CUBO A Mathematical Journal
Vol.15, No¯ 02, (33–42). June 2013

On Weak concircular Symmetries of Lorentzian Concircular
Structure Manifolds

Dhruwa Narain

D.D.U.Gorakhpur University,

Department of Mathematics & Statistics,

Gorakhpur, India.

profdndubey@yahoo.co.in

Sunil Yadav

Alwar Institute of Engineering & Technology,

Department of Mathematics,

Matsya Industrial Area, Alwar-301030,

Rajasthan, India.

prof sky16@yahoo.com

ABSTRACT

The object of the present paper is to study weakly concircular symmetric, weakly

concircular Ricci symmetric and special weakly concircular Ricci symmetric Lorentzian

concircular structure manifolds.

RESUMEN

El objetivo del presente art́ıculo es estudiar las variedades de estructura simétricas con-

circulares débiles, las simétricas Ricci concirculares débiles y concirculares Lorentzianas

simétricas Ricci concirculares débiles especiales.

Keywords and Phrases: Weakly concircular symmetric manifold, weakly concircular Ricci

symmetric manifold, concircular Ricci tensor, special weakly concircular Ricci symmetric and

Lorentzian concircular structure manifold.

2010 AMS Mathematics Subject Classification: 53C10, 53C15, 53C25.



34 Dhruwa Narain and Sunil Yadav CUBO
15, 2 (2013)

1 Introduction

The notion of weakly symmetric manifolds was introduced by Tamassy and Binh [8].A non-flat

Riemannian manifold (Mn, g) (n > 2) is called weakly symmetric manifold if its curvature tensor

R of type (0, 4) satisfies the condition

(∇XR)(Y, Z, U, V) = A(X)R(Y, Z, U, V) + B(Y)R(X, Z, U, V)

+H(Z)R(Y, X, U, V) + D(U)R(Y, Z, X, V) + E(V)R(Y, Z, U, X)
, (1.1)

for all vector fields X, Y, Z, U, V ∈ χ(Mn), χ(M) being the Lie-algebra of the smooth vector fields

of M, where A, B, H, D and E are 1−forms (not simultaneously zero) and ∇ denote the operator

of the covariant differentiations with respect to Riemannian metric g. The 1−forms are called the

associated 1−forms of the manifold and n−dimensional manifold of this kind is denoted by (WS)n.

In 1999, De and Bandyopadhyay [2] studied a (WS)n and prove that in such a manifold the

associated 1–forms B = H and D = E. Hence from (1.1) reduces to the following:

(∇XR)(Y, Z, U, V) = A(X)R(Y, Z, U, V) + B(Y)R(X, Z, U, V)

+B(Z)R(Y, X, U, V) + D(U)R(Y, Z, X, V) + D(V)R(Y, Z, U, X).
(1.2)

A transformation of n–dimensional Riemannian manifold M, which transform every geodesic circle

of M into a geodesic circle, is called a concircular transformation [11].The intersecting invariant of

a concircular transformation is the concircular curvature tensor C̃ which is defined by [11].

C̃ (Y, Z, U, V) = R (Y, Z, U, V) −
k

n (n − 1)
[ g (Z, U)g(Y, V) − g(Y, U) g(Z, V)] , (1.3)

where k is the scalar curvature of the manifold.

Recently Shaikh and Hui [5] introduced the notion of weakly concircular symmetric manifolds. A

Riemannian manifold is called weakly concircular symmetric manifold if its concircular curvature

tensor C̃ of type (0, 4) is not identically zero and satisfies the condition

(

∇XC̃
)

(Y, Z, U, V) = A (X) C̃(Y, Z, U, V) + B (Y) C̃(X, Z, U, V)

+H(Z)C̃(Y, X, U, V) + D (U) C̃(Y, Z, X, V) + E (V) C̃(Y, Z, U, X)
, (1.4)

for all vector fields X, Y, Z, U, V ∈ χ(Mn) where A, B, H, D and E are 1–form (not simultane-

ously zero) an n–dimensional manifold of this kind is denoted by
(

W C̃ S
)

n .Also it is known that

[5], in a
(

WC̃S
)

n
the associated 1–forms B = H and D = E, and hence the defining the condition

(1.4) of a
(

W, C̃S
)

n
reduces to the following form:

(

∇XC̃
)

(Y, Z, U, V) = A (X) C̃(Y, Z, U, V) + B (Y) C̃(X, Z, U, V)

+B (Z) C̃(Y, X, U, V) + D (U) C̃(Y, Z, X, V) + D (V) C̃(Y, Z, U, X)
, (1.5)



CUBO
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On Weak concircular Symmetries of Lorentzian Concircular . . . 35

where A, B and D are 1-forms (not simultaneously zero).

