I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix. 


I've seen a similar question [here](https://mathoverflow.net/questions/228964/functions-with-orthogonal-jacobian), the difference being that in my case the Jacobian can be scaled in a variable manner. 


More precisely, there exist functions $c : \mathbb{R}^d \rightarrow \mathbb{R}$, $U: \mathbb{R}^d \rightarrow \mathcal{O}\_n$, such that the Jacobian $J$ of $f$ is of the form: $J(x) = c(x)U(x)$.


My question is how can I characterize the function $f$? Is it necessarily a linear mapping? I can assume that $f$ is twice-differentiable or even smooth.



Functions with scalar times orthogonal Jacobian
Such maps are called conformal or anticonformal, depending of whether the determinants of those orthogonal matrices are positive or negative. In dimension 2 they are the same as complex analytic functions of one complex variable, or such functions composed with the complex conjugation. 


In dimensions $\geq 3$,
according to a theorem of Liouville, all conformal maps, even those defined on open subsets of $R^n$ are restrictions of Mobius transformations. Mobius transformations constitute a very small class of maps: by definition they are
compositions of finitely many inversions (reflections in spheres).


Remark. Liouville assumed that the map is sufficiently smooth. His smoothness conditions have been substantially relaxed since then.


