We study warped product pseudo-slant submanifolds of a nearly cosymplectic manifold. We obtain some characterization results on the existence or nonexistence of warped product pseudo-slant submanifolds of a nearly cosymplectic manifold in terms of the canonical structures P and F.

1. Introduction

To study the manifolds with negative curvature, Bishop and O'Neill [1] introduced the notion of warped product manifolds by homothetically warping the product metric of a product manifold N1×N2 onto the fibers p×N2 for each p∈N1. Later on, the geometrical aspect of these manifolds has been studied by many researchers (cf. [2–4]). Pseudo-slant submanifolds were introduced by Carriazo [5] as a special case of bislant submanifolds.

Almost contact manifolds with Killing structure tensors were defined in [6] as nearly cosymplectic manifolds, and it was shown that the normal nearly cosymplectic manifolds are cosymplectic (see also [7]). Later on, Blair and Showers [8] studied nearly cosymplectic structure (ϕ,ξ,η,g) on a Riemannian manifold M¯ with η closed from the topological viewpoint.

Recently, Sahin [9] studied the warped product hemislant (pseudo-slant) submanifolds of Kaehler manifolds. He proved that the warped product submanifolds of the type M=N⊥×fNθ of a Kaehler manifold M¯ do not exist and obtained some characterization results on the existence of warped product submanifold M=Nθ×fN⊥, where N⊥ and Nθ are totally real and proper slant submanifolds of a Kaehler manifold M¯, respectively. After that, we have extended this study to the more general setting of nearly Kaehler manifolds [4]. The warped product semi-invariant submanifolds of a nearly cosymplectic manifold had been studied in [10].

In this paper, we study warped product pseudo-slant submanifolds of a nearly cosymplectic manifold. We obtain some characterization results of warped product submanifolds of the types N⊥×fNθ and Nθ×fN⊥ in terms of the canonical structures P and F, where N⊥ and Nθ are anti-invariant and proper slant submanifolds of a nearly cosymplectic manifold M¯, respectively.

2. Preliminaries

A (2n+1)-dimensional C∞ manifold M¯ is said to have an almost contact structure if there exist on M¯ a tensor field ϕ of type (1,1), a vector field ξ, and a 1-form η satisfying [8]
(2.1)ϕ2=-I+η⊗ξ,ϕξ=0,η∘ϕ=0,η(ξ)=1.
There always exists a Riemannian metric g on an almost contact manifold M¯ satisfying the following compatibility condition:
(2.2)η(X)=g(X,ξ),g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),
where X and Y are vector fields on M¯ [8].

An almost contact structure (ϕ,ξ,η) is said to be normal if the almost complex structure J on the product manifold M¯×ℝ given by
(2.3)J(X,fddt)=(ϕX-fξ,η(X)ddt),
where f is a C∞-function on M¯×ℝ has no torsion, that is, J is integrable, the condition for normality in terms of ϕ,ξ and η is [ϕ,ϕ]+2dη⊗ξ=0 on M¯, where [ϕ,ϕ] is the Nijenhuis tensor of ϕ. Finally the fundamental 2-form Φ is defined by Φ(X,Y)=g(X,ϕY).

An almost contact metric structure (ϕ,ξ,η,g) is said to be cosymplectic, if it is normal and both Φ and η are closed [8]. The structure is said to be nearly cosymplectic if ϕ is Killing, that is, if
(2.4)(∇¯Xϕ)Y+(∇¯Yϕ)X=0,
for any X,Y∈TM¯, where TM¯ is the tangent bundle of M¯ and ∇¯ denotes the Riemannian connection of the metric g. Equation (2.4) is equivalent to (∇¯Xϕ)X=0, for each X∈TM¯. The structure is said to be closely cosymplectic if ϕ is Killing and η is closed. It is well known that an almost contact metric manifold is cosymplectic if and only if ∇¯ϕ vanishes identically, that is, (∇¯Xϕ)Y=0 and ∇¯Xξ=0.

Proposition 2.1 (see [<xref ref-type="bibr" rid="B4">8</xref>]).

On a nearly cosymplectic manifold the vector field ξ is Killing.

From the above proposition, one has ∇¯Xξ=0, for any vector field X tangent to M¯, where M¯ is a nearly cosymplectic manifold.

