$\DeclareMathOperator{\dx}{dx}\DeclareMathOperator{\dy}{dy}\DeclareMathOperator{\dnu}{d\nu}$ Consider in $\mathbb R^2$ the frame field $\{X=\partial_x+y\partial_y,Y=\partial_y\}.$ I computed its corresponding coframe field as $\{\omega=\dx, \theta=-y\dx+\dy\}$.
I want to show that the frame field is not holonomic.
I'm following two books.
- One defines a frame field holonomic if it comes from a local coordinate system - $\{\partial_u,\partial_v\}$.
- The other defines a coframe field $\dnu^\alpha$ as holonomic if $\dnu^\alpha=0$, and it then declares (locally) its dual frame field as holonomic.
The second definition is very practical, because for my example, differentiating the coframe shows $d\theta=\dx\wedge\dy\neq 0,$ so neither it nor the frame field are holonomic.
Is it possible (how?) to use the first definition for this example? That is, can I show that $\{X,Y\}$ cannot be expressed as $\{\partial_u,\partial_v\}$ for some local coordinates $u,v:\mathbb R^2\to \mathbb R$ directly, without going to the differentials? $ А jesuitical answer is yes:
- If $\{X,Y\}=\{\partial_u,\partial_v\}$, then the corresponding coframe field equals $\{du,dv\}$
- Since $d\theta\neq 0$, it's impossible that $\theta=dv$ is an exact form.
- Therefore, $\{X,Y\}$ is not expressible as $\{\partial_u,\partial_v\}$.
But this defeats the purpose of my question, because it simply translates the question in terms of differential forms, and doesn't show me what makes it impossible to find coordinates matching $\{X,Y\}$.
$X$ and $Y$ are coordinate vector fields if and only if their Lie bracket $[X,Y]=0$. In your case, $[X,Y]=-\partial_y$.