What can be said about a function $f : \mathbb{R}^m \to \mathbb{R}^n$ for which all singular values of the Jacobian matrix, $\mathbf{J}$, are all 1?
I have found this similar question which covers the case for functions $g : \mathbb{R}^n \to \mathbb{R}^n$, but conformal mappings and, more specifically Möbius transformations, do not seem to apply to the more general case where $m \neq n$.
One possible way of generalising the results anyway, would be to extend the domain or image of $f$ to get a function $f' : \mathbb{R}^k \to \mathbb{R}^k$ with $k = \max(m, n)$ where vectors from the domain or image of $f$ are filled up with zeros (or in some other way) in the additional dimensions. I have no idea, however, in how far this makes sense or is a valid thing to do.
Any help would be greatly appreciated.