CUBO A Mathematical Journal

Vol.19, No¯ 01, (79–87). March 2017

Three dimensional f-Kenmotsu manifold satisfying certain
curvature conditions

Venkatesha and Divyashree G.

Department of Mathematics,

Kuvempu University,

Shankaraghatta - 577 451, Shimoga, Karnataka, INDIA.

vensmath@gmail.com, gdivyashree9@gmail.com

ABSTRACT

The purpose of the present paper is to study pseudosymmetry conditions on f-Kenmotsu

manifolds.

RESUMEN

El propósito del presente art́ıculo es estudiar condiciones de pseudosimetŕıa en var-

iedades f-Kenmotsu.

Keywords and Phrases: f-Kenmotsu manifold, cyclic parallel Ricci tensor, almost pseudo Ricci

symmetry, pseudosymmetry, Ricci pseudosymmetry, Ricci generalized pseudosymmetry.

2010 AMS Mathematics Subject Classification: 47A63; 47A99.



80 Venkatesha and Divyashree G. CUBO
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1 Introduction

Let Mn be an almost contact manifold with an almost contact metric structure (φ,ξ,η, g) [1]. We

denote by Φ, the fundamental 2-form of Mn i.e., Φ(X, Y) = g(X,φY) for any vector fields X, Y ∈

χ(Mn), where χ(Mn) being the Lie algebra of differentiable vector fields on Mn. Furthermore, we

recollect the following definitions [1, 3, 8].

The manifold Mn and its structure (φ,ξ,η, g) is said to be:

i) normal if the almost complex structure defined on the product manifold Mn ×R is integrable

(equivalently, [φ,φ] + 2dη ⊗ ξ = 0),

ii) almost cosymplectic if dη = 0 and dΦ = 0,

iii) cosymplectic if it is normal and almost cosymplectic (equivalently, ∇φ = 0, where ∇ is

covariant differentiation with respect to the Levi-Civita connection).

The manifold Mn is called locally conformal almost cosymplectic (respectively, locally conformal

cosymplectic) if Mn has an open covering {Ut} endowed with differentiable functions σt : Ui −→ R

such that over each Ut the almost contact metric structure (φt,ξt,ηt, gt) defined by

φt = φ, ξt = e
σtξ, ηt = e

−σtη, gt = e
−2σtg

is almost cosymplectic (respectively, locally conformal cosymplectic).

Normal locally conformal almost cosymplectic manifold were studied by Olszak and Rosca [7].

An almost contact metric manifold is said to be f-Kenmotsu if it is normal and locally conformal

almost cosymplectic. The same type of manifold was also studied by Yildiz et al. [9] using

the projective curvature tensor. Olszak and Rosca [7] also gave a geometric interpretation of f-

Kenmotsu manifolds and studied some curvature restrictions. Among others, they proved that a

Ricci symmetric f-Kenmotsu manifold is an Einstein manifold.

Our work is structured in the following way: After introduction, we have given some basic

equations of f-Kenmotsu manifold in section 2. Section 3 deals with the study of 3-dimensional f-

Kenmotsu manifold with cyclic parallel Ricci tensor. And we study almost pseudo Ricci symmetric,

pseudosymmetric, Ricci pseudosymmetric and Ricci generalized pseudosymmetric 3-dimensional f-

Kenmotsu manifolds in sections 4, 5, 6 and 7, respectively.

2 f-Kenmotsu manifolds

Let Mn be a smooth (2n + 1)-dimensional manifold endowed with an almost contact metric struc-

ture (φ,ξ,η, g) which satisfy

φ2 = −id + η ⊗ ξ, η(ξ) = 1, η · φ = 0, (2.1)



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19, 1 (2017)

Three dimensional f-Kenmotsu manifold satisfying certain . . . 81

φξ = 0, η(X) = g(X,ξ), g(φX,φY) = g(X, Y) − η(X)η(Y), (2.2)

for any vector fields X, Y ∈ χ(Mn) where id is the identity of the tangent bundle TMn, φ is a

tensor field of type (1, 1), ξ is a vector field, η is a 1-form and g is a Riemannian metric.

We say that (Mn,φ,ξ,η, g) is an f-Kenmotsu manifold if the Levi-Civita connection ∇ of φ

satisfies the condition [6]

(∇Xφ)(Y) = f[g(φX, Y)ξ − η(Y)φX], (2.3)

where f ∈ C∞(Mn) is strictly positive and df ∧ η = 0. If f = 0, then the manifold is cosymplectic

[5]. An f-Kenmotsu manifold is called regular if f2 + f′ ̸= 0 where f′ = ξf.

