Algebraical and Topological Foundations of Geometry (eBook)

Proceedings of a Colloquium Held in Utrecht, August 1959

Hans Freudenthal (Autor)

eBook Download: PDF

2014 | 1. Auflage
216 Seiten
Elsevier Science (Verlag)
978-1-4831-8464-7 (ISBN)

Infos

Algebraical and Topological Foundations of Geometry contains the proceedings of the Colloquium on Algebraic and Topological Foundations of Geometry, held in Utrecht, the Netherlands in August 1959. The papers review the algebraical and topological foundations of geometry and cover topics ranging from the geometric algebra of the Möbius plane to the theory of parallels with applications to closed geodesies. Groups of homeomorphisms and topological descriptive planes are also discussed. Comprised of 26 chapters, this book introduces the reader to the theory of parallels with applications to closed geodesies; groups of homeomorphisms; complemented modular lattices; and topological descriptive planes. Subsequent chapters focus on collineation groups; exceptional algebras and exceptional groups; the connection between algebra and constructions with ruler and compasses; and the use of differential geometry and analytic group theory methods in foundations of geometry. Von Staudt projectivities of Moufang planes are also considered, and an axiomatic treatment of polar geometry is presented. This monograph will be of interest to students of mathematics.

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Algebraical and Topological Foundations of Geometry contains the proceedings of the Colloquium on Algebraic and Topological Foundations of Geometry, held in Utrecht, the Netherlands in August 1959. The papers review the algebraical and topological foundations of geometry and cover topics ranging from the geometric algebra of the Mobius plane to the theory of parallels with applications to closed geodesies. Groups of homeomorphisms and topological descriptive planes are also discussed. Comprised of 26 chapters, this book introduces the reader to the theory of parallels with applications to closed geodesies; groups of homeomorphisms; complemented modular lattices; and topological descriptive planes. Subsequent chapters focus on collineation groups; exceptional algebras and exceptional groups; the connection between algebra and constructions with ruler and compasses; and the use of differential geometry and analytic group theory methods in foundations of geometry. Von Staudt projectivities of Moufang planes are also considered, and an axiomatic treatment of polar geometry is presented. This monograph will be of interest to students of mathematics.

