Bieberbach Groups and Flat Manifolds

1. Front Matter

   Pages i-xiii

   PDF

2. Bieberbach’s Three Theorems

   • Leonard S. Charlap

   Pages 1-42

3. Flat Riemannian Manifolds

   • Leonard S. Charlap

   Pages 43-73

4. Classification Theorems

   • Leonard S. Charlap

   Pages 74-103

5. Holonomy Groups of Prime Order

   • Leonard S. Charlap

   Pages 104-166

6. Automorphisms

   • Leonard S. Charlap

   Pages 167-231

7. Back Matter

   Pages 233-242

   PDF

Many mathematics books suffer from schizophrenia, and this is yet another. On the one hand it tries to be a reference for the basic results on flat riemannian manifolds. On the other hand it attempts to be a textbook which can be used for a second year graduate course. My aim was to keep the second personality dominant, but the reference persona kept breaking out especially at the end of sections in the form of remarks that contain more advanced material. To satisfy this reference persona, I'll begin by telling you a little about the subject matter of the book, and then I'll talk about the textbook aspect. A flat riemannian manifold is a space in which you can talk about geometry (e. g. distance, angle, curvature, "straight lines," etc. ) and, in addition, the geometry is locally the one we all know and love, namely euclidean geometry. This means that near any point of this space one can introduce coordinates so that with respect to these coordinates, the rules of euclidean geometry hold. These coordinates are not valid in the entire space, so you can't conclude the space is euclidean space itself. In this book we are mainly concerned with compact flat riemannian manifolds, and unless we say otherwise, we use the term "flat manifold" to mean "compact flat riemannian manifold. " It turns out that the most important invariant for flat manifolds is the fundamental group.

• Algebraic structure
• Finite
• Invariant
• Morphism
• Riemannian manifold
• Topology
• algebra
• curvature
• differential topology
• geometry
• manifold
• mathematics
• theorem

• Communications Research Division, Institute for Defense Analysis, Princeton, USA

  Leonard S. Charlap

Policies and ethics