Q:

Leafwise de Rham cohomology (A true definition of differential forms along leaves)

For a foliated space $(M, \mathcal{F})$, one associate a leafwise de Rham cohomology. This cohomology and trace-class operators on this cohomology and trace interpretations for closed orbits of certain flow on $M$ is the main object of this paper "Number theory  and  dynamical  system of  foliated manifolds.
But in the later paper, I did not find a very precise definition of "Differential forms along a leaf".
So I try to find other papers or talks to find a precise definition for this concept. Then I found a definition at page 8 of this talk "Lefschetz trace 
 formula for flow  on foliated manifolds"  which gives a local representation for such forms. But my problem is the following:
I think that such representation of a differential form along leaves of a  $k$-dimensional foliation of a $n$-manifold, which is quoted below, is NOT   invariant under foliation  charts $(x,y)\mapsto (f(x,y),g(y)),\quad x\in \mathbb{R}^k, y\in \mathbb{R}^{n-k}$:
$$\omega=\sum_{\alpha_1<\alpha_2<\ldots<\alpha_k} a_{\alpha}(x,y) dx_{\alpha_1}\wedge dx_{\alpha_2}\wedge \ldots\wedge dx_{\alpha_k}.$$
Am I mistaken?
What is a precise definition and precise local representations of  "Differential forms along leaves"?

A:

The thing you are confused about is that leafwise differential forms are not invariant under foliation charts. Basic forms are invariant under change of foliation charts and foliated maps (in your notation, forms in the y-variables). One way of defining leafwise differential forms is by first choosing a metric and then choosing an adapted orthonormal frame $(f_1,\ldots, f_p, e_1,\ldots,e_q)$ of the tangent bundle near a point (where the $f_j$'s are a basis pointwise for the tangent space to the foliation), then form the dual basis $(f^1,\ldots, f^p, e^1,\ldots,e^q)$. The sections of the bundle of differential forms generated by the $f^j$'s are the leafwise differential forms. Again, the choice of metric determines this bundle, but you can check that what I wrote is independent of the choice of adapted orthonormal frame.

