This is a cross-post.

Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map.

Do there exist real-analytic structures on $M,N$ that make $f$ real-analytic?

I only assume the metrics are $C^{\infty}$. Every smooth manifold has a unique real-analytic structure (up to diffeomorphism) compatible with its smooth structure.

A reasonable starting point would be to know whether every $C^{\infty}$ conformal map between real-analytic manifolds with real-analytic metrics is real-analytic. (I don't know a reference for that; anyway, what I am asking seems harder).

*A weakly conformal map is a map whose differential at every point is either conformal or zero. (This is equivalent to $df^Tdf =(\det df)^{\frac{2}{n}} \, \text{Id}_{TM}$).

*Motivation:*

I am trying to understand if smooth weakly conformal maps whose differential vanishes at a point are constant (for dimensions $n \ge 3$). This seems to be the case for analytic maps, hence my interest in the possible analyticity of such maps.

For the Euclidean case, this follows directly by Liouville's theorem:

For $n=2$, every such map is complex-analytic. Let $\Omega \subseteq \mathbb{R}^n$ be an open subset, $n \ge 3$, and let $f:\Omega \to \mathbb{R}^n$ be a smooth conformal map. By Liouville's theorem, $f$ is of the form $$ f(x)=b+\alpha\frac{1}{|x-a|^\epsilon}A(x-a),$$

where $A$ is an orthogonal matrix, and $\epsilon \in \{0,2\}, b \in \mathbb{R}^n,\alpha \in \mathbb{R},a \in \mathbb{R}^n \setminus \Omega$.

So, up to translations and dilations, $ f(x)=\frac{A x}{|x|^2}$ (where $ 0 \notin \Omega$) which is real-analytic as a multiplication of two analytic maps. ($1/x^2$ is analytic on $\mathbb{R} \setminus \{0\}$).

areanalytic is also interesting I guess. I assume that in that case, conformal maps-in high dimensions, do need to be analytic, but I don't know a reference for that). $\endgroup$ – Asaf Shachar Oct 23 '18 at 9:11