Almostα-paracosymplectic manifolds

Abstract

MSC:primary 53B30 53C15 53C25 secondary 53D10 Keywords:Almost paracontact metric manifold Almost paracosymplectic manifold Almost para-Kenmotsu manifold Para-Kaehler manifold a b s t r a c t This paper is a complete study of almost α-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 3, almost α-paracosymplectic manifolds are locally conformal to almost paracosymplectic manifolds and locally D-homothetic to almost para-Kenmotsu manifolds. Furthermore, it is proved that characteristic (Reeb) vector field ξ is harmonic on almost α-para-Kenmotsu manifold if and only if it is an eigenvector of the Ricci operator. It is showed that almost α-para-Kenmotsu (κ, µ, ν)-space has para-Kaehler leaves. 3-dimensional almost α-paraKenmotsu manifolds are classified. As an application, it is obtained that 3-dimensional almost α-para-Kenmotsu manifold is (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.

On almost α-para-Kenmotsu manifolds satisfyıng certain conditions

“…where is a function. These manifolds are called almost -paracosymplectic [8]. They contain properly almost paracosymplectic, = 0, and almost -para-Kenmotsu, = const: 6 = 0 manifolds.…”

“…An almost paracontact metric manifold M 2n+1 , with a structure ( ; ; ; g) is said to be an almost -paracosymplectic manifold [8], if…”

“…i )g(lY; e i )] for all vector …elds X; Y on M . From the last equation, one can easily get S(Y; ) = (Y )tr(l) (5.3) On the other hand, (5.3) can be written as Q = tr(l) :So, this ends the proof.Using Theorem 9 and Theorem 13 of[8], we can give following corollary.…”

“…[8]Let (M 2n+1 ; ; ; ; g) be an almost -para-Kenmotsu manifold. Then, for any X; Y 2 (M 2n+1 );R(X; Y ) = (X)( Y + hY ) (Y )( X + hX)+(r X h)Y (r Y h)X: (4.1)Theorem 3.…”

Geometry of $${\mathcal{P}\mathcal{R}}$$ P R -semi-invariant warped product submanifolds in paracosymplectic manifold

“…From ( • an almost paracosymplectic if the forms η and Φ are closed, i.e., dη = 0 and dΦ = 0 (see [8,10]…”