New examples of compact cosymplectic solvmanifolds

J.C. Marrero, E. Padron

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