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

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

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

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

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

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 $T(x,s,\phi)$ satisfying the PDE above, then great), that would be awesome.

Thanks for taking the time to read this. :)

 

[jvicens]

Thank you for your question. I would like to provide some insight into your inquiry. A diffeomorphism is a type of mapping between differentiable manifolds that preserves the smoothness of the functions being mapped. In order for a diffeomorphism to be non-local, it would have to violate one of the properties of a diffeomorphism. This can happen if the mapping is not one-to-one, or if it is not continuously differentiable. Therefore, a sufficient condition for a diffeomorphism to be non-local would be if it does not preserve the smoothness of the functions being mapped.

In the case of the PDE you have provided, the mapping T is given by the solution to the PDE. If the perturbation term \delta_a is small enough, then T will be a perturbation of the identity map. This means that T will still preserve the smoothness of the functions being mapped and will be a valid diffeomorphism over the given domain. However, if \delta_a becomes too large, then T may no longer preserve the smoothness of the functions and will no longer be a valid diffeomorphism.

In terms of references, there are many books and resources that discuss diffeomorphisms and their properties. Some recommended books include "Introduction to Smooth Manifolds" by John M. Lee and "Differential Geometry: Connections, Curvature, and Characteristic Classes" by Loring W. Tu. These books discuss the fundamentals of diffeomorphisms and their applications in mathematics and physics.

I hope this explanation helps. Please let me know if you have any further questions or if you would like me to provide more information. Best of luck with your research!