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Sufficient condition(s) for a non-local diffeomorphism?

  1. Mar 4, 2012 #1
    I hope I'm posting this question in the right forum; anyway, here goes:

    What would be a sufficient condition for a diffeomorphism to be non-local--specifically, for it to be valid over a given domain?

    In the particular case I'm examining, the mapping I'd like to be a non-local diffeo is given by a solution to a PDE with a perturbation term. In the absence of the perturbation, I can simply take the mapping to be the identity map (which can always be shown to be a global diffeo, correct?).

    Hence, my objective is to establish that for a "small enough" perturbation term, the solution to the PDE is itself a perturbation of the identity mapping, and hence valid over the domain of interest.

    PS: For those who wish to see the actual PDE and obtain additional information, here it is:

    Given a compact set [itex]D\in\textbf{R}^{m+n}[/itex] containing the origin, I would like to find a mapping
    [itex]T:D\to \textbf{R}^m; \quad T:(x,s,\phi)\mapsto q[/itex]
    that is given by the solution to the PDE

    [itex]\frac{\partial T}{\partial s}+\left( \frac{\partial T}{\partial \phi}-\frac{\partial T}{\partial s}\,\frac{\partial N}{\partial \phi}\right)M\delta_a=0,[/itex]

    where [itex]x\in\textbf{R}^{n-1},\, s\in\textbf{R},\, \phi\in\textbf{R}^m,\, M=M(x,\phi)\in\textbf{R}^m,\, N=N(x,\phi)\in\textbf{R}[/itex] and [itex]\delta_a=\delta_a(x,s)\in\textbf{R}[/itex].

    [itex]\delta_a(x,s)[/itex] is the perturbation I'm talking about, and we see that when [itex]\delta_a=0[/itex], we can simply take [itex]q=T(\phi)=\phi[/itex]. Hence, for small enough [itex]\delta_a[/itex], we expect [itex]T[/itex] to just be a perturbation of the identity map, and I need for it to be valid for all [itex](x,s,\phi)\in D[/itex].

    If anyone could recommend some books or other references that talk about when a diffeomorphism is non-local (or global, although I don't need mine to be valid globally, but if it turns out to be the case for any [itex]T(x,s,\phi)[/itex] satisfying the PDE above, then great), that would be awesome.

    Thanks for taking the time to read this. :)
    Last edited: Mar 4, 2012
  2. jcsd
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