Compact Riemann Surfaces: An Introduction to Contemporary by Jürgen Jost

By Jürgen Jost

Even if Riemann surfaces are a time-honoured box, this ebook is novel in its vast viewpoint that systematically explores the relationship with different fields of arithmetic. it could function an creation to modern arithmetic as a complete because it develops historical past fabric from algebraic topology, differential geometry, the calculus of diversifications, elliptic PDE, and algebraic geometry. it really is targeted between textbooks on Riemann surfaces in together with an advent to Teichmüller conception. The analytic strategy is also new because it relies at the thought of harmonic maps. For this new version, the writer has elevated and rewritten a number of sections to incorporate extra fabric and to enhance the presentation.

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Proof. The derivative of expp at 0 ∈ Vp applied to v ∈ C is d γp,tv (1) dt t=0 d = γp,v (t) dt t=0 = γ˙ p,v (0) =v D expp (0)(v) = by definition of γp,v . Thus, the derivative of expp at 0 ∈ Vp is the identity. The inverse function theorem may therefore be applied to show the claim. In general, however, the map expp is not holomorphic. Thus, if we use exp−1 p as a local chart, we preserve only the differentiable, but not the conformal structure. For that reason, we need to investigate how our geometric expressions transform under differentiable coordinate transformations.

3). 6 The hyperbolic metric on D is given by 4 (1 − |z|2 ) 2 dz dz, and the isometries between H and D are again M¨ obius transformations. Proof. 3. It has of course to be kept in mind that, up to a M¨ obius transformation, w = (z − i)/(z + i) is the only transformation which carries 4 the metric y12 dzdz to the metric (1−|w| 2 )2 dwdw. 28 2 Differential Geometry of Riemann Surfaces Consider now the map z → eiz =: w of H onto D\{0}. e. the map becomes a local isometry between H with its hyperbolic metric and D\{0} with this metric.

In particular d(p, q) = v p . Proof. Let γ(t), 0 ≤ t ≤ T be any curve from p to q. Let t0 := inf t ≤ T : γ(t) ∈ expp { v p < δ} , or t0 := T if no such t ≤ T can be found. We shall show that the curve γ|[0,t0 ] is already longer than γp,v , unless it coincides with the latter one. e. if γ coincides with γp,v up to parametrization. 1 Let M be a compact metric surface. There exists ε > 0 with the property that any two points in M of distance < ε can be connected by a unique shortest geodesic (of length < ε) (up to reparametrization).

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