Geometry of crystallographic groups

Geometry of crystallographic groups / Andrzej Szczepański.

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 tw...

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Bibliographic Details
Main Author: Szczepański, Andrzej
Format: eBook
Language:English
Published: Singapore ; Hackensack, N.J. : World Scientific, ©2012.
Series:Algebra and discrete mathematics (World Scientific (Firm)) ; v. 4.
Subjects:
Symmetry groups.
Crystallography, Mathematical.
SCIENCE > Physics > Crystallography.
Crystallography, Mathematical
Symmetry groups
Online Access:Click for online access
Table of Contents:
  • 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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