Geometry of crystallographic groups:
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| Format: | Elektronisch E-Book |
| Sprache: | English |
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© 2012
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| Schriftenreihe: | Algebra and discrete mathematics (World Scientific (Firm))
v. 4 |
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| Online-Zugang: | FAW01 FAW02 Volltext |
| Beschreibung: | Includes bibliographical references and index Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. They got their name, because in three dimensions they occur as the symmetry groups of a crystal (which we imagine to extend to infinity in all directions). The book is divided into two parts. In the first part, the basic theory of crystallographic groups is developed from the very beginning, while in the second part, more advanced and more recent topics are discussed. So the first part of the book should be usable as a textbook, while the second part is more interesting to researchers in the field. There are short introductions to the theme before every chapter. At the end of this book is a list of conjectures and open problems. Moreover there are three appendices. The last one gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group. This volume omits topics about generalization of crystallographic groups to nilpotent or solvable world and classical crystallography. We want to emphasize that most theorems and facts presented in the second part are from the last two decades. This is after the book of L Charlap "Bieberbach groups and flat manifolds" was published 1. Definitions. 1.1. Exercises -- 2. Bieberbach Theorems. 2.1. The first Bieberbach Theorem. 2.2. Proof of the second Bieberbach Theorem. 2.3. Proof of the third Bieberbach Theorem. 2.4. Exercises -- 3. Classification methods. 3.1. Three methods of classification. 3.2. Classification in dimension two. 3.3. Platycosms. 3.4. Exercises -- 4. Flat manifolds with b[symbol] = 0. 4.1. Examples of (non)primitive groups. 4.2. Minimal dimension. 4.3. Exercises -- 5. Outer automorphism groups. 5.1. Some representation theory and 9-diagrams. 5.2. Infinity of outer automorphism group. 5.3. R[symbol]-groups. 5.4. Exercises -- 6. Spin structures and Dirac operator. 6.1. Spin(n) group. 6.2. Vector bundles. 6.3. Spin structure. 6.4. The Dirac operator. 6.5. Exercises -- 7. Flat manifolds with complex structures. 7.1. Kahler flat manifolds in low dimensions. 7.2. The Hodge diamond for Kahler flat manifolds. 7.3. Exercises -- 8. Crystallographic groups as isometries of H[symbol]. 8.1. Hyperbolic space H[symbol]. 8.2. Exercises -- 9. Hantzsche-Wendt groups. 9.1. Definitions. 9.2. Non-oriented GHW groups. 9.3. Graph connecting GHW manifolds. 9.4. Abelianization of HW group. 9.5. Relation with Fibonacci groups. 9.6. An invariant of GHW. 9.7. Complex Hantzsche-Wendt manifolds. 9.8. Exercises -- 10. Open problems. 10.1. The classification problems. 10.2. The Anosov relation for flat manifolds. 10.3. Generalized Hantzsche-Wendt flat manifolds. 10.4. Flat manifolds and other geometries. 10.5. The Auslander conjecture |
| Beschreibung: | 1 Online-Ressource |
| ISBN: | 1283635984 9781283635981 9789814412254 9789814412261 9814412252 9814412260 |
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| 500 | |a Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. They got their name, because in three dimensions they occur as the symmetry groups of a crystal (which we imagine to extend to infinity in all directions). The book is divided into two parts. In the first part, the basic theory of crystallographic groups is developed from the very beginning, while in the second part, more advanced and more recent topics are discussed. So the first part of the book should be usable as a textbook, while the second part is more interesting to researchers in the field. There are short introductions to the theme before every chapter. At the end of this book is a list of conjectures and open problems. Moreover there are three appendices. The last one gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group. This volume omits topics about generalization of crystallographic groups to nilpotent or solvable world and classical crystallography. We want to emphasize that most theorems and facts presented in the second part are from the last two decades. This is after the book of L Charlap "Bieberbach groups and flat manifolds" was published | ||
| 500 | |a 1. Definitions. 1.1. Exercises -- 2. Bieberbach Theorems. 2.1. The first Bieberbach Theorem. 2.2. Proof of the second Bieberbach Theorem. 2.3. Proof of the third Bieberbach Theorem. 2.4. Exercises -- 3. Classification methods. 3.1. Three methods of classification. 3.2. Classification in dimension two. 3.3. Platycosms. 3.4. Exercises -- 4. Flat manifolds with b[symbol] = 0. 4.1. Examples of (non)primitive groups. 4.2. Minimal dimension. 4.3. Exercises -- 5. Outer automorphism groups. 5.1. Some representation theory and 9-diagrams. 5.2. Infinity of outer automorphism group. 5.3. R[symbol]-groups. 5.4. Exercises -- 6. Spin structures and Dirac operator. 6.1. Spin(n) group. 6.2. Vector bundles. 6.3. Spin structure. 6.4. The Dirac operator. 6.5. Exercises -- 7. Flat manifolds with complex structures. 7.1. Kahler flat manifolds in low dimensions. 7.2. The Hodge diamond for Kahler flat manifolds. 7.3. Exercises -- 8. Crystallographic groups as isometries of H[symbol]. 8.1. Hyperbolic space H[symbol]. 8.2. Exercises -- 9. Hantzsche-Wendt groups. 9.1. Definitions. 9.2. Non-oriented GHW groups. 9.3. Graph connecting GHW manifolds. 9.4. Abelianization of HW group. 9.5. Relation with Fibonacci groups. 9.6. An invariant of GHW. 9.7. Complex Hantzsche-Wendt manifolds. 9.8. Exercises -- 10. Open problems. 10.1. The classification problems. 10.2. The Anosov relation for flat manifolds. 10.3. Generalized Hantzsche-Wendt flat manifolds. 10.4. Flat manifolds and other geometries. 10.5. The Auslander conjecture | ||
