I have a question in differential manifolds, If i have $\alpha$ a smooth 1-form on a differential manifold $M$ and we have that it is closed, and it doesn't vanish at any point in $M$. We have a distribution $D=\{ X , \alpha(X)=0\}$
I need to prove that $D$ is a regular distribution and to prove it is integrable using frobenius theorem. What would be such $\alpha$? I mean, example satisfying these conditions.
And does it really matter that it is closed to prove $D$ is integrable?
$\alpha$ is regular since it is not degenerated, its rank is $n-1$ if $dim M=n$.
Let $X$ and $Y$ be vector fields tangent to $D_x=\{u\in T_xM,\alpha_x(u)=0\}$, $d\alpha(X,Y)=X.\alpha(Y)-Y.\alpha(Y)-\alpha([X,Y])=0$, since $X$ and $Y$ are tangent to $D$, $\alpha(X)=\alpha(Y)=0$, this implies that $X.\alpha(Y)=Y.\alpha(X)=0$, we deduce that $\alpha([X,Y])=0$ and Frobenius implies that $D$ is integrable.