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?