In an f-Kenmotsu manifold, from (2.3) we have

∇Xξ = f[X − η(X)ξ]. (2.4)

The condition df ∧ η = 0 holds if dim Mn ≥ 5 but it does not hold if dim Mn = 3 [7].

(∇Xη)(Y) = f[g(X, Y) − η(X)η(Y)]. (2.5)

In a 3-dimensional Riemannian manifold, we have

R(X, Y)Z = g(Y, Z)QX − g(X, Z)QY + S(Y, Z)X − S(X, Z)Y (2.6)

−
r

2
{g(Y, Z)X − g(X, Z)Y}.

In a 3-dimensional f-Kenmotsu manifold, we see that [7]

R(X, Y)Z = (
r

2
+ 2f2 + 2f′)(X ∧ Y)Z − (

r

2
+ 3f2 + 3f′){η(X)(ξ ∧ Y)Z (2.7)

+η(Y)(X ∧ ξ)Z},

S(X, Y) = (
r

2
+ f2 + f′)g(X, Y) − (

r

2
+ 3f2 + 3f′)η(X)η(Y), (2.8)

where R, S, Q and r are the Riemannian curvature tensor, the Ricci tensor, the Ricci operator and

the scalar curvature, respectively.

Now from(2.7), we have the following:

R(X, Y)ξ = −(f2 + f′)[η(Y)X − η(X)Y], (2.9)

R(ξ, Y)Z = −(f2 + f′)[g(Y, Z)ξ − η(Z)Y], (2.10)

η(R(X, Y)Z) = −(f2 + f′)[g(Y, Z)η(X) − g(X, Z)η(Y)]. (2.11)

And from (2.8), we get

S(X,ξ) = −2(f2 + f′)η(X), (2.12)

and

Qξ = −2(f2 + f′)ξ. (2.13)



82 Venkatesha and Divyashree G. CUBO
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3 3-dimensional f-Kenmotsu manifold with cyclic parallel
Ricci tensor

Suppose the manifold Mn under consideration satisfies the cyclic parallel Ricci tensor condition

[4]. Then we have

(∇XS)(Y, Z) + (∇YS)(Z, X) + (∇ZS)(X, Y) = 0, (3.1)

for all X, Y, Z ∈ χ(Mn).

From the above equation, it is seen that r is constant. And we have

(∇XS)(Y, Z) + (∇YS)(Z, X) + (∇ZS)(X, Y) = −(
r

2
+ 3f2 + 3f′)[(∇Xη)(Y)η(Z) (3.2)

+η(Y)(∇Xη)(Z) + (∇Yη)(Z)η(X)

+η(Z)(∇Yη)(X) + (∇Zη)(X)η(Y)

+η(X)(∇Zη)(Y)].

From (3.1) and (3.2), we get

(
r

2
+ 3f2 + 3f′)[(∇Xη)(Y)η(Z) + η(Y)(∇Xη)(Z) + (∇Yη)(Z)η(X) (3.3)

+η(Z)(∇Yη)(X) + (∇Zη)(X)η(Y) + η(X)(∇Zη)(Y)] = 0.

Using (2.5) in (3.3), we get

(
r

2
+ 3f2 + 3f′)[g(X, Y)η(Z) + g(X, Z)η(Y) + g(Y, Z)η(X) + g(Y, X)η(Z) (3.4)

+g(Z, X)η(Y) + g(Z, Y)η(X) − 6η(X)η(Y)η(Z)] = 0, since f ̸= 0.

On substituting X = Y = ei in (3.4), where ei is an orthonormal basis of the tangent space at each

point of the manifold and taking summation over i, 1 ≤ i ≤ 3, which gives

4{
r

2
+ 3f2 + 3f′}η(Z) = 0. (3.5)

Hence, we get η(Z) = 0, which is a contradiction. Therefore, from (3.5) we have

r = −6(f2 + f′). (3.6)

Conversely, if r = −6(f2 + f′) then from (3.2), we obtain

(∇XS)(Y, Z) + (∇YS)(Z, X) + (∇ZS)(X, Y) = 0. (3.7)

From the above discussions we have the following:

Theorem 3.1. A 3-dimensional f-Kenmotsu manifold satisfies cyclic parallel Ricci tensor if and

only if the scalar curvature r = −6(f2 + f′), provided f ̸= 0.



