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} \begin{defn} Suppose $M$ is a manifold, and $T^\ast M$ is its cotangent bundle. Then the \PMlinkescapetext{\emph{Poincar\'e $1$-form}}, $\alpha \in \Omega^1(T^\ast M)$, is locally defined as $$ \alpha = \sum_{i=1}^n y_i dx^i $$ where $x^i, y_i$ are canonical local coordinates for $T^\ast M$. \end{defn} Let us show that the Poincar\'e $1$-form is globally defined. That is, $\alpha$ has the same expression in all local coordinates. Suppose $x^i, \tilde{x}^i$ are overlapping coordinates for $M$. Then we have overlapping local coordinates $(x^i, y_i)$, $(\tilde{x}^i, \tilde{y}_i)$ for $T^\ast M$ with the transformation rule $$ \tilde{y}_i = \frac{\partial \tilde{x}^j}{\partial x^i} y_j. $$ Hence \begin{eqnarray*} \sum_{i=1}^n \tilde{y}_i d\tilde{x}^i &=& \sum_{i=1}^n \tilde{y}_i \frac{\partial \tilde{x}^i}{\partial x^k} dx^k \\ &=& \sum_{i=1}^n \frac{\partial \tilde{x}^j}{\partial x^i} y_j \frac{\partial \tilde{x}^i}{\partial x^k} dx^k \\ &=& \sum_{k=1}^n y_k dx^k. \end{eqnarray*} \subsubsection*{Properties} \begin{enumerate} \item The Poincar\'e $1$-form play a crucial role in symplectic geometry. The form $d\alpha$ is the canonical symplectic form for $T^\ast M$. \item Suppose $\pi\colon T^\ast M\to M$ is the canonical projection. Then $$ \alpha(w) = \xi( (D\pi)(w) ),\quad w\in T_\xi(T^\ast M), $$ which is an alternative definition of $\alpha$ without local coordinates. \item The restriction of this form to the unit cotangent bundle, is a contact form. \end{enumerate}