Flat manifold
In mathematics, flat manifolds are Riemannian manifolds with constant sectional curvature zero.
Definition
A flat manifold is a complete Riemannian manifold with constant sectional curvature. ( A Riemannian metric with constant sectional curvature is called flat metric. A flat manifold is thus a manifold with a complete flat metric. )
Other characterizations
There are two other ways to define the concept of a flat manifold. Thus, it is determined
- One -dimensional flat manifold is a Riemannian manifold whose universal overlay ( that is, with the Euclidean metric ) is isometric to the Euclidean space.
- A flat manifold is a Riemannian manifold of the form, where is a discrete subgroup of the group of isometries of Euclidean space is.
These two definitions are to each other and to the definition in the section above are equivalent. The equivalence between the original definition and the first definition in this section follows from the theorem of Cartan; the equivalence of the two definitions of this section results from the superposition theory.
Bieberbach groups
In addition, if is compact, then is a crystallographic group of rank, called a space group. Because must be torsion-free, then it is a Bieberbach group.
After the first Bieberbach theorem there with a subgroup of finite index. The quotient is called the holonomy group of the flat variety.
Examples
It follows from the theorem of Chern - Gauss -Bonnet that the Euler characteristic of a flat manifold must be zero always.
2-dimensional examples
Each 2 -dimensional compact flat manifold is homeomorphic to the torus or the Klein bottle.
Examples of 3-dimensional
Up to homeomorphism, there are ten compact flat 3-manifolds, of which six orientable and four non- orientable. The six orientable examples have the Holonomiegruppen ( 3- torus ), and for ( the Hantzsch -Wendt manifold).
Generalized Hantzsche -Wendt manifolds
One -dimensional compact flat manifold is called generalized Hantzsche -Wendt manifold if the holonomy group is isomorphic to.