Let M be submanifold of an almost contact metric manifold M¯ with induced metric g and if ∇ and ∇⊥ are the induced connections on the tangent bundle TM and the normal bundle T⊥M of M, respectively, then Gauss and Weingarten formulae are given by
(2.5)∇¯XY=∇XY+h(X,Y),(2.6)∇¯XN=-ANX+∇X⊥N,
for each X,Y∈TM and N∈T⊥M, where h and AN are the second fundamental form and the shape operator (corresponding to the normal vector field N), respectively, for the immersion of M into M¯. They are related as
(2.7)g(h(X,Y),N)=g(ANX,Y),
where g denotes the Riemannian metric on M¯ as well as induced on M.

For any X∈TM, one writes
(2.8)ϕX=PX+FX,
where PX is the tangential component and FX is the normal component of ϕX.

Similarly for any N∈T⊥M, one writes
(2.9)ϕN=BN+CN,
where BN is the tangential component and CN is the normal component of ϕN.

Now, denote by 𝒫XY and 𝒬XY the tangential and normal parts of (∇¯Xϕ)Y, that is,
(2.10)(∇¯Xϕ)Y=PXY+QXY
for all X,Y∈TM. Making use of (2.8), (2.10), and the Gauss and Weingarten formulae, the following equations may easily be obtained:
(2.11)PXY=(∇¯XP)Y-AFYX-Bh(X,Y),𝒬XY=(∇¯XF)Y+h(X,PY)-Ch(X,Y).
Similarly, for any N∈T⊥M, denoting tangential and normal parts of (∇¯Xϕ)N by 𝒫XN and 𝒬XN, respectively, one obtains
(2.12)PXN=(∇¯XB)N+PANX-ACNX,𝒬XN=(∇¯XC)N+h(BN,X)+FANX,
where the covariant derivatives of P,F,B, and C are defined by
(2.13)(∇¯XP)Y=∇XPY-P∇XY,(2.14)(∇¯XF)Y=∇X⊥FY-F∇XY,(2.15)(∇¯XB)N=∇XBN-B∇X⊥N,(2.16)(∇¯XC)N=∇X⊥CN-C∇X⊥N,
for all X,Y∈TM and N∈T⊥M.

It is straightforward to verify the following properties of 𝒫 and 𝒬, which one enlists here for later use

(i) 𝒫X+YW=𝒫XW+𝒫YW, (ii) 𝒬X+YW=𝒬XW+𝒬YW,

(i) 𝒫X(Y+W)=𝒫XY+𝒫XW, (ii) 𝒬X(Y+W)=𝒬XY+𝒬XW,

(i) g(𝒫XY,W)=-g(Y,𝒫XW), (ii) g(𝒬XY,N)=-g(Y,𝒫XN),

𝒫XϕY+𝒬XϕY=-ϕ(𝒫XY+𝒬XY),

for all X,Y,W∈TM and N∈T⊥M.

On a submanifold M of a nearly cosymplectic manifold, by (2.4) and (2.10), one has
(2.17)(a)PXY+PYX=0,(b)QXY+QYX=0,

for any X,Y∈TM.

The submanifold M is said to be invariant if F is identically zero, that is, ϕX∈TM for any X∈TM. On the other hand, M is said to be anti-invariant if P is identically zero, that is, ϕX∈T⊥M, for any X∈TM.

One will always consider ξ to be tangent to the submanifold M. There is another class of submanifolds that is called the slant submanifold. For each nonzero vector X tangent to M at any x∈M, such that X is not proportional to ξx, one denotes by 0≤θ(X)≤π/2, the angle between ϕX and TxM is called the slant angle. If the slant angle θ(X) is constant for all X∈TxM-〈ξx〉 and x∈M, then M is said to be a slant submanifold [11]. Obviously, if θ=0, then M is an invariant submanifold and if θ = π/2, then M is an anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant.

One recalls the following result for a slant submanifold.

Theorem 2.2 (see [<xref ref-type="bibr" rid="B5">11</xref>]).

Let M be a submanifold of an almost contact metric manifold M¯, such that ξ∈TM. Then M is slant if and only if there exists a constant λ∈[0,1] such that
(2.18)P2=λ(-I+η⊗ξ).

Furthermore, if θ is slant angle, then λ=cos2θ.

