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Introduction to Riemann Surfaces

Edition: 
2
Publisher: 
American Mathematical Society
Number of Pages: 
309
Price: 
37.00
ISBN: 
0821831569
Date Received: 
Monday, July 10, 2006
Reviewable: 
Yes
Include In BLL Rating: 
Yes
George Springer
Publication Date: 
2002
Format: 
Hardcover
Category: 
Monograph
BLL
06/16/2009
BLL Rating: 

Introduction

  • 1-1 Algebraic functions and Riemann surfaces
  • 1-2 Plane fluid flows
  • 1-3 Fluid flows on surfaces
  • 1-4 Regular potentials
  • 1-5 Meromorphic functions
  • 1-6 Function theory on a torus

General Topology

  • 2-1 Topological spaces
  • 2-2 Functions and mappings
  • 2-3 Manifolds

Riemann Surface of an Analytic Function

  • 3-1 The complete analytic function
  • 3-2 The analytic configuration

Covering Manifolds

  • 4-1 Covering manifolds
  • 4-2 Monodromy theorem
  • 4-3 Fundamental group
  • 4-4 Covering transformations

Combinatorial Topology

  • 5-1 Triangulation
  • 5-2 Barycentric coordinates and subdivision
  • 5-3 Orientability
  • 5-4 Differentiable and analytic curves
  • 5-5 Normal forms of compact orientable surfaces
  • 5-6 Homology groups and Betti numbers
  • 5-7 Invariance of the homology groups
  • 5-8 Fundamental group and first homology group
  • 5-9 Homology on compact surfaces

Differentials and Integrals

  • 6-1 Second-order differentials and surface integrals
  • 6-2 First-order differentials and line integrals
  • 6-3 Stokes' theorem
  • 6-4 The exterior differential calculus
  • 6-5 Harmonic and analytic differentials

The Hilbert Space of Differentials

  • 7-1 Definition and properties of Hilbert space
  • 7-2 Smoothing operators
  • 7-3 Weyl's lemma and orthogonal projections

Existence of Harmonic and Analytic Differentials

  • 8-1 Existence theorems
  • 8-2 Countability of a Riemann surface

Uniformization

  • 9-1 Schlichtartig surfaces
  • 9-2 Universal covering surfaces
  • 9-3 Triangulation of a Riemann surface
  • 9-4 Mappings of a Riemann surface onto itself

Compact Riemann Surfaces

  • 10-1 Regular harmonic differentials
  • 10-2 The bilinear relations of Riemann
  • 10-3 Bilinear relations for differentials with singularities
  • 10-4 Divisors
  • 10-5 The Riemann-Roch theorem
  • 10-6 Weierstrass points
  • 10-7 Abel's theorem
  • 10-8 Jacobi inversion problem
  • 10-9 The field of algebraic functions
  • 10-10 The hyperelliptic case
  • References
  • Index
Publish Book: 
Modify Date: 
Tuesday, June 16, 2009

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