By Jürgen Jost

ISBN-10: 3540533346

ISBN-13: 9783540533344

ISBN-10: 3662034468

ISBN-13: 9783662034460

Even if Riemann surfaces are a time-honoured box, this booklet is novel in its extensive standpoint that systematically explores the relationship with different fields of arithmetic. it will possibly function an creation to modern arithmetic as an entire because it develops historical past fabric from algebraic topology, differential geometry, the calculus of adaptations, elliptic PDE, and algebraic geometry. it really is particular between textbooks on Riemann surfaces in together with an creation to Teichm?ller conception. The analytic process is also new because it relies at the idea of harmonic maps.

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**Example text**

PSL(2, JR) := SL(2, JR)j { ± Via z -+ (az + b)j(cz + d), an element of SL(2, JR) defines a Mobius transformation which maps H onto itself. Any element of PSL(2, JR) can be lifted to SL(2, JR) and thus defines a Mobius transformation which is independent of the lift. 6 A group G acts as a group of transformations or transformation group of a manifold E, if there is given a map GxE-+E (g,x) -+ gx with for all g1,g2 and ex =X E G, x E E, for all x E E where e is the identity element of G. (In particular, each 9 : E -+ E is a bijection, since the group inverse g-1 of 9 provides the inverse map).

A metric is most simply described by means of a potential. Since a potential is invariant under coordinate transformations (and hence also under isometries, cf. Def. 2 below), it provides the easiest method of studying the transformation behaviour of the metric. 3 The Laplace-Beltrami operator with respect to the metric >. 2 8z Oz = ;2 (8~2 + 8~2 ), z= x + iy. 1 log>.. Remark. With z = x + iy, we have Thus the metric differs from the Euclidian metric only by the conformal factor >. 2. In particular, the angles with respect to >.

We want to characterize the minimizers of E by a differential equation. In local coordinates, let 1(t) + S rJ(t) be a smooth variation of 1, -so:::; E, we must have 0= :s =~ so, for some So > O. ) {A2(1hi/ + 2AA"('Y"t77} dt. If the variation fixes the end points of 1, Le. 8) . 8) is called geodesic. 7) so that any geodesic is parametrized proportionally to arclength. Since the energy integral is invariant under coordinate chart transformations, so must be its critical points, the geodesics. 8) is also preserved under coordinate changes.

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

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