# Poincaré $1$-form

###### Definition 1.

Suppose $M$ is a manifold, and $T^{\ast}M$ is its cotangent bundle. Then the , $\alpha\in\Omega^{1}(T^{\ast}M)$, is locally defined as

 $\alpha=\sum_{i=1}^{n}y_{i}dx^{i}$

where $x^{i},y_{i}$ are canonical local coordinates for $T^{\ast}M$.

Let us show that the Poincaré $1$-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

 $\displaystyle\sum_{i=1}^{n}\tilde{y}_{i}d\tilde{x}^{i}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\tilde{y}_{i}\frac{\partial\tilde{x}^{i}}{\partial x% ^{k}}dx^{k}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\frac{\partial\tilde{x}^{j}}{\partial x^{i}}y_{j}% \frac{\partial\tilde{x}^{i}}{\partial x^{k}}dx^{k}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{n}y_{k}dx^{k}.$

## Properties

1. 1.

The Poincaré $1$-form play a crucial role in symplectic geometry. The form $d\alpha$ is the canonical symplectic form for $T^{\ast}M$.

2. 2.

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.

3. 3.

The restriction of this form to the unit cotangent bundle, is a contact form.

Title Poincaré $1$-form Poincare1form 2013-03-22 14:45:44 2013-03-22 14:45:44 matte (1858) matte (1858) 7 matte (1858) Definition msc 58A32 Liouville one-form