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Three dimensional f-Kenmotsu manifold satisfying certain . . . 83

4 Almost pseudo Ricci symmetric 3-dimensional f-Kenmotsu
manifold satisfying cyclic Ricci tensor

Chaki and Kawaguchi [2] introduced the concept of almost pseudo Ricci symmetric manifolds as an

extended class of pseudo symmetric manifolds. A Riemannian manifold (Mn, g) is called an almost

pseudo Ricci symmetric manifold (APRS)n, if its Ricci tensor S of type (0, 2) is not identically zero

and satisfies the following condition

(∇US)(V, W) = [A(U) + B(U)]S(V, W) + A(V)S(U, W) + A(W)S(U, V), (4.1)

where A and B are two non-zero 1-forms defined by

A(U) = g(U, P1), B(U) = g(U, P2). (4.2)

By taking the cyclic sum of (4.1), we see that

(∇US)(V, W) + (∇VS)(W, U) + (∇WS)(U, V) = [3A(U) + B(U)]S(V, W) (4.3)

+[3A(V) + B(V)]S(U, W) + [3A(W) + B(W)]S(U, V).

Let Mn admit a cyclic Ricci tensor, then (4.3) becomes

[3A(U) + B(U)]S(V, W) + [3A(V) + B(V)]S(U, W) + (4.4)

[3A(W) + B(W)]S(U, V) = 0.

Replacing W by ξ in the above equation and using (2.12) and (4.2), we get

−{2(f2 + f′)}[3A(U) + B(U)]η(V) − {2(f2 + f′)}[3A(V) + B(V)]η(U) (4.5)

+[3η(P1) + η(P2)]S(U, V) = 0.

In (4.5), substituting V = ξ and using (2.12) and (4.2), we have

−{2(f2 + f′)}[3A(U) + B(U)] − 4{2(f2 + f′)}[3η(P1) + η(P2)]η(U) = 0. (4.6)

Again treating U by ξ and using (4.2) in (4.6), we obtain

{f2 + f′}[3η(P1) + η(P2)] = 0, (4.7)

which implies

[3η(P1) + η(P2)] = 0, (4.8)

since {f2 + f′} ̸= 0.

From (4.8) and (4.6), it follows that

3A(U) + B(U) = 0. (4.9)

Thus, we can state:

Theorem 4.1. There is no almost pseudo Ricci symmetric 3-dimensional f-Kenmotsu manifold

admitting cyclic Ricci tensor, unless 3A + B vanishes everywhere.



84 Venkatesha and Divyashree G. CUBO
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5 Pseudosymmetric 3-dimensional f-Kenmotsu manifold

Let Mn be an pseudosymmetric 3-dimensional f-Kenmotsu manifold. Then we have,

(R(X, Y) · R)(U, V)W = fRQ(g, R)(U, V, W; X, Y), (5.1)

for all X, Y, U, V, W ∈ χ(Mn).

From the above relation it follows that

R(X, Y)R(U, V)W − R(R(X, Y)U, V)W − R(U, R(X, Y)V)W (5.2)

−R(U, V)R(X, Y)W = fR[(X ∧g Y)R(U, V)W − R((X ∧g Y)U, V)W

−R(U, (X ∧g Y)V)W − R(U, V)(X ∧g Y)W],

where

(X ∧g Y)Z = g(Y, Z)X − g(X, Z)Y. (5.3)

Substituting X by ξ and using (2.10) and (5.3), (5.2) yields

[(f2 + f′) + fR]{g(Y, R(U, V)W)ξ − η(R(U, V)W)Y − g(Y, U)R(ξ, V)W (5.4)

+η(U)R(Y, V)W − g(Y, V)R(U,ξ)W + η(V)R(U, Y)W − g(Y, W)R(U, V)ξ

+η(W)R(U, V)Y} = 0.

Taking inner product of (5.4) with ξ, we get

[(f2 + f′) + fR]{R(U, V, W, Y) − η(Y)η(R(U, V)W) − g(Y, U)η(R(ξ, V)W) (5.5)

+η(U)η(R(Y, V)W) − g(Y, V)η(R(U,ξ)W) + η(V)η(R(U, Y)W)

−g(Y, W)η(R(U, V)ξ) + η(W)η(R(U, V)Y)} = 0.

By using (2.11), (5.5) becomes

[(f2 + f′) + fR]{R(U, V, W, Y) − (f
2 + f′)[−g(V, W)η(Y)η(U) (5.6)

+g(U, W)η(Y)η(V) − g(Y, U)g(V, W) + g(Y, U)η(V)η(W) + g(V, W)η(U)η(Y)

−g(Y, W)η(U)η(V) − g(Y, V)η(W)η(U) + g(Y, V)g(U, W) + g(Y, W)η(V)η(U)

−g(U, W)η(Y)η(V) + g(V, Y)η(W)η(U) − g(U, Y)η(V)η(W)]} = 0.

Contracting the above equation, we obtain

[(f2 + f′) + fR]{S(V, W) + 2(f
2 + f′)g(V, W)} = 0. (5.7)

The above equation can hold only if either

(i) (f2 + f′) = −fR, or



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Three dimensional f-Kenmotsu manifold satisfying certain . . . 85

(ii) S(V, W) = αg(V, W), where α = −2(f2 + f′).

