# Terminology for a type of smooth foliation

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 $\hat{\Sigma}$.

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.

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.

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Willie, I once needed something similar. In my paper, Local solvability of overdetermined systems defined by commuting first-order differential operators. Communications on Pure and Applied Mathematics, 39 (1986) 401–421, I mention that a "lens-shaped domain" can be used to solve a certain class of overdetermined elliptic systems, and show that more generally a "ravioli-shaped domain" can be used to solve a more general class of systems. I don't know of anyone else who has ever used such domains and foliations. – Deane Yang Sep 5 '11 at 18:33