Being knowingly sloppy because I know I don't know how to be precise (hence the question, and expecting brickbats despite the apology)...
Suppose I have a very ordinary differentiable manifold $M$ diffeomorphic to $\mathbb{R}^n$ equipped with a Cartesian coordinate frame such that $M$ can be foliated by slices $\Sigma$ of $\mathbb{R}^{n-1}$, indexed by, say, the x-axis (though x is continuous I believe indexed is the right term), such that there is a diffeomorphism $\phi_{x_i, x_i}:\Sigma(x_i)\rightarrow\Sigma(x_j)$ for any $i,j \in$ (some interval) $I$.
1/ What is the terminology and notation for expressing the idea that there is such a set of diffeomorphisms (did I just do it?)
2/ If I excise some region $\mathscr{V_{x_0}}$ from $\Sigma(x_0)$ and require the metric on the $\Sigma(x_i)$ to be fixed (otherwise I would be able to define a new diffeomorphism that generated new metrics via the pushforward/pullback), how do I describe the image of $\mathscr{V_{x_i}}$ throughout $M$...
a) Under $\phi$, and
b) If $\mathscr{V_{x_i}}$ follows some curve $\gamma$ in $M$ that intersects $\Sigma(x_i)$ at $p$ and whose points are not all images of $p$ under $\phi$ (i.e. $\phi$ ceases to be a diffeomorphism as initially defined)
I guess one part of the answer should be about defining diffeomophisms as a continuous function of some variable(s)...
