Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology of such a sheaf?


I am pretty sure that such a question was studied (and maybe even has a complete answer), but I don't know a reference. 


A more specific question is: what happen when $F$ is 1-dimensional, given by integral trajectories of a non-vanishing vector field? Or even more specifically, suppose $H^1(M^n)=0$ and we consider a Killing vector field $v$ on $M^n$ (i.e. $v$ is preserving a metric). Is it true the the sheaf of functions $\cal F$ locally constant along trajectories of $v$ is acyclic? (we need $H^1(M^n)=0$, otherwise $S^1$ will be an obvious counterexample).


**An example of a foliation.** Consider the unit sphere $S^3$ in $\mathbb C^2$ and conisder the action of $\mathbb R$ via diagonal matrixes : $(z,w)\to (e^{ita}z, e^{itb}w)$ with $\frac{a}{b}$ irrational. 



Cohomology of a sheaf of functions locally constant along a foliation
Nikita Markarian just explained to me (if there is a mistake below, it is mine), that the last and more specific question about acyclicity has 100% negative answer. Namely, we can consider the case $M^3=S^3$ ($H^1(S^3)=0$) and the foliation is given by the fibers of the Hopf fibration $S^3\to S^2$. In this chase the sheaf of functions locally constant on the fibers has a two-term resolution (by soft sheaves). The first term is given by all functions on $S^3$ and the second by $1$-forms on $S^3$, restricted to fibers. The differential is just the differential along the fibers. In this case it is clear, that the first cohomology is huge, it is parameterised by all functions on the base $S^2$.


So this condition $H^1(M^n)=0$ does not help at all. 


It is a good exercise to apply the same reasoning to the other foliation on $S^3$, described in the question. 


