Flat manifold

In mathematics , flat manifolds are Riemannian manifolds with sectional curvature that are constantly zero.

definition

A flat manifold is a complete Riemannian manifold with constant intersection curvature . (A Riemannian metric with constant cutting curvature is called a flat metric . A flat manifold is therefore a manifold with a complete flat metric.) ${\displaystyle 0}$ ${\displaystyle 0}$

Other characterizations

There are two other ways to define the concept of the flat manifold. So it is determined

• a -dimensional flat manifold is a Riemannian manifold whose universal superposition is isometric to Euclidean space (that is, the one with the Euclidean metric ). ${\displaystyle n}$ ${\displaystyle \mathbb {E} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle dx_{1}^{2}+\ldots +dx_{n}^{2}}$

• a flat manifold is a Riemannian manifold of form , being a discrete subgroup of the group of isometries of Euclidean space. ${\displaystyle \Gamma \backslash \mathbb {E} ^{n}}$ ${\displaystyle \Gamma \subset Isom(\mathbb {E} ^{n})\cong O(n)\ltimes \mathbb {R} ^{n}}$

These two definitions are equivalent to each other and to the definition in the section above. The equivalence between the original definition and the first definition in this section follows from Cartan's theorem ; the equivalence of the two definitions from this section results from the superposition theory .

In particular, a simply connected flat manifold is isometric to Euclidean space.

Bieberbach groups

If is a flat manifold then it must be torsion free. The group is then isomorphic to the fundamental group of . ${\displaystyle M=\Gamma \backslash \mathbb {E} ^{n}}$ ${\displaystyle \Gamma \subset \operatorname {Isom} (\mathbb {E} ^{n})}$ ${\displaystyle \Gamma }$ ${\displaystyle M}$

If in addition is compact, then a crystallographic group is of rank , a so-called space group . Because it has to be torsion-free, it is then a Bieberbach group . ${\displaystyle M}$ ${\displaystyle \Gamma }$ ${\displaystyle n}$ ${\displaystyle \Gamma }$

According to Bieberbach's 1st theorem, there is a subgroup of finite index with . The quotient is called the holonomy group of the flat manifold. ${\displaystyle A\subset \Gamma }$ ${\displaystyle A\cong \mathbb {Z} ^{n}}$ ${\displaystyle H:=\Gamma /A}$

Examples

From the Chern-Gauss-Bonnet theorem it follows that the Euler characteristic of a flat manifold must always be zero.

Two-dimensional examples

Every two-dimensional compact flat manifold is homeomorphic to the torus or Klein's bottle .

Three-dimensional examples

Except for homeomorphy, there are ten compact flat 3-manifolds , six of which are orientable and four are non-orientable. The six orientable examples have the holonomy groups (the 3-torus), for and (the Hantzsche-Wendt manifold). ${\displaystyle H=1}$ ${\displaystyle H=\mathbb {Z} /k\mathbb {Z} }$ ${\displaystyle k=2,3,4,6}$ ${\displaystyle H=\mathbb {Z} /2\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z} }$

Generalized Hantzsche-Wendt manifolds

A -dimensional compact flat manifold is called a generalized Hantzsche-Wendt manifold if the holonomy group is isomorphic to . ${\displaystyle n}$ ${\displaystyle H}$ ${\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n-1}}$

Web links

• Hantzsche-Wendt flat manifolds (PDF; 456 kB)

literature

• Wolf, Joseph A .: Spaces of constant curvature. Sixth edition. AMS Chelsea Publishing, Providence, RI, 2011. ISBN 978-0-8218-5282-8

swell

1. ↑ Hantzsche-Wendt: "Three-dimensional Euclidean spatial forms"