This leads to the following:

Theorem 5.1. A 3-dimensional pseudosymmetric f-Kenmotsu manifold with never vanishing

function {(f2 + f′) = −fR} is an Einstein manifold.

6 Ricci pseudosymmetric 3-dimensional f-Kenmotsu mani-
fold

Suppose (Mn, g) be a 3-dimensional Ricci pseudosymmetric f-Kenmotsu manifold. Then we have,

(R(X, Y) · S)(U, V) = fSQ(g, S)(U, V; X, Y), (6.1)

for all X, Y, U, V, W ∈ χ(Mn). From the above relation it follows that

(R(X, Y) · S)(U, V) = fS((X ∧g Y) · S)(U, V),

or

−S(R(X, Y)U, V) − S(U, R(X, Y)V) = f[−g(Y, U)S(X, V) + g(X, U)S(Y, V) (6.2)

−g(Y, V)S(U, X) + g(X, V)S(U, Y)].

Replacing X and U by ξ and using (2.1), (2.10) and (2.12) in the above equation, we get

[(f2 + f′) + fS]{S(Y, V) + 2(f
2 + f′)g(Y, V)} = 0, (6.3)

which follows that either [(f2 + f′) + fS] = 0 or

S(Y, V) = αg(Y, V), (6.4)

where α = −2(f2 + f′).

Thus we can state:

Theorem 6.1. If a 3-dimensional f-Kenmotsu manifold Mn is Ricci pseudosymmetric with re-

strictions X = U = ξ, then either [(f2 + f′) + fS] = 0 or the manifold is an Einstein manifold.

7 Ricci generalized pseudosymmetric 3-dimensional f-Kenmotsu
manifold

Consider a Ricci generalized pseudosymmetric 3-dimensional f-Kenmotsu manifold. Then we have

(R(X, Y) · R)(U, V)W = f((X ∧S Y) · R)(U, V)W, (7.1)



86 Venkatesha and Divyashree G. CUBO
19, 1 (2017)

for all X, Y, U, V, W ∈ χ(Mn).

We can write the above form as

R(X, Y)R(U, V)W − R(R(X, Y)U, V)W − R(U, R(X, Y)V)W (7.2)

−R(U, V)R(X, Y)W = f[S(Y, R(U, V)W)X − S(X, R(U, V)W)Y

−S(Y, U)R(X, V)W + S(X, U)R(Y, V)W − S(Y, V)R(U, X)W

+S(X, V)R(U, Y)W − S(Y, W)R(U, V)X + S(X, W)R(U, V)Y].

On substituting X = U = ξ and using (2.10) and (2.12), (7.2) reduces to

−(f2 + f′)[(f2 + f′)g(V, W)Y + R(Y, V)W − (f2 + f′)g(Y, W)V] (7.3)

= f[(f2 + f′)S(Y, V)η(W)ξ − 2(f2 + f′)2g(V, W)Y − 2(f2 + f′)R(Y, V)W

+2(f2 + f′)2g(Y, W)η(V)ξ + (f2 + f′)S(Y, W)η(V)ξ − (f2 + f′)S(Y, W)V

+2(f2 + f′)2g(V, Y)η(W)ξ].

Taking inner product of the above equation with Z, we get

−(f2 + f′)[(f2 + f′)g(V, W)g(Y, Z) + g(R(Y, V)W, Z) − (f2 + f′)g(Y, W)g(V, Z)] (7.4)

= f[(f2 + f′)S(Y, V)η(W)η(Z) − 2(f2 + f′)2g(V, W)g(Y, Z)

−2(f2 + f′)g(R(Y, V)W, Z) + 2(f2 + f′)2g(Y, W)η(V)η(Z)

+(f2 + f′)S(Y, W)η(V)η(Z) − (f2 + f′)S(Y, W)g(V, Z)

+2(f2 + f′)2g(V, Y)η(W)η(Z)].

Contracting (7.4) and simplifying gives

(f2 + f′)(3f − 1)[S(Y, Z) + 2(f2 + f′)g(Y, Z)] = 0, (7.5)

which means that either (f2 + f′)(3f − 1) = 0 or S(Y, Z) = αg(Y, Z), where α = −2(f2 + f′).

Hence we can state the following:

Theorem 7.1. If a 3-dimensional f-Kenmotsu manifold is Ricci generalized pseudosymmetric then

either

(i) (f2 + f′)(3f − 1) = 0, or

(ii) it is an Einstein manifold.

Acknowledgement: The second author is thankful to UGC for financial support in the form

of Rajiv Gandhi National Fellowship (F1-17.1/2015-16/RGNF-2015-17-SC-KAR-26367).

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