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# Decomposition of flat manifolds

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Let M be a compact flat Riemannian manifold of dimension n, and Γ its fundamental group. Then we have the following exact sequence (see [1])

where Zn is a maximal abelian subgroup of Γ and G is a finite group isomorphic to the holonomy group of M. We shall call Γ a Bieberbach group. Let T be a flat torus, and let Ggr act via isometries on T; then ┌ acts isometrically on × T where is the universal covering of M and yields a flat Riemannian structure on ( × T)/Γ. A flat-toral extension (see [9, p. 371]) of the Riemannian manifold M is any Riemannian manifold isometric to ( × T)/Γ where T is a flat torus on which Γ acts via isometries. It is convenient to adopt the convention that a single point is a 0-dimensional flat torus. If this is done, M is itself among the flat toral extensions of M. Roughly speaking, this is a way of putting together a compact flat manifold and a flat torus to make a new flat manifold the dimension of which is the sum of the dimensions of its constituents. It is, more precisely, a fibre bundle over the flat manifold with a flat torus as fibre.

## References

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1.Charlap, L. S.. Bieherbach Groups and Flat Manifolds (Springer-Verlag, New York, 1986).
2.Cliff, G. and Weiss, A.. Torsion free space groups and permutation lattices for finite groups. Contemporary Mathem., 93 (1989), 123132.
3.Curtis, C. W. and Reiner, I.. Methods of Representation Theory (John Wiley, 1987).
4.Hamrick, G. C. and Royster, D. C.. Flat Riemannian manifolds are boundaries. Invent. Math., 66 (1982), 405413.
5.Hiller, H. and Sah, C. H.. Holonomy of flat manifolds with b 1=0, I. Quarterly J. Math. Oxford, 37 (1986), 177187.
6.Kargapolov, M. I. and Merzljakov, Ju. I.. Fundamentals of the Theory of Groups (Springer-Verlag, 1979).
7.Plesken, W.. Some applications of representation theory. In Representation theory of finite groups and finite dimensional algebras. Progress in Mathematics, 95 (Birkhauser, Basel, 1991).
8.Symonds, P.. Localization in the classification of flat manifolds. Pac. J. of Math., 127 (1987), 389399.
9.Vasquez, A. T.. Flat Riemannian manifolds. J. Diff. Geom., 4 (1970), 367382.
10.Yau, S. T.. Compact flat Riemannian manifolds, J. Diff. Geom., 6 (1972), 395402.
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# Decomposition of flat manifolds

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