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New examples of compact cosymplectic solvmanifolds

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J.C. Marrero, E. Padron
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** Address.**
Depto. Matematica Fundamental,
Facultad de Matematicas,
Universidad de la Laguna,
Tenerife,
Canary Islands, SPAIN

** E-mail:**
jcmarrer@ull.es, mepadron@ull.es

**Abstract.**
In this paper we present new examples of
$(2n+1)$-dimensional compact cosymplectic
manifolds which are not
topologically equivalent to the canonical examples,
i.e., to the pro\-duct of the $(2m+1)$-dimensional real torus and the
$r$-dimensional complex projective space, with $m,r\geq
0$ and $m+r=n.$ These new examples are compact solvmanifolds and they are
constructed as suspensions with fibre the $2n$-dimensional real torus.
In the particular case $n=1,$ using the examples obtained, we
conclude that a $3$-dimensional compact flat orientable Riemannian
manifold with non-zero first Betti number admits a cosymplectic
structure. Furthermore, if the first Betti number is equal to $1$
then such a manifold is not topologically equivalent
to the global product of a compact K\"ahler manifold with the circle $S^1.$

**AMSclassification.**
Primary 53C15, 53C55; Secondary 22E25

**Keywords.**
Cosymplectic manifolds, solvmanifolds, Kahler manifolds,
suspensions, flat Riemannian manifolds