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 $\mathbb{C}^{k}$?

2. Are the above two properties, equivalents?(A map is $HP_{1}$ iff it is $HP_{2}$?)

Motivation: The answer to the first question is "yes" if we replace $HP_{1}$ by $HP_{2}$. On the other hand a holomorphic pull back of a conformal metric on $\mathbb{C}$ is again conformal.

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 $HP_{2}$?)

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 $\mathbb{C}^{k}$?

2. Are the above two properties, equivalents?(A map is $HP_{1}$ iff it is $HP_{2}$?)

Motivation: The answer to the first question is "yes" if we replace $HP_{1}$ by $HP_{2}$. On the other hand a holomorphic pull back of a conformal metric on $\mathbb{C}$ is again conformal.

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 $\mathbb{C}^{k}$?

2. Are the above two properties, equivalents?(A map is $HP_{1}$ iff it is $HP_{2}$?)

Motivation: The answer to the first question is "yes" if we replace $HP_{1}$ by $HP_{2}$. On the other hand a holomorphic pull back of a conformal metric on $\mathbb{C}$ is again conformal.