CHAPTER 17

Metric Geometry

Eberhard M. SCHRÖDER

Mathematisches Seminar der Universität Hamburg,
Bundesstraß e 55, D-20146 Hamburg, Germany

Contents

Introduction

1. Quadratic Forms

1.1. Definition of quadratic forms

1.2. Orthogonality

1.3. The kernel of a quadratic form

1.4. Orthogonal decompositions of quadratic forms

1.5. Isomorphisms

1.6. Isometries

1.7. Geometric equivalence of quadratic forms

1.8. Metric vector spaces of dimension 2

1.9. Metric vector spaces of dimension 3 and 4

2. Quadrics

2.1. Quadratic sets

2.2. Geometric characterizations of quadrics

2.3. Symmetries

2.4. Geometric characterizations of oval quadrics

3. Affine metric geometry

3.1. Concepts of affine metric geometry

3.2. Isomorphisms of affine metric geometry

3.3. Motions and similarities

3.4. Affine metric planes

3.5. Geometric characterizations of affine metric spaces

4. Projective metric geometry

4.1. Isomorphisms of projective metric geometry

4.2. Quadric circle geometry

4.3. The stereographic projection

4.4. Affine metric geometry and quadric circle geometry

4.5. Subgeometries of projective metric spaces

4.6. Geometric characterizations of subgeometries of projective metric spaces

References