# When does a non singular integrable differential one-form define a regular foliation?

Let $\mathcal{M}$ be a smooth manifold of dimension $m<+\infty$. Let $\theta$ be a nowhere vanishing (non-singular) differential one-form on $\mathcal{M}$ such that $\theta\wedge d\theta=0$. According to Frobenious' theorem, the kernel of $\theta$ is an involutive distribution and thus generates a codimension-one foliation $\mathcal{F}$ of $\mathcal{M}$.

The foliation $\mathcal{F}$ is defined to be regular if all of its leaves are regular submanifolds of $\mathcal{M}$. I would like to know if there are some conditions on $\theta$ assuring that the foliation $\mathcal{F}$ is regular without the need to explicitely find the leaves of $\mathcal{F}$.

Thamk You

• I would be surprised if there was something reasonable to test. As a test case, try irrational lines on the torus, given as the kernel of $d\theta_1 + rd\theta_2$. Those submanifolds are regular iff $r$ is rational. – user98602 Jan 26 '16 at 18:34