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 , whereas is a foliation chart.
I see that is a basis for . 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: , since is the kernel of the linear map above and therefore a linear subspace of .
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?