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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, 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.

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    $\begingroup$ Your question is answered in the accepted answer to the linked question. $\endgroup$
    – Steven Gubkin
    Commented May 5, 2016 at 20:45
  • $\begingroup$ I might have misunderstood that answer: I thought it only refers to functions whose Jacobian is an orthogonal matrix. My question regards Jacobians which are scaled orthogonal matrices, where the scaling may depend on the variable $x \in \mathbb{R}^d$. $\endgroup$
    – ualex
    Commented May 5, 2016 at 21:30
  • $\begingroup$ It is true that the linked question is not exactly the same as the one you ask here. However, Alexandre Eremenko, in a prescient move across the time continuum, does answer the question you asked ("conformal" = "angle preserving" should be formalized by your condition). $\endgroup$
    – Christian Remling
    Commented May 5, 2016 at 23:26

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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.

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  • $\begingroup$ Thank you very much @AlexandreEremenko! Is there any recommended resource about real Mobius transformations? Most of what I see is about complex spaces. $\endgroup$
    – ualex
    Commented May 6, 2016 at 20:16
  • $\begingroup$ @ualex: Ahlfors, MR0725161 Mobius transformations in several dimensions, U. Minnesota, 1981. $\endgroup$
    – Alexandre Eremenko
    Commented May 6, 2016 at 22:53
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Such a function would be conformal, and these functions are Mobius transformations.

In dimension two, you obtain functions which are either holomorphic or antiholomorphic.

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  • $\begingroup$ Hmm. I see that what I write is already contained in the answer to the linked question. I think OP should probably be closed as a duplicate, since this question is actually answered there. $\endgroup$
    – Steven Gubkin
    Commented May 5, 2016 at 20:45
  • $\begingroup$ I might have misunderstood that answer: I thought it only refers to functions whose Jacobian is an orthogonal matrix. My question regards Jacobians which are scaled orthogonal matrices, where the scaling may depend on the variable $x \in \mathbb{R}^d$ @StevenGubkin $\endgroup$
    – ualex
    Commented May 5, 2016 at 21:49
  • $\begingroup$ @Steven Gubkin: not necessary linear but Mobius. Mobius transformations are compositions of reflections in spheres. Most of them are not linear. $\endgroup$
    – Alexandre Eremenko
    Commented May 6, 2016 at 0:59

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