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Examples of compact almost contact manifolds admitting neither Sasakian nor cosymplectic structures. (English) Zbl 0616.53032

The paper starts with the explicit construction of two families of compact odd dimensional manifolds, denoted by M(1,r) and M(r,1), and realized as quotients of some generalized Heisenberg groups. The authors describe the cohomology rings of these manifolds and then, by using various topological obstructions to the existence of a Sasakian or a cosymplectic structure on a compact odd dimensional manifold, the following nonexistence results are proved:
(1) M(1,r) for r even or \(r=4s+1\), \(s\geq 1\), and \(M(1,1)\times T^{2r}\) for \(r\geq 2\), can have no Sasakian structures, where \(T^{2r}\) is the torus of dimension 2r;
(2) M(1,r), M(r,1) and \(M(1,1)\times T^{2r}\) can have no cosymplectic structures.
In spite of the results (1) and (2) the authors establish the existence of some interesting almost contact metric structures. For instance, they construct a Sasakian structure on M(r,1) and almost cosymplectic structures on M(1,r) and \(M(1,1)\times T^{2r}\).
Reviewer: M.Martin

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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