**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