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"?

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    $\begingroup$ Given a foliation $\mathcal F$, differential forms along the leaves of $\mathcal F$ are sections of the graded vector bundle $\Lambda^*\mathcal F^*$ which is a quotient of $\Lambda^* T^*M$. 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/… $\endgroup$ Jun 4 '19 at 11:39
  • $\begingroup$ @BertramArnold Thank you for your comment. May you give a more precise definition of differential forms along leaves in terms of materials of the linked ncatlab. I think $\mathcal{F}$ is a subbundle of $TM$. so you are considering the dual map $*$ then we have a map $TM^* \to \mathcal{F}^*$. So how you relate to the linked you mentioned? BTW in the linked you mentioned, a few lines above the point you indicated, I think that there is a typos, it is written $d\rho$ preserve the Lie bracket but I think the true statement is that $\rho$ preserve the Lie bracket. $\endgroup$ Jun 4 '19 at 18:42

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.

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    $\begingroup$ I emphasize that my answer above refers to the set of leafwise differential forms as a subspace of the space of all differential forms $\Omega(M)$. If, however, you don't need to think of leafwise forms as living inside $\Omega(M)$, you can define the space of differential forms as mentioned before by @BertramArnold; the tangent bundle to the foliation is well-defined, and thus the dual of the tangent bundle is well defined, and so you can use that to generate leafwise differential forms (that are not in $\Omega(M)$). The construction above gives an isomorphism with a subspace of $\Omega(M)$. $\endgroup$ Jun 4 '19 at 20:21
  • $\begingroup$ Thank you for your answer. Just a few minutes ago I was thinking to the answer of @BertramArnold. My understanding is the following: There is a natural map $T^* M \to T^*\mathcal{F}$. This gives us a natural restriction map $\Lambda^*(M) \to \Lambda^* \mathcal{F}$. The the IMAGE(RANGE) of the later is the space of differential forms along leaves. obviously the diferertial operator $d$ is well defined.Am I correct? Now I try to understand the metric interpretation you described. Thanks for your attention to my question. $\endgroup$ Jun 4 '19 at 20:45
  • $\begingroup$ I correct: Then we have a natural map from $\Gamma \Lambda^*T^* M \to \Gamma \Lambda^* F^*$. The range(image) of the Later is the differential forms along leaves. Then d is well defined. Here is the point that we use integrability of the distribution. $\endgroup$ Jun 4 '19 at 20:56
  • $\begingroup$ Am I correct in understanding the space of differential forms? BTW, it is amazing that there are some foliation with dense orbit for which this cohomology is infinite $\endgroup$ Jun 4 '19 at 20:59
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    $\begingroup$ I think you have the right idea. You can decompose the differential into three pieces (dependent on the metric). The leafwise part automatically squares to zero, so you can get leafwise cohomology. I suggest looking at the papers by Alvarez-Lopez and coauthors: "Dimension of the Leafwise Reduced Cohomology", "Hodge decomposition along the leaves of a Riemannian foliation", "Long time behavior of leafwise heat flow for Riemannian foliations". You do not need special metrics to define the leafwise cohomology, but when you have a bundle-like metric, you get extra results like the Hodge theorem. $\endgroup$ Jun 5 '19 at 21:08

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