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"?

quotientof $\Lambda^* T^*M$. In your local formula, you essentially choose a splitting of this quotient map which is not invariant under a change of foliation chart. The quotient map amounts to using the usual formula for the coordinate change of a differential form and discarding all terms containing a $\mathrm dy_i$. A generalization: ncatlab.org/nlab/show/… $\endgroup$ – Bertram Arnold Jun 4 '19 at 11:39