The following relations are straightforward consequence of (2.18):
(2.19)g(PX,PY)=cos2θ(g(X,Y)-η(Y)η(X)),(2.20)g(FX,FY)=sin2θ(g(X,Y)-η(Y)η(X)),
for all X,Y∈TM.

A submanifold M of an almost contact manifold M¯ is said to be a pseudo-slant submanifold if there exist two orthogonal complementary distributions D1 and D2 satisfying:

TM=D1⊕D2⊕〈ξ〉,

D1 is a slant distribution with slant angle θ≠π/2,

D2 is anti-invariant that is, ϕD2⊆T⊥M.

A pseudo-slant submanifold M of an almost contact manifold M¯ is mixed geodesic if
(2.21)h(X,Z)=0,
for any X∈D1 and Z∈D2.

If μ is the invariant subspace of the normal bundle T⊥M, then in the case of pseudo-slant submanifold, the normal bundle T⊥M can be decomposed as follows:
(2.22)T⊥M=FD1⊕FD2⊕μ.

3. Warped Product Pseudo-Slant Submanifolds

Bishop and O'Neill [1] introduced the notion of warped product manifolds. These manifolds are the natural generalizations of Riemannian product manifolds. They defined these manifolds as follows Let (N1,g1) and (N2,g2) be two Riemannian manifolds and f, a positive differentiable function on N1. The warped product of N1 and N2 is the Riemannian manifold N1×fN2=(N1×N2,g), where
(3.1)g=g1+f2g2.
A warped product manifold N1×fN2 is said to be trivial if the warping function f is constant. We recall the following general formula on a warped product manifold [1]:
(3.2)∇XZ=∇ZX=(Xlnf)Z,
where X is tangential to N1 and Z is tangential to N2.

Let M=N1×fN2 be a warped product manifold. This means that N1 is totally geodesic and N2 is a totally umbilical submanifold of M, respectively [1].

Throughout this section, we consider the warped product pseudo-slant submanifolds which are either in the form N⊥×fNθ or Nθ×fN⊥ in a nearly cosymplectic manifold M¯, where Nθ and N⊥ are proper slant and anti-invariant submanifolds of a nearly cosymplectic manifold M¯, respectively. On a warped product submanifold M=N1×fN2 of a nearly cosymplectic manifold M¯, we have the following result.

Theorem 3.1 (see [<xref ref-type="bibr" rid="B12">10</xref>]).

A warped product submanifold M=N1×fN2 of a nearly cosymplectic manifold M¯ is an usual Riemannian product if the structure vector field ξ is tangential to M2, where M1 and M2 are the Riemannian submanifolds of M¯.

Now, one considers the warped product pseudo-slant submanifolds in the form M=N⊥×fNθ of a nearly cosymplectic manifold M¯. If one considers the structure vector field ξ∈TNθ then by Theorem 3.1, the warping function f is constant and hence one will considers ξ∈TN⊥.

Proposition 3.2.

Let M=N⊥×fNθ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M¯. Then,
(3.3)g(∇PX⊥FPX-∇X⊥FX,FZ)=(Zlnf)sin2θ‖X‖2+(1+cos2θ)g(h(X,PX),FZ),
for any X∈TNθ and Z∈TN⊥, where Nθ and N⊥ are proper slant and anti-invariant submanifolds of M¯, respectively.

Proof.

For any X∈TNθ and Z∈TN⊥, by (2.8), we have
(3.4)g(∇¯XϕX,FZ)=g(∇¯XPX,FZ)+g(∇¯XFX,FZ).
Using (2.5), (2.6), and the covariant derivative property of ϕ, we obtain
(3.5)g((∇¯Xϕ)X,FZ)+g(ϕ∇¯XX,ϕZ)=g(h(X,PX),FZ)+g(∇X⊥FX,FZ).
Then from (2.2), (2.4), and the fact that ξ is a Killing vector field on M¯, thus we obtain
(3.6)g(∇¯XX,Z)=g(h(X,PX),FZ)+g(∇X⊥FX,FZ).
Using the property of ∇¯, we get
(3.7)-g(X,∇¯XZ)=g(h(X,PX),FZ)+g(∇X⊥FX,FZ).
Then by (2.5) and (3.2), we derive
(3.8)-(Zlnf)‖X‖2=g(h(PX,X),FZ)+g(∇X⊥FX,FZ).
Interchanging X by PX in (3.8) and using (2.18), (2.19), and the fact that ξ∈TN⊥, we obtain
(3.9)-(Zlnf)cos2θ‖X‖2=-cos2θg(h(X,PX),FZ)+g(∇PX⊥FPX,FZ).
Thus, the result follows from (3.8) and (3.9).

