Hello,

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) $\displaystyle \in E_p$) 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?

Claim:

If X,Y are smooth vector fields to E, =>[X,Y] is also a vect. field:

Pf:

[X,Y](p)=$\displaystyle X_p (Y) - Y_p (X)$. 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.

Regards