**Edit** according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question


We consider the following two classes of smooth maps on $\mathbb{R}^{n}.$ ( $n$ is not necessarily even):


**$HP\_{1}$**: $\;$A smooth map $f:\mathbb{R}^{n} \to \mathbb{R^{n}}$ satisfies property $HP\_{1}$ if the pull back metric $f^{\*}(g)$ is a conformal metric for every conformal metric $g$ on $\mathbb{R}^{n}.$ ( By conformality I mean $g=e^{h} \sum dx\_{i}^{2})$


**$HP\_{2}$**: $\;$A smooth map $X$ on $\mathbb{R}^{n}$ with coordinates $X=(P\_{1},P\_{2},\ldots,P\_{n})$ satisfies $HP\_{2}$ if the space of harmonic functions is invariant under the derivational operator $U \mapsto X.U=\sum P\_{i} \partial U/ \partial x\_{i}$


**Questions:**



> 
> 1. Assume $n=2k$, identify $\mathbb{R}^{2k}$ with $\mathbb{C}^{k}$. Let $f$ satisfies $HP\_{1}$. Is $f$ a holomorphic map on $\mathbb{C}^{k}$? The same question for $HP\_{2}$?
> 2. Are the above two properties, equivalents?(A map is $HP\_{1}$ iff it is $HP\_{2}$?)
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Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)
This is an extended comment. The first property $HP\_1$ simply says that $f$ is conformal.
It is not true that holomorphic maps of $C^n=R^{2n}$ are conformal, except when $n=1$.
In fact there are very few conformal maps in $R^n$ for $n\geq 3$: they are only Mobius transfomrations (compositions of reflections).