Proposition 3.3.

Let M=N⊥×fNθ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M¯. Then,
(3.10)g((∇¯XF)X,FZ)=sec2θg((∇¯PXF)PX,FZ)
for any X∈TNθ and Z∈TN⊥, where Nθ and N⊥ are proper slant and anti-invariant submanifolds of M¯, respectively.

Proof.

For any X∈TNθ and Z∈TN⊥ by (2.14), we have
(3.11)g(∇X⊥FX,FZ)=g((∇¯XF)X,FZ)+g(F∇XX,FZ).
Using (2.20), (2.5), and the fact that ξ is killing vector field, we obtain
(3.12)g(∇X⊥FX,FZ)=g((∇¯XF)X,FZ)-sin2θg(X,∇XZ).
Then from (3.2), we derive
(3.13)g(∇X⊥FX,FZ)=g((∇¯XF)X,FZ)-(Zlnf)sin2θ‖X‖2.
Now, from (3.8) and (3.13), we obtain
(3.14)g((∇¯XF)X,FZ)=-(Zlnf)cos2θ‖X‖2-g(h(X,PX),FZ).
Interchanging X by PX in (3.14) and then using (2.18), (2.19), and the fact that ξ∈TN⊥, we get
(3.15)g((∇¯PXF)PX,FZ)=-(Zlnf)cos4θ‖X‖2-cos2θg(h(X,PX),FZ).
From (3.14) and (3.15), we arrive at
(3.16)g((∇¯XF)X,FZ)=sec2θg((∇¯PXF)PX,FZ).
Hence, the result is proved.

Lemma 3.4.

Let M=N⊥×fNθ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M¯. Then,
(3.17)g(PXPX,Z)=g(h(X,Z),FPX)-g(h(PX,Z),FX)
for any X∈TNθ and Z∈TN⊥, where Nθ and N⊥ are proper slant and anti-invariant submanifolds of M¯, respectively.

Proof.

For any X∈TNθ and Z∈TN⊥ by (2.5), we have
(3.18)g(h(PX,Z),FX)=g(∇¯ZPX,FX)=-g(PX,∇¯ZFX).
Then from (2.8), we derive
(3.19)g(h(PX,Z),FX)=g(PX,∇¯ZPX)-g(PX,∇¯ZϕX).
From the covariant derivative property of ϕ and (2.5), we obtain
(3.20)g(h(PX,Z),FX)=g(PX,∇ZPX)-g(PX,(∇¯Zϕ)X)-g(PX,ϕ∇¯ZX).
By (2.2), (2.10), and (3.2), we derive
(3.21)g(h(PX,Z),FX)=(Zlnf)g(PX,PX)-g(PX,PZX)+g(ϕPX,∇¯ZX).
Using (2.5), (2.8), (2.17)(a), (2.19) and the fact that ξ∈TN⊥, we get
(3.22)g(h(PX,Z),FX)=(Zlnf)cos2θ‖X‖2+g(PX,PXZ)+g(P2X,∇ZX)+g(h(X,Z),FPX).
Thus, by property (p3)(i), (2.18), and (3.2) and the fact that ξ∈TN⊥, we obtain
(3.23)g(h(PX,Z),FX)=(Zlnf)cos2θ‖X‖2-g(PXPX,Z)-(Zlnf)cos2θ‖X‖2+g(h(X,Z),FPX).
Hence, the above equation takes the form
(3.24)g(PXPX,Z)=g(h(X,Z),FPX)-g(h(PX,Z),FX),
which proves our assertion.

Theorem 3.5.

Let M=N⊥×fNθ be a warped product submanifold of a nearly cosymplectic manifold M¯. Then M is Riemannian product of N⊥ and Nθ if and only if 𝒫XTX∈TNθ, for any X∈TNθ, where Nθ and N⊥ are proper slant and anti-invariant submanifolds of M¯, respectively.

Proof.

