Let $M$ be a smooth manifold and let $\Delta \_i$ for $i=1,...,k$ be distributions of $TM$ which are integrable such that $\Delta\_i \cap \Delta \_j$ is zero distribution for $i \neq j$. Suppose $\Delta = \Delta\_1 \oplus ...\oplus \Delta \_k$. Is $\Delta$ integrable?



Integrability of direct sum of some integrable distributions
No. On the 3-sphere, take the usual metric and the 2-plane field $\Delta$ perpendicular to the fibers of the Hopf fibration. It splits as a sum of two line fields $\Delta\_1$ and $\Delta\_2$. I will write out the details when I have a moment.


Edit: some details. You think of $S^3$ as the unit quaternions, and so as a Lie group, with Maurer-Cartan form $\omega=\omega\_1 i + \omega\_2 j + \omega\_3 k$ valued in the imaginary quaternions. (Or use $SO(3)$ if you prefer; it has the same Lie algebra). Then $\omega\_1=0$ is a contact plane field $\Delta$, by the structure equations $d \omega\_1 = -2 \omega\_2 \wedge \omega\_3$. But this is the sum of the plane fields $\Delta\_1=(\omega\_1=\omega\_2=0)$ and $\Delta\_2=(\omega\_1=\omega\_3=0)$.


