Note: This question is probably a duplicate but difficult to Google, so please feel free to close and link to the original question if this is a duplicate. /Note
Question 1: Is there a term for functions with the following property?
- Let $U, V$ be open subsets of $\mathbb{R}^n$, and let $f: U \to V$ be a continuously (Frechet) differentiable function such that, for each $u \in U$, the derivative $D f (u) = a(u) T(u)$ for some functions $a: U \to \mathbb{R}_{\ge 0}$ and $T: U \to SO(n)$.
- I.e. at every point in the domain of $f$, the derivative of $f$ is a linear transformation that is a composition of a rotation ("twist", "Drehung") and multiplication by a nonnegative constant ("amplification", "Streckung").
In the case of $n=2$, with $\mathbb{R}^2$ interpreted as $\mathbb{C}$, this condition is supposed to be equivalent to satisfying the Cauchy-Riemann equations in an open region, and thus to being holomorphic. (Cf. e.g. Visual Complex Analysis by Tristan Needham.)
Question 2: If functions with this property do have a name including when $n > 2$, then is it known whether such functions still have any extra "nice" properties even in the case where $n > 2$? Or is $n=2$ special?
For example in the case that $n=2$, the aforementioned textbook argues that this property can be used to prove Cauchy's formula for holomorphic functions and with it their infinite differentiability. Do any analogous results hold for $n > 2$?
Bonus question: Can the condition that $f$ is continuously differentiable be omitted, i.e. is it implied by the remaining conditions?
I'm unclear on whether it can be omitted in the $n=2$ case, and hence even more so when $n > 2$.
Related questions: Again this question is difficult to Google, but this question on MathOverflow might be related: generalisation of Cauchy-Riemann equations to 3D
In particular one of the answers states "a mapping has the title property in dimension at least three if and only if it is either constant or a Riemannian covering up to homothety. For $\mathbb{R}^n$ this means a composition of translations, rotations, and homothety ($x \to a x$, for some $a > 0$)".
Now I'm not sure whether the property asked about in that question (preserving harmonicity) is equivalent to the property above. (Honestly I don't think it is.) But the property above is requiring that the derivative at every point be the composition of a rotation and a homothety, so based on that answer it sounds like it could be related.
The reference used by this math.se question Equivalent characterizations of holomorphic functions appears to state the result in $n=2$ whose generalization I am asking about: "$f \colon \mathbb C \to \mathbb C$ is holomorphic if it is is $C^1$ as a function from $\mathbb R^2$ to $\mathbb R^2$ and the Jacobian is a rotation-dilation".
Also one of the answers to that question seems to the answer the bonus question in the case that $n=2$, namely that the extra hypothesis of continuous differentiability is unnecessary due to Goursat's theorem.