TEICHMÜLLER SPACES OF NONORIENTABLE 3-DIMENSIONAL FLAT MANIFOLDS

The various deformation spaces associated with maximal geometric structures on closed oriented 3-manifolds was studied in [2], leaving out the geometry of $\mathbb{R}^3$. In this paper, we study the Weil spaces and Teichm$\ddot{u}$ller spaces of non-orientable 3-dimensional flat Riemannian manifolds. In particular, we find the Teichm$\ddot{u}$ller spaces are homeomorphic to the Euclidean spaces $\mathbb{R}^4$ or $\mathbb{R}^3$ depending on the holonomy group $\mathbb{Z}_2$ or $\mathbb{Z}_2{\times}\mathbb{Z}_2$ respectively.