an injective map can not take several intersecting arcs onto line segment I read a result in the theory of harmonic mappings, and i think it might be true in general setting as well. But i am unable to get a proof of this. Can anyone help me with proving it. The statement is 

If a harmonic function $f$ is loally injective at a point $z_0$, then it can not take a set consisting of several arcs intersecting at $z_0$ onto a line segment.

 A: Let's assume for now $f:\mathbb{R}^m\to\mathbb{R}^n$ is a continuous map between Euclidean spaces, where the objects you mentioned (arcs & line segment) naturally make sense. We will prove that if $f$ is locally injective at $z_0$, then $f$ is a local homeomorphism at $z_0$. The claim you made then follows.
By assumption, $z_0$ has a neighborhood $U$ such that $f|_U$ is injective. Without loss of generality we may take $U$ to be an open ball centered at $z_0$; then $U$ has a compact subset $K$ and an open subset $V$ with $z_0\in V\subset K\subset U$ (say, take $K$ to be a closed ball of smaller radius, and $V$ the interior of $K$). Now consider the map $f|_K:K\to f(K)$. this is a map from a compact space to a Hausdorff space (as a subset of the Hausdorff space $\mathbb{R}^n$, and is bijective by construction. By a well-known theorem, $f|_K$, as well as $f|_V:V\to f(V)$, is a homeomorphism.
I think only local Euclidean properties are used in my argument above, so my guess is that your claim holds for mappings between manifolds.