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Datensatz im Suchindex
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| any_adam_object | |
| author | Szczepański, Andrzej 1954- |
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| dewey-full | 548/.81 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 548 - Crystallography |
| dewey-raw | 548/.81 |
| dewey-search | 548/.81 |
| dewey-sort | 3548 281 |
| dewey-tens | 540 - Chemistry and allied sciences |
| discipline | Chemie / Pharmazie |
| format | Electronic eBook |
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| id | DE-604.BV043138782 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:18:38Z |
| institution | BVB |
| isbn | 1283635984 9781283635981 9789814412254 9789814412261 9814412252 9814412260 |
| language | English |
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| spelling | Szczepański, Andrzej 1954- Verfasser (DE-588)1225127718 aut Geometry of crystallographic groups Andrzej Szczepański Singapore World Scientific © 2012 1 Online-Ressource txt rdacontent c rdamedia cr rdacarrier Algebra and discrete mathematics (World Scientific (Firm)) v. 4 Includes bibliographical references and index Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. They got their name, because in three dimensions they occur as the symmetry groups of a crystal (which we imagine to extend to infinity in all directions). The book is divided into two parts. In the first part, the basic theory of crystallographic groups is developed from the very beginning, while in the second part, more advanced and more recent topics are discussed. So the first part of the book should be usable as a textbook, while the second part is more interesting to researchers in the field. There are short introductions to the theme before every chapter. At the end of this book is a list of conjectures and open problems. Moreover there are three appendices. The last one gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group. This volume omits topics about generalization of crystallographic groups to nilpotent or solvable world and classical crystallography. We want to emphasize that most theorems and facts presented in the second part are from the last two decades. This is after the book of L Charlap "Bieberbach groups and flat manifolds" was published 1. Definitions. 1.1. Exercises -- 2. Bieberbach Theorems. 2.1. The first Bieberbach Theorem. 2.2. Proof of the second Bieberbach Theorem. 2.3. Proof of the third Bieberbach Theorem. 2.4. Exercises -- 3. Classification methods. 3.1. Three methods of classification. 3.2. Classification in dimension two. 3.3. Platycosms. 3.4. Exercises -- 4. Flat manifolds with b[symbol] = 0. 4.1. Examples of (non)primitive groups. 4.2. Minimal dimension. 4.3. Exercises -- 5. Outer automorphism groups. 5.1. Some representation theory and 9-diagrams. 5.2. Infinity of outer automorphism group. 5.3. R[symbol]-groups. 5.4. Exercises -- 6. Spin structures and Dirac operator. 6.1. Spin(n) group. 6.2. Vector bundles. 6.3. Spin structure. 6.4. The Dirac operator. 6.5. Exercises -- 7. Flat manifolds with complex structures. 7.1. Kahler flat manifolds in low dimensions. 7.2. The Hodge diamond for Kahler flat manifolds. 7.3. Exercises -- 8. Crystallographic groups as isometries of H[symbol]. 8.1. Hyperbolic space H[symbol]. 8.2. Exercises -- 9. Hantzsche-Wendt groups. 9.1. Definitions. 9.2. Non-oriented GHW groups. 9.3. Graph connecting GHW manifolds. 9.4. Abelianization of HW group. 9.5. Relation with Fibonacci groups. 9.6. An invariant of GHW. 9.7. Complex Hantzsche-Wendt manifolds. 9.8. Exercises -- 10. Open problems. 10.1. The classification problems. 10.2. The Anosov relation for flat manifolds. 10.3. Generalized Hantzsche-Wendt flat manifolds. 10.4. Flat manifolds and other geometries. 10.5. The Auslander conjecture SCIENCE / Physics / Crystallography bisacsh Crystallography, Mathematical fast Symmetry groups fast Symmetry groups Crystallography, Mathematical Raumgruppe (DE-588)4177070-5 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Raumgruppe (DE-588)4177070-5 s Geometrie (DE-588)4020236-7 s 1\p DE-604 http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=491503 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Szczepański, Andrzej 1954- Geometry of crystallographic groups SCIENCE / Physics / Crystallography bisacsh Crystallography, Mathematical fast Symmetry groups fast Symmetry groups Crystallography, Mathematical Raumgruppe (DE-588)4177070-5 gnd Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4177070-5 (DE-588)4020236-7 |
| title | Geometry of crystallographic groups |
| title_auth | Geometry of crystallographic groups |
| title_exact_search | Geometry of crystallographic groups |
| title_full | Geometry of crystallographic groups Andrzej Szczepański |
| title_fullStr | Geometry of crystallographic groups Andrzej Szczepański |
| title_full_unstemmed | Geometry of crystallographic groups Andrzej Szczepański |
| title_short | Geometry of crystallographic groups |
| title_sort | geometry of crystallographic groups |
| topic | SCIENCE / Physics / Crystallography bisacsh Crystallography, Mathematical fast Symmetry groups fast Symmetry groups Crystallography, Mathematical Raumgruppe (DE-588)4177070-5 gnd Geometrie (DE-588)4020236-7 gnd |
| topic_facet | SCIENCE / Physics / Crystallography Crystallography, Mathematical Symmetry groups Raumgruppe Geometrie |
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| work_keys_str_mv | AT szczepanskiandrzej geometryofcrystallographicgroups |