1 - Eugenio Calabi affirms that every closed non singular 1-forms in closed manifolds are intrinsecally harmonic ("An instrinsic characterization of harmonic 1-forms", 1969). This is easy proved observing first that such forms are transitive. The dual case is a open problem, but I believe that is true. A closed $(n-1)$-form non singular and non null in cohomology is intrinsecally harmonic iff the voluming-preserving flow induced by this form admit cross section (closed comdimension one submanifold cutting every orbit of the flow) or iff admit complementar foliation (foliation transversal to the flow; at leas $C^2$). Is possible proof this fact in the spirit of Calabi's work. This problem have consequences in flat characterization of circle bundles.

2 - In the case of closed $p$-forms of the rank $p$, the problem have a translate for foliation theory. Is possible to show that if a closed $p$ form $\omega$ of the rank $p$ is transitive, then exists complementar form $\eta$, that is, $\eta$ is a closed form such that $\omega\wedge\eta$ is volume form. This is proved using the theory of foliations cycles (see the Sullivan's paper "Cycles for the dynamical study of foliated manifolds and complex manifolds"). However, in a fiber bundle $\xi=(\pi,F,E,M)$ with symple connected base, compact total space, if $\Omega_M$ is any volume form in $M$, the form $\pi^*\Omega$ is intrinsecally harmonic iff $\xi$ is trivial. We can run away from trivial cases showing that $[F]\neq 0\in H_{\dim F}(E;\mathbb{R})$ and exists exemples those bundles with section. This give us the examples cited by Dan Fox. The problem is the dimension of the kernel of $\eta$. In the cases $p=1$ or $p=n-1$ (without singularities), in the condition of transitivity, the dimension of the kernel of $\eta$ is $n-1$ (case $p=1$) or 1 (case $p=n-1$), and the Calabi argument applies.

3 - Is too a open problem to show that harmonic forms are transitive, as observed by Katz ("Harmonic forms and near-minimal singular foliations"). We can to show that if $\omega$ is a harmonic $p$-form of rank $p$, then the have in $M$ two complementary $SL(*)$-foliations induced by $\ker\omega$ and $\ker *\omega$.

4 - I have studied the problem of decomposable forms. By the Tischler's argument (and others considerations; see "On fibering certain foliated manifolds over $\mathbb{S}^1$") is sufficient considering bundles $\xi=(F,\pi,E,\mathbb{T}^{p})$. If this bundle admit transversal foliation with holonomy group contained in $SL(*)$, then the form $\pi^*(\Omega_{\mathbb{T}^{p}})$ is intrinsecally harmonic. The ideia of study particular examples is know if we can rule out the hypothesis of transitivity. In any bundle $\xi=(\pi,F,E,M)$, with compact total space, $[F]\neq 0$ and $\pi_1(M)$ finite, the form $\pi^*(\Omega)$ is intrinsecally harmonic.

Ps. The above remarks is part of development of my doctoral project and are is under analysis.