Induced map on cotangent space is isom means the map is locally analytic isomo

Question

Let $X$ and $Y$ be Riemann surfaces. $Φ:X→Y$ be map between them.

I want to prove or confirm the following statement,

Suppose induced map $Φ*$ on cotangent space of $X,Y$ is isomorphic,$Φ$ is local analytic isomorphism.

I couldn't find any references. Any references are welcome, thank you in advance.