# Article

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Keywords:
cosymplectic manifolds; solvmanifolds; Kähler manifolds; suspensions; flat Riemannian manifolds
Summary:
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 Kähler manifold with the circle \$S^1.\$
References:
[1] Blair D. E.: Contact manifolds in Riemannian geometry. Lecture Notes in Math., 509, Springer-Verlag, Berlin, (1976). MR 0467588 | Zbl 0319.53026
[2] Blair D. E., Goldberg S. I.: Topology of almost contact manifolds. J. Diff. Geometry, 1, 347-354 (1967). MR 0226539 | Zbl 0163.43902
[3] Chinea D., León M. de, Marrero J. C.: Topology of cosymplectic manifolds. J. Math. Pures Appl., 72, 567-591 (1993). MR 1249410 | Zbl 0845.53025
[4] Hector G., Hirsch U.: Introduction to the Geometry of Foliations. Part A. Aspects of Math., Friedr. Vieweg and Sohn, (1981). MR 0639738 | Zbl 0486.57002
[5] León M. de, Marrero J. C.: Compact cosymplectic manifolds with transversally positive definite Ricci tensor. Rendiconti di Matematica, Serie VII, 17 Roma, 607-624 (1997). MR 1620868 | Zbl 0897.53026
[6] Wolf J. A.: Spaces of constant curvature. 5nd ed., Publish or Perish, Inc., Wilmington, Delaware, (1984). MR 0928600 | Zbl 0556.53033

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