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Non-existence of affine structures on Seifert fibre spaces. (Inexistence de structures affines sur les fibrés de Seifert.) (French) Zbl 0793.57006

Let \(M\) be a circle bundle over a surface of genus \(\geq 2\). We prove that \(M\) cannot be an affinely flat unimodular manifold. By passing to finite covering, the same result is true if \(M\) is a Seifert fibre space with hyperbolic base. As a corollary if furthermore the first Betti number \(b_ 1(M)\) is zero then there is no affine structure on \(M\). This is for example the case of the Brieskorn manifolds. The main question of existence of affine structures on a nontrivial circle bundle over a surface of genus \(\geq 2\) remains open.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
53C05 Connections (general theory)
57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
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