Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is denotedb by $\pi_D$.

Is there a a constant $\delta >0$ such that every distribution with the following property $P_\delta$ is necessarily an integrable distribution?

Property $P_\delta$: For every two smooth local vector fields $X,Y$ tangent to $D$ we have $$\left|[X,Y]- \pi_D[X,Y] |\leq \delta |X|Y\right|$$

What about non compact case?

Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is denotedb by $\pi_D$.

Is there a a constant $\delta >0$ such that every distribution with the following property $P_\delta$ is necessarily an integrable distribution?

Property $P_\delta$: For every two smooth local vector fields $X,Y$ tangent to $D$ we have $$\left|[X,Y]- \pi_D[X,Y] |\leq \delta |X||Y|\right|$$

What about non compact case?