Poincaré $1$-form

Definition 1.

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

$$\alpha =\sum _{i=1}^n{y}_{i}d{x}^i$$

where $x^i\mathrm{,}{y}_i$ are canonical local coordinates for $T^{\mathrm{\ast }}\mathit{}M$.

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 haveoverlapping 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 \sum _{i=1}^n}{\tilde{y}}_i{\displaystyle \frac{\partial {\tilde{x}}^i}{\partial x^k}}d{x}^k$

$=$${\displaystyle \sum _{i=1}^n}{\displaystyle \frac{\partial {\tilde{x}}^j}{\partial x^i}}{y}_j{\displaystyle \frac{\partial {\tilde{x}}^i}{\partial x^k}}d{x}^k$

$=$${\displaystyle \sum _{k=1}^n}{y}_{k}d{x}^k.$

Properties

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. Suppose $\pi :T^{\ast }M\to M$ is the canonical projection.Then

   $$\alpha (w)=\xi ((D\pi )(w)),w\in T_{\xi }(T^{\ast }M),$$

   which is an alternative definition of $\alpha$ without local coordinates.

3. The restriction of this form to the unit cotangent bundle, is acontact form.