Again Tamassy and Binh [9] introduced the notion of weakly Ricci symmetric manifolds. A Rie-

mannian manifold (Mn, g), (n > 2) is called weakly Ricci symmetric manifold if its Ricci tensor S

of type(0, 2) is not identically zero and satisfies the condition:

(∇XS) (Y, Z) = A(X)S(Y, Z) + B(Y)S(X, Z) + D(Z)S(Y, X), (1.6)

where A, B and D are three non-zero 1–forms called the associate 1-forms of the manifold, and ∇

is the operator of covariant differentiation with respect to metric g. Such n–dimensional manifold

is denoted by (W R S) n. If A = B = D then is called pseudo Ricci symmetric.

Let{ei : i = 1, 2, ......n} be an orthonormal basis of the tangent space at each point of the manifold

and let

S̃ (Y, V) =

n∑

i=1

C̃ (Y, ei, ei, Y)

Then from (1.3), we have

S̃ (Y, V) = S (Y, V) −
k

n
g (Y, V), (1.7)

The tensor S̃ is called the concircular Ricci symmetric tensor which is symmetric tensor of type

(0, 2). In [1] De and Ghose introduced the notion of weakly concircular Ricci symmetric manifolds.

A Riemannian manifold (Mn, g), (n > 2) is called weakly concircular Ricci symmetric manifolds

[1] if its concircular Ricci tensor S̃ of type (0, 2) is not identically zero satisfies the condition:

(

∇XS̃
)

(Y, Z) = A (X) S̃(Y, Z) + B (Y) S̃(X, Z) + D (Z) S̃(Y, X), (1.8)

whereA, B and D are three 1-form (not simultaneously zero).If A = B = D then Mn is called

pseudo concircular Ricci symmetric. A Riemannian manifold is called special weakly Ricci sym-

metric manifold if

(∇XS) (Y, Z) = 2A(X)S(Y, Z) + A(Y)S(X, Z) + A(Z)S(Y, X), (1.9)

where A is a 1–form and is defined by

A(X) = g(X, ρ). (1.10)

where ρ is the associated vector field.

Motivated by above studied we define and syudy special weakly concircular Ricci symmetric mani-

fold. An n−dimensional Riemannian manifold is called special weakly concircular Ricci symmetric

manifolds. If
(

∇XS̃
)

(Y, Z) = 2 A (X) S̃(Y, Z) + A (Y) S̃(X, Z) + A (Z) S̃(Y, X). (1.11)

where A is a 1–form and is defined by (1.10).



36 Dhruwa Narain and Sunil Yadav CUBO
15, 2 (2013)

An (2n + 1)-dimensional Lorentzian manifold M is smooth connected Para contact Hausdorff

manifold with Lorentzian metric g, that is, M admits a smooth symmetric tensor field g of type

(0, 2) such that for each point p ∈ M, the tensor gp : TpM × TpM → R is a non degenerate inner

product of signature (−, +, .....+)where TpMdenotes the tangent space of M at p and R is the real

number space. In a Lorentzian manifold (M, g) a vector field ρ defined by

g(X, ρ) = A(X)

for any vector field X ∈ χ(M) is said to be concircular vector field [5] if

(∇XA)(Y) = α [g(X, Y) + ω(X)A(Y)]

where α is a non zero scalar function, A is a 1-form and ω is a closed 1-form.

