Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 23F (Riemann Surfaces)

A Riemann surface is a Hausdorff, second-countable, one-dimensional complex manifold, with charts to open subsets of $\mathbb{C}$ whose transition maps are holomorphic. Connectedness is often included in the convention. A map between Riemann surfaces is holomorphic when its expression in every compatible pair of charts is holomorphic. It is biholomorphic when it is bijective and both it and its inverse are holomorphic.

Let $A$ be the set of points with a neighbourhood on which $f=g$. It is nonempty and open. If $p$ is in its closure, continuity and Hausdorffness give $f(p)=g(p)$. Choose a common target chart and a source disc around $p$ on which both images lie in that chart. Their coordinate difference is holomorphic and vanishes at accumulating points from $A$, so the identity theorem makes it vanish on the disc. Thus $p\in A$. Therefore $A$ is also closed, and connectedness implies $f=g$ on all of $R$.

Choose centred charts so the coordinate expression $F$ satisfies $F(0)=0$. On a component where the map is nonconstant, the identity theorem makes $F$ nonconstant. Its Taylor expansion factors as $F(z)=z^{n}h(z)$, with $n\ge 1$ and $h(0)\ne 0$. In a sufficiently small disc, choose a holomorphic logarithm of $h$, and hence a holomorphic $n$th root $q$ with $q^n=h$. The function $zq(z)$ has nonzero derivative at zero. The holomorphic inverse function theorem, which says that a holomorphic function with nonzero derivative has a holomorphic local inverse, makes it a valid new source coordinate. In that coordinate,If the map is bijective, local injectivity forces $n=1$: for $n>1$, a small nonzero value has $n$ distinct preimages in this local disc. The inverse is therefore holomorphic in a neighbourhood of every point, and the local inverses agree, proving the map is biholomorphic. The nonconstant hypothesis must apply to the component containing $p$ if disconnected Riemann surfaces are allowed; otherwise a constant component is an exception to the proposed local form.