By Jürgen Jost
Even supposing Riemann surfaces are a time-honoured box, this e-book is novel in its large standpoint 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 adaptations, elliptic PDE, and algebraic geometry. it's special between textbooks on Riemann surfaces in together with an advent to Teichmüller conception. The analytic strategy is also new because it relies at the conception of harmonic maps. For this re-creation, the writer has improved and rewritten a number of sections to incorporate extra fabric and to enhance the presentation.
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Extra resources for Compact Riemann Surfaces: An Introduction to Contemporary Mathematics
Proof. 3). e. z → λ2 z, and there is precisely one geodesic of H which is invariant under the action of γ, namely the imaginary axis. A moment’s reﬂection shows that the closed geodesics on H/Γ are precisely the projections of geodesics on H which are invariant under some nontrivial element of Γ , and this element of Γ of course determines the homotopy class of the geodesic. 3, we also observe that the length of that closed geodesic on H/Γ is log λ2 because γ identiﬁes points that distance apart on the imaginary axis.
Now, ∂F ⊂ F ∪ g(F ), ∂F ⊂ F ∪ g(F ), 50 2 Diﬀerential Geometry of Riemann Surfaces and g∈Γ g(F ) is the whole of H, while g1 (F ) and g2 (F ) have no interior point in common if g1 = g2 . Thus if F and F had non-empty interior, we would have F = g (F ) and F = g (F ) for some g , g = g; in particular the interiors of F and F would be fundamental domains. But g(a1 ) ⊂ ∂F , hence either F or F would have at least two sides fewer than g(F ). This is clearly impossible, since all images of F by elements of Γ of course have the same number of sides.
Some Basic Concepts of Surface Topology and Geometry. e. , with gU1 ∩ U2 = ∅ for all g ∈ G. e. has no accumulation point in E). We now wish to study properly discontinuous subgroups Γ of PSL(2, R); Γ acts on H as a group of isometries. 1. In particular, Γ is countable, because every uncountable set in R4 , and hence also any such subset of SL(2, R) or PSL(2, R), has an accumulation point. 2 Two points z1 , z2 of H are said to be equivalent with respect to the action of Γ if there exists g ∈ Γ with gz1 = z2 .