Let M be a Lorentzian manifold admitting a unit time like concircular vector field ξ, called the

characteristic vector field of the manifold. Then we have

g(ξ, ξ) = −1, (1.12)

Since ξ is the unit concircular vector field, there exist a non zero 1-form η such that

g(X, ξ) = η(X), (1.13)

the equation (1.13) of the following form holds:

(∇Xη)(Y) = α [g(X, Y) + η(X)η(Y)] (α 6= 0), (1.14)

for all vector field X, Y, where ∇denotes the operator of covariant differentiation with respect to

Lorentzian metric g and α is a non zero scalar function satisfying

(∇Xα) = (Xα) = ρη(X), (1.15)

where ρ being a scalar function. If we put

φX =
1

α
∇Xξ, (1.16)

Then from (1.14) and (1.16), we have

φ2X = X + η(X)ξ, (1.17)

from which it follows that φ is a symmetric (1, 1)-tensor. Thus the Lorentzian manifold M together

with unit time like concircular vector field ξ, it’s associate 1-form η and (1, 1)–tensor field φ is said

to be Lorentzian concircular structure manifolds (briefly (LCS)2n+1-manifold) [6]. In particular if

α = 1, then the manifold becomes LP-Sasakian structure of Matsumoto [3].



CUBO
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On Weak concircular Symmetries of Lorentzian Concircular . . . 37

2 Lorentzian Concircular Structure manifolds

A differentiable manifold M of dimension (2n+1) is called (LCS)2n+1-manifold if it admits a (1, 1)

-tensor φ, a contravarient vector field ξ, a covariant vector field η and a Lorentzian metric g which

satisfy the following

η(ξ) = −1, (2.1)

φ2 = I + η ⊗ ξ, (2.2)

g(φX, φY) = g(X, Y) + η(X)η(Y), (2.3)

g(X, ξ) = η(X), (2.4)

φξ = 0 , η(φX) = 0, (2.5)

for all X, Y ∈ TM. Also in a (LCS)2n+1-manifold the following relations are satisfied [7].

η(R(X, Y)Z) = (α2 − ρ) [g(Y, Z)η(X) − g(X, Z)η(Y)] , (2.6)

R(X, Y)ξ = (α2 − ρ) [η(Y)X − η(X)Y] , (2.7)

R(ξ, X)Y = (α2 − ρ) [g(X, Y)ξ − η(Y)X] , (2.8)

R(ξ, X)ξ = (α2 − ρ) [η(X)ξ + X] , (2.9)

(∇Xφ)(Y) = α [g(X, Y)ξ + 2η(X)η(Y)ξ + η(Y)X] , (2.10)

S(X, ξ) = 2n(α2 − ρ)η(X), (2.11)

S(φX, φY) = S(X, Y) + 2n(α2 − ρ)η(X)η(Y), (2.12)

Definition 2.1 A Lorentzian concircular structure manifold is said to be η–Einstein if the

Ricci operator Q satisfies

Q = aId + bη ⊗ ξ,

where a and b are smooth functions on the manifolds, In particular if b = 0, then M is an

Einstein manifold.



38 Dhruwa Narain and Sunil Yadav CUBO
15, 2 (2013)

3 Main Results

Definition 3.1A Lorentzian concircular structure manifold(M2n+1, g) (n > 1) is said to be weakly

concircular symmetric if its concircular curvature tensor C̃ of type (0, 4) satisfies (1.5)

Substituting Y = V = ei in (1.5) and taking summation over i , 1 ≤ i ≤ 2n + 1, we get

(∇XS) (Z, U) −
dκ(X)

n
g(Z, U) = A(X)

[

S(Z, U) − κ
n
g(Z, U)

]

+ B(Z) [S(X, U)

− κ
n
g(X, U) + D(U)

[

S(X, Z) − κ
n
g(X, Z)

]

+ B(R(X, Z)U) + D(R(X, U)Z)

− κ
n(n−1)

[{B(X) + D(X) g(Z, U) − B(Z)g(X, U) − D(U)g(Z, X)]

(3.1)

Again setting X = Z = U = ξ in (3.1) and using (2.7)and (2.11), we have

A(ξ) + B(ξ) + D(ξ) =
dk(ξ)

k − 2n2(α2 − ρ)
(3.2)

This leads to the following result.

Theorem 3.1.In a weakly concircular symmetric Lorentzian concircular structure manifold (M2n+1, g) (n >

1) the relation (3.2) holds.

Corollary 3.1 In a weakly concircular symmetric Lorentzian concircular structure manifold (M2n+1, g) (n >

1) the sum of 1-forms A, B and D is zero everywhere if and only if the scalar curvature κ of the

manifold is constant.

