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Geometry on surfaces, a source for mathematical developments

Norbert A'Campo, Athanase Papadopoulos

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 usingreal 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 graphs, stratifications of spaces of monic polynomials, and others. Classical methods and new techniques complement each other. The final version of this paper will appear as a chapter in the Volume Surveys in Geometry. II (ed. A. Papadopoulos), Springer Nature Switzerland, 2024.

Geometry on surfaces, a source for mathematical developments

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 usingreal 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 graphs, stratifications of spaces of monic polynomials, and others. Classical methods and new techniques complement each other. The final version of this paper will appear as a chapter in the Volume Surveys in Geometry. II (ed. A. Papadopoulos), Springer Nature Switzerland, 2024.
Paper Structure (37 sections, 21 theorems, 31 equations, 9 figures)

This paper contains 37 sections, 21 theorems, 31 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Rigidity of geometric structures
  3. Volume, symplectic and contact forms
  4. Almost complex structures
  5. Almost complex structures on $n$-spheres
  6. The first compact Riemann surface
  7. The Riemann sphere
  8. The group ${\rm SU}(2)$ and its action on the Riemann sphere.
  9. All three planar geometries and hyperbolic 3-space simultaneously
  10. A stratification of the Riemann sphere arising from algebra
  11. A note on the field with one element
  12. The Riemann sphere and shadow numbers
  13. $J$-compatible metrics
  14. Spherical geometry
  15. Models for hyperbolic 3-space and metrics of constant positive curvature on the sphere
  16. ...and 22 more sections

Key Result

Theorem 2.1

Let $M$ be a compact connected oriented manifold of dimension $n$ equipped with two smooth volume forms $\omega_0$ and $\omega_1$ of equal total volume. Then there exists an isotopy $\phi_t,\, t\in [0,1],$ satisfying $\phi_t^*(t\omega_1+(1-t)\omega_0)=\omega_0$. In particular, we have $\omega_0=\phi

Figures (9)

  • Figure 1: Figure for Exercise \ref{['ex:1']}
  • Figure 2: Figure for Exercise \ref{['ex:2']}
  • Figure 3: Three examples of regular line complexes with their coloring. The associated branched coverings, restricted to the complement of the branch points, are universal coverings of the sphere with 2, 3 and 4 punctures respectively
  • Figure 4: The line complex of the Riemann surface of the function $f(z)=ze^{-z}$
  • Figure 5: From Kobayashi's paper Kobayashi1 on the type problem
  • ...and 4 more figures

Theorems & Definitions (21)

  • Theorem 2.1: J. Moser Mos65
  • Theorem 2.2: J. Moser
  • Theorem 2.3: J. W. Gray Gray1959
  • Theorem 2.4: Local Darboux rigidities
  • Theorem 2.5
  • Theorem 2.6: Kirchhoff K
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1: The Riemann Mapping Theorem Riemann1851
  • Theorem 5.2: Uniformization theorem
  • ...and 11 more