CHAPTER 14

Chain Geometries

Armin HERZER*

Fachbereich Mathematik (a.D.), Joh. Gutenberg Universität, D-55099 Mainz, Germany

*Private address: Prof. Dr. Armin Herzer, Im Gries 13, D-78351 Bodman-Ludwigshafen, Germany

Contents

Introduction

1. The projective line over a ring R

1.1. Free modules over a ring

1.2. The projective line over a ring

1.3. Transitivity properties of the group of projectivities

1.4. The projective line over special rings

2. The chain geometry

2.1. Algebras

2.2. Chains on the projective line

2.3. Properties of the chain geometry

2.4. Parallelism of points

2.5. Geometries of Möbius, Laguerre and Minkowski

3. The affine chain geometry

3.1. Weak chain spaces

3.2. Residual spaces

3.3. Chain spaces

3.4. Contact spaces

3.5. Chain geometries are chain spaces

3.6. Investigation of the affine chains in

3.7. Cremonian geometries

4. The chain geometry

5. The chain geometry

5.1. Generalized reguli

5.2. Intrinsic characterization of the distance on

5.3. Representation of chain geometries on the Grassmannian

6. Rational representations of chain geometries

6.1. The general case

6.2. Chain geometries over quadratic algebras

6.3. Chain spaces associated to quadratic sets

6.4. Stereographic projection

7. Chain geometries over commutative algebras

7.1. Using determinants

7.2. Angles between chains

7.3. Two -configurations

8. Characterizations and direct products of chain geometries

8.1. Möbius geometries

8.2. Strong chain spaces

8.3. Minkowski geometries

8.4. Direct products of chain geometries

9. Isomorphisms of chain geometries

9.1. Jordan homomorphisms

9.2. The structure theorem for chain geometries

10. n-chain geometries

10.1. Short historical review

10.2. The general concept

10.3. Burau geometries

11. Bibliographical remarks

References