Almost αα-paracosymplectic manifolds

This paper is a complete study of almost αα-paracosymplectic manifolds. Basic properties of such manifolds are obtained and general curvature identities are proved. The manifolds with para-Kaehler leaves are characterized. It is proved that, for dimensions greater than 33, almost αα-paracosymplectic manifolds are locally conformal to almost paracosymplectic manifolds and locally DD-homothetic to almost para-Kenmotsu manifolds. Furthermore, it is proved that characteristic (Reeb) vector field ξξ is harmonic on almost αα-para-Kenmotsu manifold if and only if it is an eigenvector of the Ricci operator. It is showed that almost αα-para-Kenmotsu (κ,μ,ν)(κ,μ,ν)-space has para-Kaehler leaves. 33-dimensional almost αα-para-Kenmotsu manifolds are classified. As an application, it is obtained that 33-dimensional almost αα-para-Kenmotsu manifold is (κ,μ,ν)(κ,μ,ν)-space on an every open and dense subset of the manifold if and only if Reeb vector field is harmonic. Furthermore, examples are constructed.