Let $\omega$ be smooth closed 1-form on smooth manifold.such that $\omega_p \ne 0$ .prove that there exist some local coordinate $(x_1,...,x_n)$: such that $\omega|_U = dx^1$.
Which is somewhat dual to the problem of canonical form of vector field near regular point.
My attempt:
First by Poincare lemma for covector field there is some $f\in C^\infty(U)$ such that $\omega|_U = df$
If we choose $x^1 =f$ then we find the first coordinate component. I have no idea how to find the rest of coordinate component.
I can complete the local section of cotangent bundle $df:U \to T^*M$ to local frame $(df,\sigma_2,...,\sigma_n)$. Maybe this idea works,I don't know how to make this local section back to coordinate function $(\sigma_2,...,\sigma_n)\mapsto (x_2,...,x_n)$
There are two question here :
- How to find other coordinate chart component $(x_2,...,x_n)$
- where do we use the assumption that $\omega_p \ne 0$?(Oh I find the point ,to make $df$ one of local section in the local frame we need $df$ no where vanishing,by continuity of $\omega$ ,regular point is open subset ,there exist a neiborhood near $p$ such that $\omega = df$ is no where vanishing on $U$)
First $\omega|_U = df$ on some neiborhood of $U$.where $\omega_p \ne 0$ means $df_p \ne 0$ hence $f$ is regular at $p$.
Choose the coordinate chart $(x_1,...,x_n)$,idea is replace one $x^i \mapsto f$.We select one of $i$ such that $\frac{\partial f}{\partial x^i}(p) \ne 0$
Then construct the map $$\Phi:\Bbb{R}^n \to \Bbb{R}^n\\(x_1,..x_i..,x_n) \mapsto (x_1,..,f,...,x_n)$$
If we can prove it's local diffeomorphism then we are done.
Compute the Jacobian matrix is diagonal has determinant $\frac{\partial f}{\partial x^i} (p) \ne 0$
So it's local diffeomorphism. If we shrink the chart a liitle bit $\Phi$ will be diffeomorphism hence,$(V,\Phi\circ \varphi)$ then is again a chart with i-th coordinate function $f$.