Inhalt

Front Cover 1
Algebraical and Topological Foundations of Geometry 4
Copyright Page 5
Table of Contents 7
PREFACE 6
CHAPTER 1. HJELMSLEVSCHE GEOMETRIE 10
CHAPTER 2. ZUR GEOMETRISCHEN ALGEBRADER MÖBIUSEBENEN 14
LITERATUR 15
CHAPTER 3. THE THEORY OF PARALLELS WITH APPLICATIONS TO CLOSED GEODESICS 16
1. THE AXIOMS 16
2. EXAMPLES OF NON-SYMMETRIC SPACES 17
3. PARALLELS 19
4. THE RELATION TO CLOSED GEODESICS 20
REFERENCES 22
CHAPTER 4. TACTICAL DECOMPOSITIONS OF .-SPACES 24
1. INTRODUCTION 24
2. DEFINITIONS AND EXAMPLES 25
3. BASIC PROPERTIES OF TACTICAL DECOMPOSITIONS OF .-SPACES 26
4. RESULTS ON PROBLEM (I) 28
5. RESULTS ON PROBLEM (II) 29
6. A RESULT ON PROBLEM (III) 30
BIBLIOGRAPHY 32
CHAPTER 5. SCHWACH PROJEKTIVE RÄUME ÜBER DREIFACHEN TERNÄRKÖRPERN 34
LITERATUR 37
CHAPTER 6. SYMPLEKTISCHE UND METASYMPLEKTISCHEGEOMETRIEN 38
CHAPTER 7. BERICHT ÜBER DIE THEORIE DER ROSENFELDSCHEN ELLIPTISCHEN EBENEN 44
CHAPTER 8. ON GROUPS OF HOMEOMORPHISMS 48
(1) THE MAIN THEOREM 49
2. A COUNTER-EXAMPLE 52
REFERENCES 53
CHAPTER 9. TOPOLOGICAL DESCRIPTIVE PLANES 54
CHAPTER 10. COMPLEMENTED MODULAR LATTICES 56
A. COORDINATIZATION THEORY 56
B. INDEPENDENCE, FINITENESS AND CONTINUITY 60
REFERENCES 64
CHAPTER 11. ÜBER DIE TOPOLOGISCHE UND ALGEBRAISCHE STRUKTUR TOPOLOGISCHER DOPPELLOOPS UND EINIGER TOPOLOGISCHER PROJEKTIVER EBENEN 66
1. DIE TOPOLOGISCHE STRUKTUR TOPOLOGISCHER DOPPELLOOPS 67
2. DIE ALGEBRAISCHE STRUKTUR TOPOLOGISCHER DOPPELLOOPS 68
3. DIE TOPOLOGISOHE STRUKTUR GEWISSER TOPOLOGISCHER PROJEKTIVER EBENEN 73
LITERATUR 75
CHAPTER 12. SOME RESULTS ON COLLINEATION GROUPS 78
BIBLIOGRAPHY 79
CHAPTER 13. ANORDNUNGSFRAGEN IN TERNÄREN RINGEN UND ALLGEMEINEN PROJEKTIVEN UND AFFINEN EBENEN 80
1. BESCHREIBUNG DER ANORDNUNG DURCH ORDNUNGSFUNKTIONEN 81
2. ANORDNUNGEN DES TERNÄREN RINGES 84
3. EIGENSCHAFTEN DER STARKEN HALBORDNUNG 92
4. SPEZIELLE UND VOLLE ANORDNUNGEN 93
LITERATUR 95
CHAPTER 14. CONSTRUCTION GEOMETRIES AND CONSTRUCTION FIELDS 96
1. INTRODUCTION 96
2. CONSTRUCTION OPERATIONS AND CONSTRUCTION FIELDS 98
3. RATIONAL OPERATIONS 101
4. QUADRATIC CONSTRUCTIONS 103
5. THE MOHR-MASCHERONI THEOREM 106
REFERENCES 106
CHAPTER 15. PROJEKTIVE GEOMETRIE UND LINEARE ALGEBRA ÜBER VERALLGEMEINERTEN BEWERTUNGSRINGEN 108
1. PROJEKTIVE RÄUME ÜBER LOKALEN RINGEN 108
2. DIE STRUKTUR DER LINEAREN GRUPPE ÜBER VERALLGEMEINERTEN BEWERTUNGSRINGEN 111
LITERATUR 115
CHAPTER 16. VERALLGEMEINERTE METRISCHE EBENEN UND ORTHOGONALE GRUPPEN 118
LITERATUR 130
CHAPTER 17. ANNEAUX TERNAIRES ET CORPS GÉNÉRALISES LIES AUX GÉOMÉTRIES NON-ARGUÉSIENNESt 132
I. DEMIGROUPES TERNAIRES (DEMI-AMAS) 132
II. TREILLIS TERNAIRES 136
III. CORPS TERNAIRES ET «CORPS GÉNÉRALISÉS» 137
BIBLIOGRAPHIE 138
CHAPTER 18. DIFFERENTIAL GEOMETRY AND ANALYTIC GROUP THEORY METHODS IN FOUNDATIONS OF GEOMETRY 140
1. THE MAIN THEOBEMS 140
2. PRELIMINARIES 141
3. AN OUTLINE OF THE PROOF OF THEOREM 142
4. THE HOMOGENEOUS CASE 143
BIBLIOGRAPHY 143
CHAPTER 19. EINFACHE LIE-GRUPPEN UND NICHTEUKLIDISCHE GEOMETRIEN 144
1. ALGEBREN 145
2. PROJEKTIVE RÄUME 146
3. ELLIPTISCHE RÄUME 148
4. DIE BEWEGUNGSGRUPPEN DER KOMPAKTEN ELLIPTISCHEN RÄUME 149
5. DIE BEWEGUNGSGRUPPEN DER NICHTKOMPAKTEN ELLIPTISCHEN UND HYPERBOLISCHEN RÄUME 154
6. SYMMETRIEGEBILDE 155
7. PROJEKTIVE ALS ELLIPTISCHE GEOMETRIEN 155
8. SYMPLEKTISCHE ALS ELLIPTISCHE GEOMETRIE 157
9. METASYMPLEKTISCHE ALS ELLIPTISCHE GEOMETRIE 158
10. EINFACHHEITSGEBILDE 159
11. FRAGEN 163
LITERATUR 164
CHAPTER 20. TOPOLOGISCHE PROJEKTIVE EBENEN 166
LITERATUR 172
CHAPTER 21. VON STAUDT PROJECTIVITIESOF MOUFANG PLANES 174
REFERENCES 175
CHAPTER 22. VERALLGEMEINERTE AFPINE RÄUME UNDIHRE ALGEBRAISCHE DARSTELLUNG 176
CHAPTER 23. THE PROJECTIVE OCTAVE PLANE 182
CHAPTER 24. GROUPES ALGÉBRIQUES SEMI-SIMPLES ET GEOMETRIES ASSOCIÉES 184
§ 1. Définitions préliminaires. Géométries §§ 184
§ 2. Géométries et diagrammes de Dynkin † d'un groupe 186
§ 3. Classification 190
§ 4. Geometries de type .. Tableau caractéristique 195
BIBLIOGRAPHIE 200
CHAPTER 25. AN AXIOMATIC TREATMENT OF POLARGEOMETRY 202
REFERENCES 207
CHAPTER 26. PERSPECTIVITIES AND THE LITTLEPROJECTIVE GROUP 208
§ 1. INTRODUCTION 208
§ 2. DEFINITIONS AND BASIC CONCEPTS 208
§ 3. COLLINEATIONS OF FlNITE .-SPACES 210
§ 4. PERSPECTIVITIES OF PROJECTIVE SPACES 211
§ 5. PERSPECTIVITIES OF FINITE PROJECTIVE SPACES 212
§ 6. PERSPECTIVITIES OF FINITE PROJECTIVE PLANES 213
§ 7. COLLINEATION GROUPS OF FINITE PROJECTIVE SPACES 215
§ 8. COLLINEATION GROUPS OF INFINITE PLANES 215
REFERENCES 216

Produkt-Details

• Erscheint lt. Verlag: 9.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4831-8464-1 / 1483184641
• ISBN-13: 978-1-4831-8464-7 / 9781483184647