If the structure vector field ξ∈TNθ, then, by Theorem 3.1, M is Riemannian product of N⊥ and Nθ. Now, we consider ξ∈TN⊥, then for any X∈TNθ and Z∈TN⊥ from (2.5), we have
(3.25)g(h(X,PX),FZ)=g(∇¯PXX,ϕZ).
Then by (2.2), we get
(3.26)g(h(X,PX),FZ)=-g(ϕ∇¯PXX,Z).
Using the covariant derivative formula of ϕ, we derive
(3.27)g(h(X,PX),FZ)=g((∇¯PXϕ)X,Z)-g(∇¯PXϕX,Z).
Then from (2.10) and the property of ∇¯, we obtain
(3.28)g(h(X,PX),FZ)=g(PPXX,Z)+g(ϕX,∇¯PXZ).
Thus by (2.5), (2.8), and (2.17)(a), we arrive at
(3.29)g(h(X,PX),FZ)=-g(PXPX,Z)+g(PX,∇PXZ)+g(h(PX,Z),FX).
Using (3.2) and then (2.19) and the fact that ξ∈TN⊥, we get
(3.30)g(h(X,PX),FZ)=-g(PXPX,Z)+(Zlnf)cos2θ‖X‖2+g(h(PX,Z),FX).
By property (p3)(i), we derive
(3.31)g(h(X,PX),FZ)=g(PX,PXZ)+(Zlnf)cos2θ‖X‖2+g(h(PX,Z),FX).
Interchanging X by PX in (3.30) and then using (2.18), (2.19), and the fact that ξ∈TN⊥, we obtain
(3.32)-cos2θg(h(X,PX),FZ)=-cos2θg(X,PPXZ)+(Zlnf)cos4θ‖X‖2-cos2θg(h(X,Z),FPX).
Using the property (p3)(i) and then (2.17)(a), we arrive at
(3.33)-g(h(X,PX),FZ)=-g(PXPX,Z)+(Zlnf)cos2θ‖X‖2-g(h(X,Z),FPX).

Then from (3.30) and (3.33), we obtain
(3.34)2(Zlnf)cos2θ‖X‖2=2g(PXPX,Z)+g(h(X,Z),FPX)-g(h(PX,Z),FX).

Thus, by Lemma 3.4, we conclude that
(3.35)(Zlnf)cos2θ‖X‖2=32g(PXPX,Z).
Since Nθ is proper slant, thus we get (Zlnf)=0, if and only if 𝒫XPX lies in TNθ for all X∈TNθ and Z∈TN⊥. This proves the theorem completely.

Now, we discuss the other case, that is, the warped product submanifold M=Nθ×fN⊥ of a nearly cosymplectic manifold M¯. In this case also, if the structure vector filed ξ∈TN⊥ then the warping function f is constant (by Theorem 3.1), thus we consider ξ∈TNθ.

Proposition 3.6.

Let M=Nθ×fN⊥ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M¯. Then,
(3.36)g((∇¯XF)Z,FX)+g((∇¯PXF)Z,FPX)=sin2θg(h(X,PX),FZ)+(1+cos2θ)g(PXZ,PX)-cos2θη(X)g(PξZ,PX)-g(QZX,FX)-g(QZPX,FPX)
for any X∈TNθ and Z∈TN⊥, where Nθ and N⊥ are proper slant and anti-invariant submanifolds of M¯, respectively.

Proof.

