Download Compact Riemann Surfaces: An Introduction to Contemporary by Jürgen Jost PDF

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.

Show description

Read or Download Compact Riemann Surfaces: An Introduction to Contemporary Mathematics PDF

Similar differential geometry books

An Introduction to Noncommutative Geometry

Noncommutative geometry, encouraged by way of quantum physics, describes singular areas via their noncommutative coordinate algebras and metric buildings via Dirac-like operators. Such metric geometries are defined mathematically through Connes' conception of spectral triples. those lectures, added at an EMS summer season institution on noncommutative geometry and its purposes, offer an summary of spectral triples in accordance with examples.

Geometry, Topology and Quantization

This can be a monograph on geometrical and topological gains which come up in a variety of quantization tactics. Quantization schemes contemplate the feasibility of arriving at a quantum approach from a classical one and those contain 3 significant techniques viz. i) geometric quantization, ii) Klauder quantization, and iii) stochastic quanti­ zation.

Complex Spaces in Finsler, Lagrange and Hamilton Geometries

From a ancient standpoint, the idea we undergo the current learn has its origins within the well-known dissertation of P. Finsler from 1918 ([Fi]). In a the classical proposal additionally traditional category, Finsler geometry has along with a few generalizations, which use an analogous paintings procedure and which might be thought of self-geometries: Lagrange and Hamilton areas.

Introductory Differential Geometry For Physicists

This e-book develops the math of differential geometry in a much more intelligible to physicists and different scientists drawn to this box. This booklet is essentially divided into three degrees; point zero, the closest to instinct and geometrical event, is a quick precis of the speculation of curves and surfaces; point 1 repeats, reviews and develops upon the conventional tools of tensor algebra research and point 2 is an advent to the language of contemporary differential geometry.

Extra resources for Compact Riemann Surfaces: An Introduction to Contemporary Mathematics

Example text

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 .

Download PDF sample

Rated 4.42 of 5 – based on 37 votes