i have a question about involutive subbundles.
If M is a manifold and E is a subbundle. We say E is involutive, if for each smooth vector fields X,Y of E, (that is X(p),Y(p) ) also the lie bracket [X,Y] is a smooth vector field to E.
I think that not any subbundle is involutive, but i don't know why?
Whats wrong with this proof?
If X,Y are smooth vector fields to E, =>[X,Y] is also a vect. field:
[X,Y](p)= . since X,Y are vector fields and E_p is a linear subspace, therefore the sum above is also in E_p??
Where is the mistake? I don't see it.