Let $X$ and $Y$ be Riemann surfaces. $f:X→Y$ be analytic map. Suppose $f$ induces isomorphism $f*$ between (co)tangent spaces, then, $∀p∈X$, $f'(p)≠0$.
How can I prove(understand) this?
P.S. Such $f$ is called local analytic isomorphism, and if $f$ is bijective, then from holomorphic inverse function theorem, $f$ is proved to be isom between $X$ and $Y$ as Riemann surfaces.