Let $(M,\mathcal F)$ be a smooth foliated manifold. An automorphism of $(M,\mathcal F)$ is a diffeomorphism of $M$ that takes leaves of $\mathcal F$ onto leaves. Let now $L$ be a leaf of $\mathcal F$. It may happen that $L$ has an open neighborhood $U$ which is a sum of leaves and such that for every leaf $L'\subset U$ there exists an automorphism $\phi$ taking $L$ onto $L'$.


My questions are:



> 
> Does every $(M,\mathcal F)$ have a leaf $L$ with the described property?
> 
> 
> If the answer is no, how much can we hope for instead? And is there a simple counterexample? Are there some natural classes of foliations which still have this property?
> 
> 
> 


Context:  

Ideally, I would like to consider a smooth foliated manifold $(M,\mathcal F)$ such that in some flat chart $\psi\colon U\to \mathbb R^n$ there exists an open set $V\subset U$ and a family $\tau\_x$ of automorphisms of $(M,\mathcal F)$ indexed by a neighborhood of 0 in $\mathbb R^n$, where $\tau\_x$ acts on $V$ like a translation by $x$ in the chart $\psi$. This condition is satisfied for instance when there is a neighborhood $U$ of a leaf $L$ and a foliation-preserving diffeomorphism $U\to L\times W$, where $L\times W$ has the corresponding product foliation.



A local transitivity property of the automorphism group of a foliated manifold
No. There exist foliations $F$ of the plane that are rigid in the sense that no automorphism of $F$ can permute the leaves. See the following paper:


MR2407104 (2009e:54055)
Gartside, Paul(1-PITT); Gauld, David(NZ-AUCK); Greenwood, Sina(NZ-AUCK)
Homogeneous and inhomogeneous manifolds. (English summary)
Proc. Amer. Math. Soc. 136 (2008), no. 9, 3363--3373. 


