Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.


Consider the manifold $S^{n-2}\times S^1\times S^1$ with coordinates $(x\_1,...,x\_{n-1},\varphi,\theta)$ such that $\sum\_{i=1}^{n-1}x\_i^2 = 1$ , and two differential one-forms 


$$\eta\_1 = d\theta \quad\text{ and }\quad\eta\_2= ((1-\sin(\theta))^2 + x\_1^2) d\varphi + \sin(\theta) dx\_1 .$$ 


Frobenius' integrability condition is satisfied as can be easily checked, so there's an integral manifold of codimension two for these differential forms. 


There are some compact leaves homeomorphic to $S^{n-2}$ given by fixing $\theta = 0$ and $\varphi = \mathit{constant}$.


On the other hand, there are leaves of a second kind given by $\theta = \pi/2$ and $\varphi = \mathit{constant} + \frac{1}{x\_1}$ and these are said to be non-compact.


**Question 1:** Is there any simple argument for non-compactness of these leaves?


**Intuition:** Some sequence in one of these leaves for which the first coordinate converges to zero, cannot converge in the same leaf "because it is not given by the same equation ($\varphi = \frac{1}{x\_1}$)". I'm not sure if this is correct, and even if it was, I couldn't find some neat way to explain it.


**Question 2:** We have that $d\eta\_2 = 2x\_1 dx\_1\wedge d\varphi $ on the leaves of the second kind but is zero when $x\_1$ is. Does this give some topological difference for leaves? 



Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves
The precise description of the leaves which are not spheres is as follows.
The leaves passing through points with $x\_1\neq 0, \theta=\pi/2, \theta=constant+1/x\_1$ are homeomorphic to $\mathbb{R}^{n-2}$ (you can work out an explicit parametrization).
The leaves passing through points with $x\_1=0, \theta=\pi/2$ are homeomorphic to $S^{n-3}\times S^1$.
When $x\_1$ tends to $0$ the non-compact leaves spiral around $S^{n-3}\times S^1$ as in the classical Reeb foliation.


In fact, the example is originally due to Reeb himself [CRAS 226 (1948), 1337-1339], as you can see here (in French): <https://gallica.bnf.fr/ark:/12148/bpt6k31787/f1338.image> (but few details are given). For a more recent exposition (when $n=4$, but the general case is similar) I recommend to have a look at Section 1.6 "Reeb Transition" in the paper by Pablo Lessa [Asian J. Math. 19 (2015), 433-464].


