1. ## involutive subbundle

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

I see that $(\frac{d}{dx^i})_{|p} \;, i=1,...,n-q$ is a basis for $E_p$. 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: $[X,Y](p)=X_p Y - Y_p X. \in E_p \subset T_p M$, since $E_p$ is the kernel of the linear map above and therefore a linear subspace of $T_p M$.

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

2. As a geometry object, the definition of a foliation doesn't depend on the choice of the charts. The foliation defines a collection of n-q dimensional sub-manifolds. In each chart (x,y), a plaque is defined as y=c where c is constant, though in another chart (x1,y1) the same plaque is represented by the same type of equation y1=c1, with a (possible) different constant c1. So the plaque( a part of a submanifold) is defined independent of charts. Then the tangent space of this plaque is well-defined. Piece together all those tangent spaces we get the sub-bundle.
To show it's involutive you need the following equation:
$\langle X\wedge Y, d\omega\rangle=X\langle Y, \omega \rangle - Y \langle X, \omega \rangle - \langle [X,Y], \omega\rangle$
Let $\omega=dy^j$ and notice that $\langle Y, dy^j\rangle=\langle X, dy^j \rangle = 0$, we get $\langle [X,Y], dy^j\rangle=0$, which shows [X,Y] belongs to the sub-space.

3. Hello,

What do you mean by "<.,.>"?

Regards

4. Tensor contraction - Wikipedia, the free encyclopedia. < X, df > = df(X) = X(f) = the directional derivative of f along X.

5. Hello. Thank you for your help. now i understand you better. One more question:
X,Y are vector fields, not alternating tensors. Why $X \wedge Y$ makes sense? what is its definition?

Regards