Next, putting X and Z by ξ in (3.1) and using (2.4), (2.7)and (2.11) we obtain

(∇XS) (ξ, U) −
dκ(ξ)

n
η(U) = [A(ξ) + B(ξ)]

{
2n(α2 − ρ) − κ

n

}
η(U)+

[

κ
n
− 2n(α2 − ρ) − κ

n(n−1)
+ 1

]

D(U) + D(ξ)
[

(α2 − ρ) − κ
n(n−1)

]

η(U).
(3.3)

Also from (2.11), we have

(∇ξS) (ξ, U) = 0. (3.4)

In view of (3.2) and (3.4), equation (3.3) reduces to

D(U) =

[

k + (n − 1)
{
k − 2n2(α2 − ρ)

}

−k + (n − 1) {k − 2n2(α2 − ρ)}

]

D(ξ) η(U). (3.5)

Next setting X = U = ξ in (3.1) and proceeding in the similar manner as above, we have

B(Z) =

[

k − (n − 1)
{
k − 2n2(α2 − ρ)

}

−k + (n − 1) {k − 2n2(α2 − ρ)}

]

B(ξ) η(Z), (3.6)



CUBO
15, 2 (2013)

On Weak concircular Symmetries of Lorentzian Concircular . . . 39

Again, substituting Z = U = ξ (3.1), we obtain

(∇XS) (ξ, ξ) +
dκ(X)

n
= A(X)

[

κ
n
− 2n(α2 − ρ)

]

+
[

κ
n(n−1)

− (α2 − ρ)
]

{B(X) + D(X) + (B(ξ) + D(ξ))η(X)} +
[

2n(α2 − ρ) − κ
n

]

{B(ξ) + D(ξ)} η(X)
(3.7)

On the other hand we have

(∇ξS) (ξ, ξ) = ∇XS(ξ, ξ) − 2S(∇Xξ, ξ),

which yield by using (1.16) and (2.1) that.

(∇ξS) (ξ, ξ) = −2n(α
2 − ρ)ξ, (3.8)

In view of (3.7) and (3.8), we get

A(X) =
[

dk(X)−2n
2
(α

2
−ρ)ξ

k−2n2(α2−ρ)

]

−
[

k
(n−1){(k−2n2(α2−ρ)}

]

{B(X) + D(X)} − {D(ξ) + B(ξ)}η(X) −
{

dk(ξ)

k−2n2(α2−ρ)
− A(ξ)

}
η(X)

(3.9)

This leads to the following result.

Theorem 3.2. In a weakly concircular symmetric Lorentzian concircular structure manifold

(M2n+1, g) (n > 1) the associated 1-forms D, B and A are given by (3.5) (3.6) and (3.9) re-

spectively.

Definition 3.2 A Lorentzian concircular structure manifold (M2n+1, g) (n > 1) is said to be

weakly concircular Ricci symmetric if its concircular Ricci tensor S of type (0, 2) satisfies (1.8).

In view of (1.8) and (1.9) yield

(∇XS) (Y, Z) −
dk(X)

n
g(Y, Z) = A(X)

[

S(Y, Z) − k
n
g(Y, Z)

]

+ B(Y)
[

S(X, Z) − k
n
g(X, Z)

]

+ D(Z)
[

S(X, Y) − k
n
g(X, Y)

] (3.10)

Setting X = Y = Z = ξ in above we get the relation (3.2). Hence we can state the following

Theorem 3.3. In a weakly concircular Ricci symmetric Lorentzian concircular structure manifold

(M2n+1, g) (n > 1) the relations (3.2) holds

Corollary 3.2. In a weakly concircular Ricci symmetric Lorentzian concircular structure manifold

(M2n+1, g) (n > 1) the sum of 1-forms A, B and D is zero everywhere and only if the scalar

curvature k of the manifold is constant.

Now, taking X and Y by ξ in (3.10), we have



40 Dhruwa Narain and Sunil Yadav CUBO
15, 2 (2013)

(∇ξS) (ξ, Z) −
dκ(X)

n
η(Z) = {A(ξ) + B(ξ)} + D(Z)

[

S(ξ, ξ) +
κ

n

]

(3.11)

In view of (3.2) and (3.4), equation (3.11) yields.

