Hello,

I have two questions about involutive subbundles and foliations.

If we have a foliation on a manifold M of codimension q, then this submits an involutive subbundle E of T(M).

We can take $\displaystyle E_{|U_i}=ker(d(pr_2\circ\phi_i))$, whereas $\displaystyle (U_i,\phi_i )$ is a foliation chart.

I see that $\displaystyle (\frac{d}{dx^i})_{|p} \;, i=1,...,n-q$ is a basis for $\displaystyle E_p$. But now i don't know why it is involutive.

My guess is, that if X,Y are smooth sections for E, then also [X,Y] is a smooth section since: $\displaystyle [X,Y](p)=X_p Y - Y_p X. \in E_p \subset T_p M$, since $\displaystyle E_p$ is the kernel of the linear map above and therefore a linear subspace of $\displaystyle T_p M$.

Is this correct? I ask because in my book it is shown very complicated, s.t. i don't understand it really.

My second question is, why the induced subbundle E is unique? what happens if i choose a different chart?

I think that the elm. of the basis changes of course. But why is the subspace the same?

Regards