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Spaces of Constant Curvature

Joseph A. Wolf
Publish or Perish, Inc.
Publication Date: 
Number of Pages: 
[Reviewed by
Gizem Karaali
, on

This book was first published in 1967, and has been revised and republished a few times. The reviewer got a copy of the fifth edition, from Publish or Perish Inc. It is typeset in a very readable format, with a large and friendly font, allowing the reader to comfortably enjoy the beauty of the mathematical content and the lucidity of the exposition simultaneously.

The book provides a self contained study of spaces of constant curvature. However, if one is interested specifically on symmetric spaces, or fixed point free groups and their representations instead, there is also an exhaustive description of these topics to satisfy one's curiosity. A basic introduction to the representation theory of finite groups is included, as well as a summary of the Dynkin diagrams of simple Lie algebras. In short there is much that can be of interest to any student of geometry in this book.

Wolf assumes his readers have some fluency with the basic language of differential geometry. The short introduction in the first five pages of the first chapter would not be sufficient for anyone with no previous exposure to the notions of differentiable manifolds and vector fields. The book is more appropriate as a text for a second course in geometry; a typical semester long course on differentiable manifolds should be more than enough to follow and enjoy the contents of the book. For readers with this minimal background, there are many possible routes through the text. One such path is already suggested by the author in the preface; the first two chapters together with Chapter 8 would make for a self contained course in Riemannian geometry.

Joseph Wolf's Spaces of Constant Curvature is a classic. Many generations of students interested in differential geometry, and more specifically, symmetric spaces, have learned from it. If that sounds like your cup of tea, and you have the right mathematical prerequisites, this will be a most satisfactory read.

Gizem Karaali is assistant professor of mathematics at Pomona College.

Part I    Riemannian Geometry
  1.1  Differentiable manifolds
     1.2  Vector fields
     1.3  Differential forms
     1.4  Maps
     1.5  Lie groups
     1.6  The frame bundle: parallelism and geodesics
     1.7  Curvature, torsion and the structure equations
     1.8  Covering spaces
     1.9  The Cartan-Ambrose-Hicks Theorem

 2. Riemannian Curvature
     2.1  The Levi-Cività connections
     2.2  Sectional curvature
     2.3  Isometries and curvature
     2.4  Models for spaces of constant curvature
     2.5  The 2-dimensional space forms
     2.6  Finite rotation groups
     2.7  Homogeneous space forms
     2.8  Appendix: The metric space structure of a riemannian manifold

Part II    The Euclidean Space Form Problem

 3. Flat Riemannian Manifolds
     3.1  Discontinuous groups on euclidean space
     3.2  The Bieberbach Theorems on crystallographic groups
     3.3  Application to euclidean space forms
     3.4  Questions of holonomy
     3.5  Three dimensional euclidean space forms
     3.6 Three attacks on the classification problem for flat compact manifolds
     3.7  Flat homogeneous pseudo-riemannian manifolds

Part III    The Spherical Space Form Problem

 4. Representations of Finite Groups
     4.1  Basic definitions
     4.2  The Frobenius-Schur relations
     4.3  Frobenius reciprocity and the group algebra
     4.4  Divisibility
     4.5  Tensor products and dual representations
     4.6  Two lemmas on representations over algebraically non-closed fields
     4.7  Unitary and orthogonal representations

 5. Vincent's Work on the Spherical Space Form Problem
     5.1  Vincent's program
     5.2  Preliminaries on p-groups
     5.3  Necessary conditions on fixed point free groups
     5.4  Classification of the simplest type of fixed point free groups
     5.5  Representations of finite groups in which every Sylow subgroup is cyclic
     5.6  A partial solution to the spherical space form problem

 6. The Classification of Fixed Point Free Groups
     6.1  Zassenhaus' work on solvable groups with cyclic odd Sylow subgroups
     6.2  The binary icosahedral group
     6.3  Non-solvable fixed point free groups

 7. The Solution to the Spherical Space Form Problem
     7.1  Representations of binary polyhedral groups
     7.2  Fixed point free complex representations
     7.3  The action of automorphisms on representations
     7.4  The classification of spherical space forms
     7.5  Spherical space forms of low dimension
     7.6  Clifford translations

Part IV    Space Form Problems on Symmetric Spaces

 8. Riemannian Symmetric Spaces
     8.1  Lie formulation of locally symmetric spaces
     8.2  Structure of orthogonal involutive Lie algebras
     8.3  Globally symmetric spaces and orthogonal involutive Lie algebras
     8.4  Curvature
     8.5  Cohomology
     8.6  Cartan subalgebras, rank and maximal tori
     8.7  Hermitian symmetric spaces
     8.8  The full group of isometries
     8.9  Extended Schläfli-Dynkin diagrams
    8.10  Subgroups of maximal rank
    8.11  The classification of symmetric spaces
    8.12  Two point homogeneous spaces
    8.13  Appendix: Manifolds with irreducible linear isotropy group

 9. Space Forms of Irreducible Symmetric Spaces
     9.1  Feasibility of space form problems
     9.2  Grassmann manifolds as symmetric spaces
     9.3  Grassmann manifolds of even dimension
     9.4  Grassmann manifolds of odd dimension
     9.5  Symmetric spaces of positive characteristic
     9.6  An isolated manifold

10. Locally Symmetric Spaces of Non-negative Curvature
    10.1  The structure theorems
    10.2  Application of the structure theorems

Part V    Space Form Problems on Indefinite Metric Manifolds

11. Spaces of Constant Curvature
    11.1  The classification of finite space forms
    11.2  The geometry of pseudo-spherical space forms
    11.3 Homogeneous finite space forms
    11.4  The lattice space forms
    11.5  A wild Lorentz signature
    11.6  The classification for homogeneous manifolds of constant curvature

12. Locally Isotropic Manifolds
    12.1  Reductive Lie groups
    12.2  Examples of locally isotropic manifolds
    12.3  Structure of locally isotropic spaces
    12.4  A partial classification of complete locally isotropic manifolds