Geometry on Surfaces, a Source for Mathematical Developments

Norbert A’Campo () and Athanase Papadopoulos ()
Additional contact information
Norbert A’Campo: Universität Basel, Departement Mathematik und Informatik, Fachbereich Mathematik
Athanase Papadopoulos: Institut de Recherche Mathématique Avancée and Centre de Recherche et Expérimentation sur l’Acte Artistique

Chapter Chapter 2 in Surveys in Geometry II, 2024, pp 7-70 from Springer

Abstract: Abstract We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results for such structures. Higher-dimensional analogues are also discussed. Some constructions with Riemann surfaces lead, by analogy, to notions that hold for arbitrary fields, and not only the field of complex numbers. The Riemann sphere is also defined using surjective homomorphisms of real algebras from the ring of real univariate polynomials to (arbitrary) fields, in which the field with one element is interpreted as the point at infinity of the Gaussian plane of complex numbers. Several models of the hyperbolic plane and hyperbolic 3-space appear, defined in terms of complex structures on surfaces, and in particular also a rather elementary construction of the hyperbolic plane using real monic univariate polynomials of degree two without real roots. Several notions and problems connected with conformal structures in dimension 2 are discussed, including dessins d’enfants, the combinatorial characterization of polynomials and rational maps of the sphere, the type problem, uniformization, quasiconformal mappings, Thurston’s characterization of Speiser nets, stratifications of spaces of monic polynomials, and others. Classical methods and new techniques complement each other.

Keywords: Geometric structure; Conformal structure; Almost complex structure (J-field); Riemann sphere; Uniformization; The type problem; Rigidity; Model for hyperbolic space; Cross ratio; Belyi’s theorem; Riemann–Hurwitz formula; Chasles 3-point function; Branched covering; Type problem; Dessin d’enfants; Slalom polynomial; Slalom curve; Space of monic polynomials; Stratification; Fibered link; Divide; Speiser curve; Speiser graph; Line complex; Quasiconformal map; Almost analytic function; Net; Speiser net; 12D10; 26C10; 14H55; 30F10; 30F20; 30F30; 53A35; 53D30; 57K10 (search for similar items in EconPapers)
Date: 2024
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-43510-2_2

Ordering information: This item can be ordered from
http://www.springer.com/9783031435102

DOI: 10.1007/978-3-031-43510-2_2

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().