By Jürgen Jost

ISBN-10: 3540533346

ISBN-13: 9783540533344

ISBN-10: 3662034468

ISBN-13: 9783662034460

Even supposing Riemann surfaces are a time-honoured box, this booklet is novel in its large viewpoint that systematically explores the relationship with different fields of arithmetic. it could actually function an advent to modern arithmetic as a complete because it develops historical past fabric from algebraic topology, differential geometry, the calculus of adaptations, elliptic PDE, and algebraic geometry. it's distinct between textbooks on Riemann surfaces in together with an creation to Teichm?ller concept. The analytic procedure is also new because it relies at the conception of harmonic maps.

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**Extra info for Compact Riemann Surfaces: An Introduction to Contemporary Mathematics**

**Example text**

X=coy2, (co=const. ). -:L -1 ( ~)2 x - c6 y2 . This equation is satisfied by the circle 32 2. 1. -c5 - that intersects the real axis orthogonally. A careful analysis of the preceding reasoning shows that we have thus obtained all geodesics of the hyperbolic metric. 0 Correspondingly, the geodesics on the model D of hyperbolic geometry are the subarcs of circles and straight lines intersecting the unit circle orthogonally. For our metric on the sphere 8 2 , the geodesics are the great circles on 2 8 C 1R3 or (in our representation) their images under stereographic projection.

4) If S is a Riemann surface with conformal charts {U,8, z,8}, and 7r : S' -+ Sa local homeomorphism, then there is a unique way of making S' a Riemann surface such that 7r becomes holomorphic. The charts {U~, z~} for S' are constructed such that 7r I U~ is bijective, and the Z,8 0 7r 0 z,~l are holomorphic wherever they are defined. Thus h 0 7r will be holomorphic on S' if and only if h is holomorphic on S. 5) If 7r : S' -+ S is a (holomorphic) local homeomorphism of Riemann surfaces, then every covering transformation cp is conformal.

A holomorphic map h with nowhere vanishing derivative ~~ is called conformal. We shall usually identify Ua C 8 with za(Ua). The subscript is usually unnecessary, and we shall then identify p E U with z(p) E C. This will not cause any difficulties, since we only study local objects and concepts which are invariant under conformal maps. For example, this holds for holomorphic functions and maps, for meromorphic functions, for harmonic and subharmonic 2 functions, and for differentiable or rectifiable curves.

### Compact Riemann Surfaces: An Introduction to Contemporary Mathematics by Jürgen Jost

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