# DEFORMATION SPACES OF 3-DIMENSIONAL FLAT MANIFOLDS

Kang, Eun-Sook;Kim, Ju-Young

• Published : 2003.01.01
• 41 8

#### Abstract

The deformation spaces of the six orientable 3-dimensional flat Riemannian manifolds are studies. It is proved that the Teichmuller spaces are homeomorphic to the Euclidean spaces. To state more precisely, let $\Phi$ denote the holonomy group of the manifold. Then the Teichmuller space is homeomorphic to (1) ${\mathbb{R}}^6\;if\;\Phi$ is trivial, (2) ${\mathbb{R}}^4\;if\;\Phi$ is cyclic with order two, (3) ${\mathbb{R}}^2\;if\;\Phi$ is cyclic of order 3, 4 or 6, and (4) ${\mathbb{R}}^3\;if\;\Phi\;\cong\;{\mathbb{Z}_2}\;\times\;{\mathbb{Z}_2}$.

#### Keywords

Weil space;Teichmuller space;Bieberbach group

#### References

1. Bieberbach Groups and Flat Manifolds L. S. Charlap
2. Crystal-lographic groups of four-dimensional space H. Brown;R. Bulow;J. Neubuser;H. Wondratschek;H. Zassenhaus
3. Proceedings Conf. Geometry and Topology, Univ. of Maryland 1983-1984, Springer Lecture Notes in Mathematics v.1167 Deformation Space for Seifert Manifolds R. Kulkarni;K. B. Lee;F. Raymond;J. Alexander(ed.);J. Harer(ed.)
4. Spaces of Constant Curvature J. Wolf

#### Cited by

1. Teichmüller theory and collapse of flat manifolds vol.197, pp.4, 2018, https://doi.org/10.1007/s10231-017-0723-7