In writing a paper I need to refer to the following type of objects, does anybody know if it has an established name?


**What we are given**


We have $M$ a smooth manifold and $\Sigma \subset M$ a codimension 1, smooth compact submanifold with boundary $\partial\Sigma$ and interior $\hat{\Sigma}$. 


**What I am interested in**


An open subset $D\subset M$ and a smooth map $\Phi: (-1,1)\times \Sigma \to M$ such that


* $\Phi$ restricted to $(-1,1)\times \hat{\Sigma}$ is a diffeomorphism onto $D$
* $\Phi(0,\cdot)$ is the identity map on $\Sigma$.
* $\Phi(t,\cdot)|\_{\partial\Sigma}$ is the identity map (independent of $t$).


The second condition roughly states that we are looking at some sort of smooth homotopies on maps from the manifold with boundary $\Sigma$ to the manifold $M$. The third condition restricts to considering homotopies between maps that fixes the boundary $\partial\Sigma$ (Is there a commonly used terminology for this condition alone?) 


What's most important, however, is the first condition. This allows us to construct a smooth foliation of $D$ by submanifolds diffeomorphic to $\hat{\Sigma}$. 




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If I add some further analytic conditions on it, and abuse terminology a little bit, I can call it a *development* a la PDE theory. But it turns out that for the paper I am writing I need to talk about some properties of pairs $(D,\Phi)$ relative to a fixed $\Sigma$ with the above definition independently of the PDE/analysis assumptions. I would prefer not to invent a new name if a construction like above has been used in the literature before.




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A simple example of the object: let $M = \mathbf{R}^2$ with the usual coordinates $(x,y)$, Let $\Sigma = [-1,1]\times\{0\}$ with coordinate $x$. 


Then $\Phi(s,x) = (x,s(x-1)(x+1))$ defined on $(-1,1)\times [-1,1]$ is smooth. It sends $(s,\pm 1) \mapsto (\pm 1, 0)$. It sends $(0,x) \to (x,0)$. And the Jacobian determinant $|d\Phi| = (x-1)(x+1) > 0$ if $|x| < 1$. So it is a diffeomorphism from $(-1,1)\times(-1,1)$ to its image. 



Terminology for a type of smooth foliation
(Just to get this off the unanswered list)


Deane's suggestion of "lens-shaped domain" and "ravioli-shaped domain" are actually quite descriptive. And they are sufficiently close in their original definitions to what I have in mind that I don't feel uncomfortable abusing them. :-)