D(Z) =
−dk(ξ)

k − 2n2(α2 − ρ)
η(Z) +

[

dk(ξ)

k − 2n2(α2 − ρ)
− D(ξ)

]

η(Z), (3.12)

Again putting X = Z = ξ in (3.11) and proceeding in a similar manner as above, we get

B(Y) =
−dk(ξ)

k − 2n2(α2 − ρ)
η(Y) +

[

dk(ξ)

k − 2n2(α2 − ρ)
− B(ξ)

]

η(Y), (3.13)

A(X) =
−2n2(α2 − ρ)ξη(x)

k − 2n2(α2 − ρ)
+

dk(X)

k − 2n2(α2 − ρ)
+

[

dk(ξ)

k − 2n2(α2 − ρ)
− A(ξ)

]

η(X), (3.14)

This leads to the following result.

Theorem 3.4. In a weakly concircular Ricci symmetric Lorentzian concircular structure mani-

fold (M2n+1, g) (n > 1) the associated 1-form D, B and A are given by (3.12) (3.13) and (3.14)

respectively

Adding equations (3.12) (3. 13) and (3.14), using (3.3) we obtain

A(X) + B(X) + D(X) =
dk(X) − 2n2(α2 − ρ)ξ

k − 2n2(α2 − ρ)
(3.15)

for any vector field X .

This leads to the following result.

Theorem 3.5. In a weakly concircular Ricci symmetric Lorentzian concircular structure manifold

(M2n+1, g) (n > 1) the sum of the associated 1-form A, B and D is given by (3.15)

Corollary 3.3 There is no weakly concircular Ricci symmetric Lorentzian concircular structure

manifold (M2n+1, g)(n > 1) unless the sum of the 1-forms is everywhere zero if dk(X) = 2n2(α2−

ρ)ξ.

Also taking cyclic sum of (1.11), we get

(

∇XS̃
)

(Y, Z) +
(

∇YS̃
)

(Z, Y) +
(

∇ZS̃
)

(X, Y) = 4 A (X) S̃(Y, Z)

+A (Y) S̃(X, Z) + A (Z) S̃(Y, X), (3.16)

Let M2n+1 admits a cyclic Ricci tensor. Then (3.16) reduces to



CUBO
15, 2 (2013)

On Weak concircular Symmetries of Lorentzian Concircular . . . 41

A (X) S̃(Y, Z) + A (Y) S̃(X, Z) + A (Z) S̃(Y, X) = 0.

Taking Z = ξ in above and then using (1.7), (1.10) and (2.11), we obtain

[

2n2(α2 − ρ) −
κ

n

]

{A(X)η(Y) + A(Y)η(X)} + η(ρ)S(X, Y) = 0. (3.17)

Again taking Z = ξ in (3.17), we get

2η(ρ)η(X) = A(X) (3.18)

Taking X = ξ in (3.18)and using (1.7), we yields

η(ρ) = 0. (3.19)

In view of (3.18) and (3.19), we get A(X) = 0, ∀X.

This leads to the following result.

Theorem 3.6.If a special weakly concircular Ricci symmetric Lorentzian concircular structure

manifold (M2n+1, g) (n > 1) admits Cyclic Ricci tensor then the 1-form A must vanishes.

Finally for Einstein manifold (∇XS) (Y, Z) = 0 and S(Y, Z) = ag(Y, Z). Then (1.7) and (1.11),

we get

−
dκ(X)

n
g(Y, Z) = 2A(X)

[(

a −
κ

n

)

g(Y, Z)
]

+ A(Y)
[(

a −
κ

n

)

g(X, Z)
]

+A(Z)
[(

a −
κ

n

)

g(X, Y)
]

, (3.20)

Plugging Z = X = Y = ξ in (3.20), we obtain that

4 η (ρ) (an − κ) = dκ (ξ)

which implies that if κ is constant then η(ρ) = 0,that is A(Y) = 0, ∀Y. Therefore we state the

results

Theorem 3.7. A special weakly concircular Ricci symmetric Lorentzian concircular structure

manifold (M2n+1, g) (n > 1)can not Einstein manifold if the scalar curvature of the manifold is

constant.

Corollary 3.4.In a special weakly concircular Ricci symmetric Lorentzian concircular structure

manifold (M2n+1, g) (n > 1) the 1-form A is given by
[

A (ξ) =
d k (ξ) − 2n2 (α2 − ρ)

2 { k − 2n2 (α2 − ρ) + n }
.

]



42 Dhruwa Narain and Sunil Yadav CUBO
15, 2 (2013)

Received: January 2012. Accepted: October 2012.

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