For any X∈TNθ and Z∈TN⊥, by (2.2) we have
(3.37)g(ϕ∇¯XZ,ϕX)=g(∇¯XZ,X)-η(X)g(∇¯XZ,ξ).
Using the property of the connection ∇¯ and the fact that ξ is a Killing vector field, then, from (2.5), we obtain
(3.38)g(ϕ∇¯XZ,ϕX)=g(∇XZ,X).
Thus by (3.2) and the covariant derivative formula of ϕ, we derive
(3.39)g(∇¯XϕZ,ϕX)-g((∇¯Xϕ)Z,ϕX)=(Xlnf)g(Z,X).
Then form (2.6), (2.8), (2.10), and by the orthogonality of two distributions, we get
(3.40)-g(AFZX,PX)+g(∇X⊥FZ,FX)-g(PXZ,PX)-g(QXZ,FX)=0.
Thus, on using (2.7) and (2.17)(b), the above equation takes the form
(3.41)g(∇X⊥FZ,FX)=g(h(X,PX),FZ)+g(PXZ,PX)-g(QZX,FX).
Now, for any X∈TNθ and Z∈TN⊥ from (2.14), we have
(3.42)g(∇X⊥FZ,FX)=g((∇¯XF)Z,FX)+g(F∇XZ,FX).
Using (3.2), we obtain
(3.43)g(∇X⊥FZ,FX)=g((∇¯XF)Z,FX)+(Xlnf)g(FZ,FX).
By orthogonality of two normal distributions, we get
(3.44)g(∇X⊥FZ,FX)=g((∇¯XF)Z,FX).
Then, from (3.41) and (3.44), we obtain
(3.45)g((∇¯XF)Z,FX)=g(h(X,PX),FZ)+g(PXZ,PX)-g(QZX,FX).
Interchanging X by PX in (3.45) and using (2.18) and the fact that h(X,ξ)=0, for any X on a nearly cosymplectic manifold M¯, hence we get
(3.46)g((∇¯PXF)Z,FPX)=-cos2θg(h(X,PX),FZ)-cos2θg(PPXZ,X)+cos2θη(X)g(PPXZ,ξ)-g(QZPX,FPX).
Using property (p3)(i) and (2.17), we derive
(3.47)g((∇¯PXF)Z,FPX)=-cos2θg(h(X,PX),FZ)-cos2θg(PXPX,Z)+cos2θη(X)g(PξPX,Z)-g(QZPX,FPX).
Again, by property (p3)(i), we obtain
(3.48)g((∇¯PXF)Z,FPX)=-cos2θg(h(X,PX),FZ)+cos2θg(PXZ,PX)-cos2θη(X)g(PξZ,PX)-g(QZPX,FPX).
Thus, the result follows from (3.45) and (3.48).

Theorem 3.7.

Let M=Nθ×fN⊥ be a warped product submanifold of a nearly cosymplectic manifold M¯. Then M is Riemannian product of Nθ and N⊥ if and only if
(3.49)g(h(X,Z),FZ)=g(h(Z,Z),FX),
for any X∈TNθ and Z∈TN⊥, where Nθ and N⊥ are proper slant and anti-invariant submanifolds of M¯, respectively.

Proof.

If ξ∈TN⊥, then by Theorem 3.1, f is constant on M. Now, we consider ξ∈TNθ. In this case, for any X∈TNθ and Z∈TN⊥ by (2.5), we have
(3.50)g(h(PX,Z),FZ)=g(∇¯ZPX,ϕZ).
Using (2.2), we get
(3.51)g(h(PX,Z),FZ)=-g(ϕ∇¯ZPX,Z).
Thus, on using the covariant derivative property of ϕ, we obtain
(3.52)g(h(PX,Z),FZ)=g((∇¯Zϕ)PX,Z)-g(∇¯ZϕPX,Z).
Then from (2.8) and (2.10), we get
(3.53)g(h(PX,Z),FZ)=g(PZPX,Z)-g(∇¯ZP2X,Z)-g(∇¯ZFPX,Z).
Using property (p3)(i) and the property of the connection ∇¯, we derive
(3.54)g(h(PX,Z),FZ)=-g(PZZ,PX)+g(P2X,∇ZZ)+g(FPX∇¯ZZ).
As we have 𝒫ZZ=0 from (2.4) and (2.10), then by (2.18) the above equation reduced to
(3.55)g(h(PX,Z),FZ)=-cos2θg(X,∇¯ZZ)+cos2θη(X)g(ξ,∇¯ZZ)+g(h(Z,Z),FPX).
Since ξ is a Killing vector field on M¯, then by (2.5), (3.2), and the property of the connection ∇¯, the above equation takes the form
(3.56)g(h(PX,Z),FZ)=(Xlnf)cos2θ‖Z‖2+g(h(Z,Z),FPX).
Interchanging X by PX in (3.56) and using (2.18), we obtain
(3.57)cos2θg(h(X,Z),FZ)+cos2θη(X)g(h(Z,ξ),FZ)=-(PXlnf)cos2θ‖Z‖2+cos2θg(h(Z,Z),FX).
Since h(Z,ξ)=0, for nearly cosymplectic, then the above equation reduces to
(3.58)(PXlnf)‖Z‖2=g(h(Z,Z),FX)-g(h(X,Z),FZ).
Thus, from (3.58), we obtain (PXlnf)=0 if and only if g(h(Z,Z),FX)=g(h(X,Z),FZ). This proves the theorem completely.

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