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Compact flat manifolds with holonomy group \(\mathbb Z_2\oplus\mathbb Z_2\). II. (English) Zbl 0970.53029

The classification of compact flat manifolds (up to affine equivalence) is equivalent, after Bieberbach’s results, to the classification of their fundamental groups (up to isomorphism). In this paper the authors give a complete classification of compact flat manifolds with holonomy group \({\mathbb Z}_2\oplus{\mathbb Z}_2\), with the property that the holonomy representation is a direct sum of \({\mathbb Z}_2\oplus{\mathbb Z}_2\)-indecomposable representations of \({\mathbb Z}\)-rank equal to 1 or 2. The compact flat manifolds in this class which have first Betti number zero are exactly those considered in [the authors, Proc. Am. Math. Soc. 124, No. 8, 2491–2499 (1996; Zbl 0864.53027)].

MSC:

53C29 Issues of holonomy in differential geometry
57S30 Discontinuous groups of transformations
53C20 Global Riemannian geometry, including pinching
57M05 Fundamental group, presentations, free differential calculus

Citations:

Zbl 0864.53027
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References:

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