Ivancevic_Applied-Diff-Geom

18 Applied Differential Geometry: A Modern IntroductionThe main point of Riemann surfaces is that holomorphic (analytic complex)functions may be defined between them. Riemann surfaces are nowadaysconsidered the natural setting for studying the global behavior of thesefunctions, especially multi–valued functions such as the square root or thelogarithm.Every Riemann surface is a 2D real analytic manifold (i.e., a surface),but it contains more structure (specifically, a complex structure) which isneeded for the unambiguous definition of holomorphic functions. A 2D realmanifold can be turned into a Riemann surface (usually in several inequivalentways) iff it is orientable. So the sphere and torus admit complexstructures, but the Möbius strip, Klein bottle and projective plane do not.Geometrical facts about Riemann surfaces are as ‘nice’ as possible, andthey often provide the intuition and motivation for generalizations to othercurves and manifolds. The Riemann–Roch Theorem is a prime example ofthis influence. 17Examples of Riemann surfaces include: the complex plane 18 , open subsetsof the complex plane 19 , Riemann sphere 20 , and many others.Riemann surfaces naturally arise in string theory as models of string17 Formally, let X be a Hausdorff space. A homeomorphism from an open subsetU ⊂ X to a subset of C is a chart. Two charts f and g whose domains intersect are saidto be compatible if the maps f ◦ g −1 and g ◦ f −1 are holomorphic over their domains. IfA is a collection of compatible charts and if any x ∈ X is in the domain of some f ∈ A,then we say that A is an atlas. When we endow X with an atlas A, we say that (X, A)is a Riemann surface.Different atlases can give rise to essentially the same Riemann surface structure on X;to avoid this ambiguity, one sometimes demands that the given atlas on X be maximal,in the sense that it is not contained in any other atlas. Every atlas A is contained in aunique maximal one by Zorn’s lemma.18 The complex plane C is perhaps the most trivial Riemann surface. The map f(z) = z(the identity map) defines a chart for C, and f is an atlas for C. The map g(z) = z* (theconjugate map) also defines a chart on C and g is an atlas for C. The charts f and g arenot compatible, so this endows C with two distinct Riemann surface structures.19 In a fashion analogous to the complex plane, every open subset of the complex planecan be viewed as a Riemann surface in a natural way. More generally, every open subsetof a Riemann surface is a Riemann surface.20 The Riemann sphere is a useful visualization of the extended complex plane, whichis the complex plane plus a point at infinity. It is obtained by imagining that all therays emanating from the origin of the complex plane eventually meet again at a pointcalled the point at infinity, in the same way that all the meridians from the south poleof a sphere get to meet each other at the north pole.Formally, the Riemann sphere is obtained via a one–point compactification of the complexplane. This gives it the topology of a 2–sphere. The sphere admits a unique complexstructure turning it into a Riemann surface. The Riemann sphere can be characterizedas the unique simply–connected, compact Riemann surface.