This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes’ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss–Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow’s Theorem on compact holomorphic submanifolds in complex projective spaces.
Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.
Author(s): Norbert A'Campo
Series: Universitext
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 284
City: Cham, Switzerland
Tags: Differential Geometry, Manifolds, Riemann Surfaces
Preface
Acknowledgements
Contents
Chapter 1 Basic Differential Geometry
1.1 Fields on Open Sets in Real Vector Spaces
1.2 Closed Forms are Locally Exact
1.3 Fixed Point Theorems
1.4 The Abstract Field C Versus the R-Algebra C of Complex Numbers
1.5 Coordinates and Locally Smooth Rigidity Theorems
1.6 Differentiation in Banach Spaces
1.7 Sard's Theorem
1.8 The Morse Lemma and Morse Functions
Chapter 2 The Geometry of Manifolds
2.1 Differentiable Manifolds
2.2 Fields on Manifolds
2.3 Frobenius' Integrability Condition
2.4 Foliations on Manifolds
2.5 The Topology of Connected, Compact Surfaces
2.6 Thoughts
Chapter 3 Hyperbolic Geometry
3.1 The Hyperbolic Plane H = HI
3.2 Intermezzo: Higher Cross-Ratios
3.3 Hyperbolic Trigonometry
3.4 Hyperbolic Area
3.5 A Compact Hyperbolic Surface of Genus g ≥ 2
3.6 The Riemann Sphere C U {∞}
Chapter 4 Some Examples and Sources of Geometry
4.1 The Space of Norms
4.2 Combinatorial Geometry
4.3 Spaces of Involutions
4.4 Conflicts and Dynamics
Chapter 5 Differential Topology of Surfaces
5.1 0- and 1-de Rham Cohomology of Surfaces
5.2 The Hyperbolic Plane Again, Now H = HJ
5.3 Reminder: Multi-Linear Algebra
5.4 Reminder: Holomorphic Functions in One Complex Variable
5.5 J-Laplace Operator and Metric
5.6 J-Surfaces
Chapter 6 Riemann Surfaces
6.1 Riemann Surfaces as z- and as J-Surfaces
6.2 Natural Structures on the Space J(TS)
6.3 J-Fields and Integrability in Higher Dimensions
6.4 Integrability of Fibred J-Fields
6.5 Analysis of Laplace Operators on J-Surfaces
6.6 Topology of the Two-Point Green Function
Chapter 7 Surfaces of Genus g = 0
7.1 The Uniformization Theorem, the Genus g = 0 Case
7.2 Strong J-Rigidity
7.3 Strong J-Rigidity and Volume Stretching
Chapter 8 Surfaces with Riemannian Metric
8.1 Riemannian Curvature
8.2 Topology of Surfaces and Curvature
8.3 Hyperbolic Length and Extremal Length
Chapter 9 Outline: Uniformization by Spectral Determinant
9.1 A Theorem of Mueller–Wendland and Osgood–Phillips–Sarnak
9.2 Uniformization by Spectral Determinant, g ≥ 0
9.3 Polyakov's String Dynamics
Chapter 10 Uniformization by Energy
10.1 Energy and Curvature
10.2 The Uniformization Theorem, Case g ≥ 1, By Energy
10.3 The Uniformization Theorem, Case g = 1
10.4 Comments About Uniformization, g = 0,1 or g ≥ 2
10.5 Consequences of the Uniformization Theorem for Surfaces of Genus ≥1
10.6 The ''Turn'' M(S) → J(TS)
Chapter 11 Families of Spaces
11.1 What Do Locally Trivial, Trivial and Constant Mean?
11.2 The Legendre Family
Chapter 12 Functions on Riemann Surfaces
12.1 Meromorphic Functions on Riemann Surfaces
12.2 J-Harmonic 1-Differential Forms on J-Surfaces
12.3 Riemann's Theorem About the Sub-Space Holo(S,J) of Closed Forms Ω1,0 J (S,C)
12.4 Explicit Basis of Hol(S,J) for the Hyperelliptic Surface Defined By y2 = –x2g+1 + 1
12.5 Why Functions?
12.6 The Field K(S) of Meromorphic Functions
12.7 Reconstruction of the Riemann Surface S From K(S) and its Subfield K0(S)
Chapter 13 Line Bundles and Cohomology
13.1 Divisors and Line Bundles
13.2 Cech and Dolbeault Cohomology
13.3 Computations of Cohomology
13.4 More General Computation of Cohomology
13.5 Roch's Inequality
13.6 Line Bundles, Degree and Exact Cech Cohomology Sequences
13.7 Intermezzo: Global Infinitesimal Deformations of Locally Rigid Structures
13.8 Hyperelliptic Curves
Chapter 14 Moduli Spaces and Teichmüller Spaces
14.1 Teichmüller Spaces as Smooth Manifolds
14.2 The Space Jμ(TSg) as a Symplectic Product
14.3 The Space J(TS) as a Product With Three Factors
14.4 The Geometry of Tangent Vectors to a Teichmüller Space
Chapter 15 Dimensions of Spaces of Holomorphic Sections
15.1 The Riemann–Roch Theorem
15.2 Consequences of the Riemann–Roch Theorem
15.3 The Birth of Serre Duality
Chapter 16 The Teichmüller Curve and its Universal Property
Chapter 17 Riemann Surfaces and Algebraic Curves
17.1 Chow’s Theorem
17.2 Riemann Surfaces as Projective Curves
Chapter 18 The Jacobian of a Riemann Surface
18.1 Vector Spaces Attached to a Riemann Surface
18.2 The Period Matrix and Riemann's Bilinear Relations
18.3 The Jacobian Jac(S)
18.4 The Abel–Jacobi Map
Chapter 19 Special Metrics on J-Surfaces
19.1 The Bergman Metric
19.2 Special Metrics and Covering Spaces
19.3 The Energy of Canonical Embeddings
Chapter 20 The Fundamental Group and Coverings
20.1 Simply Connected Riemann Surfaces and the Universal Uniformization Theorem
20.2 The Universal Cover and Uniformization of Riemann Surfaces
Appendix A Reminder: Topology
A.1 Topological Properties
A.2 The Fundamental Group
A.3 Covering Spaces
A.4 Tessellations and Coverings
References
Index
