Let $A$ and $B$ be Riemann surfaces. If we want to prove $f$:$A$→$B$ is isomorphism of Riemann surfaces, what we should do?
My book reads like this,
①Computing $f$'s effect on the cotangent spaces of $A$ and $B$ ②To show $f$ is local analytic isomorphism ( I don't know it's definition). ③To show $f$ is bijective, then, we can conclude that $f$ is global isomorphism.
My question : What is the definition of local analytic isomorphism? And why we can say $f$ is global isomorphism if we could show $f$ is also bijective?