By Nigel Hitchin
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Extra info for Geometry Of Surfaces B3 Course 2004 (Lecture notes)
A If a closed surface X is triangulated so that each face lies in a coordinate neighbourhood, then we can define the area of X as the sum of the areas of the faces by the formula above. It is independent of the choice of triangulation. 3 Isometric surfaces Definition 19 Two surfaces X, X are isometric if there is a smooth homeomorphism f : X → X which maps curves in X to curves in X of the same length. 53 A practical example of this is to take a piece of paper and bend it: the lengths of curves in the paper do not change.
3 Multi-valued functions The Riemann-Hurwitz formula is useful for determining the Euler characteristic of a Riemann surface defined in terms of a multi-valued function, like g(z) = z 1/n . We look for a closed surface on which z and g(z) are meromorphic functions. The example above is easy: if w = z 1/n then wn = z, and using the coordinate z = 1/z on a neighbourhood of ∞ on the Riemann sphere S, if w = 1/w then w n = z . Thus w and w are standard coordinates on S, and g(z) is the identity map S → S.
Subdivide the triangulation into smaller triangles such that each one is contained in one of the sets V . Then the inverse images of the vertices and edges of S form the vertices and edges of a triangulation of X. If the triangulation of S has V vertices, E edges and F faces, then clearly the triangulation of X has dE edges and dF faces. It has fewer vertices, though — in a neighbourhood where f is of the form w → wm the origin is a single vertex instead of m of them. For each ramification point of order mk we therefore have one vertex instead of mk .