Topological, Differential and Conformal Geometry of Surfaces

Description

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. 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