EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 7, No. 3, 2014, 246-255
ISSN 1307-5543 – www.ejpam.com

Pseudo Conharmonically Symmetric Manifolds

Füsun Özen Zengin 1,∗, Ayşe Yavuz Taşcı 2

1 Department of Mathematics, Faculty of Sciences and Letters, Istanbul Technical University, Istan-
bul, Turkey
2 Department of Mathematics, Faculty of Sciences and Letters, Piri Reis University, Istanbul, Turkey

Abstract. The object of the present paper is to study pseudo conharmonically symmetric manifold
which is a type of non-flat Riemannian manifold. In the first section, we give the definition of a pseudo
conharmonically symmetric manifold. In the second section, some theorems about this manifold are
proved. In the last section, we give an example for the existence of this manifold.

2010 Mathematics Subject Classifications: 53B20, 53C25,53B30,53C15

Key Words and Phrases: Conharmonic Curvature Tensor, Pseudo Conharmonically Symmetric Mani-
fold, Recurrence Vector Field.

1. Introduction

As we know, in differential geometry, symmetric spaces play an important role. In the late
twenties, Cartan [3] initiated Riemannian symmetric spaces and obtained a classification of
those spaces. Let (M , g) be an n-dimensional Riemannian manifold with the Riemannian met-
ric g and the Levi-Civita connection ∇. If the Riemannian curvature tensor of a Riemannian
manifold satisfies the condition ∇R = 0 then this manifold is called locally symmetric [3].
For every point P of this manifold, this symmetry condition is equivalent to the fact that the
local geodesic symmetry F(P) is an isometry [13]. The class of Riemannian symmetric mani-
folds is very natural generalization of the class of manifolds of constant curvature. Many au-
thors have been studied the notion of locally symmetric manifolds extending several manifolds
such as conformally symmetric manifolds [5], recurrent manifolds [24], conformally recur-
rent manifolds [2], conformally symmetric Ricci-recurrent spaces [18], pseudo-Riemannian
manifold with recurrent concircular curvature tensor [12], semi-symmetric manifolds [22],
pseudo symmetric manifolds [4, 14, 15], weakly symmetric manifolds [23], projective sym-
metric manifolds [21], almost pseudo concircularly symmetric manifolds [9], decomposable

∗Corresponding author.

Email addresses: fozen@itu.edu.tr (F. Zengin), aytasci@pirireis.edu.tr ( A. Taşcı)

http://www.ejpam.com 246 c© 2014 EJPAM All rights reserved.



F. Zengin, A. Taşcı / Eur. J. Pure Appl. Math, 7 (2014), 246-255 247

almost pseudo conharmonically symmetric manifolds [25], etc. A non-flat Riemannian man-
ifold (M , g) (n > 2) is said to be a pseudo symmetric manifold [4] if its curvature tensor R
satisfies the condition

(∇X R)(Y, Z)W =2A(X )R(Y, Z)W + A(Y )R(X , Z)W + A(Z)R(Y, X )W

+ A(W )R(Y, Z)X + g(R(Y, Z)W, X )ρ (1)

where A is a non-zero 1-form, ρ is a vector field defined by

g(X ,ρ) = A(X ) (2)

for all X and∇ denotes the operator of the covariant differentiation with respect to the metric
tensor g. The 1-form A is called the associated 1-form of the manifold. If A = 0, then the
manifold reduces to a symmetric manifold in the sense of E.Cartan. An n-dimensional pseudo
symmetric manifold is denoted by (PS)n. This is to be noted that the notion of pseudo symmet-
ric manifold studied in particular by Deszcz [10] is different from that Chaki [4]. The notion
of weakly symmetric manifolds was introduced by Tamassy and Binh [23]. If the curvature
tensor R of type (1,3) of an n-dimensional Riemannian manifold (n> 2) satisfies the condition

(∇X R)(Y, Z)W =A(X )R(Y, Z)W + B(Y )R(X , Z)W + D(Z)R(Y, X )W

+ E(W )R(Y, Z)X + g(R(Y, Z)W, X )ρ (3)

where ∇ denotes the Levi-Civita connection on (M , g) and A,B,D,E and ρ are 1-forms and a
vector field respectively, which are non-zero simultaneously, then this manifold is denoted by
(WS)n. Many authors have been studied weakly symmetric manifolds [6, 7, 11, 16, 17], etc.

Conformal transformation of a Riemannian structure is an important object of study in
differential geometry. The conharmonic transformation which is a special type of conformal
transformations preserves the harmonicity of smooth functions. Such transformation has an
invariant tensor which is called the conharmonic curvature tensor. It is easy to verify that this
tensor is an algebraic curvature tensor, that is, it possesses the classical symmetry properties
of the Riemannian curvature tensor.

Let M and N be two Riemannian manifolds with the metrics g and g, respectively related
by

ḡ = e2σg (4)

where σ is a real function. Then M and N are called conformally related manifolds, and the
correspondence M and N is known as conformal transformation [20]. It is known that a har-
monic function is defined as a function whose Laplacian vanishes. In generally, the harmonic
function is not invariant. In 1957, Ishii obtained the conditions at which a harmonic function
remains invariant and he introduced the conharmonic transformation as a subgroup of the
conformal transformation (4) satisfying the condition

σh
,h +σ,hσ

h
, = 0 (5)



F. Zengin, A. Taşcı / Eur. J. Pure Appl. Math, 7 (2014), 246-255 248

where comma denotes the covariant differentiation with respect to the metric g. A rank-four
tensor H that remains invariant under conharmonic transformation of a Riemannian manifold
(M , g) is given by

H(X , Y, Z , U) =R(X , Y, Z , U)−
1

n− 2
[g(Y, Z)S(X , U)− g(X , Z)S(Y, U)

+ g(X , U)S(Y, Z)− g(Y, U)S(X , Z)] (6)

where R and S denote the Riemannian curvature tensor of type (0,4) defined by
R(X , Y, Z , U) = g(R(X , Y )Z , U) and the Ricci tensor of type (0,2), respectively. The curvature
tensor defined by (6) is known as conharmonic curvature tensor. A manifold whose conhar-
monic curvature tensor vanishes at every point of the manifold is called conharmonically flat.
Thus, this tensor represents the deviation of the manifold from conharmonic flatness. Many
authors have been studied the conharmonic curvature tensor [1, 20]. The present paper deals
with an n-dimensional pseudo conharmonically symmetric Riemannian manifold (M , g) (non-
conharmonically flat) whose conharmonic curvature tensor H satisfies the condition

(∇X H)(Y, Z , U , V ) =2A(X )H(Y, Z , U , V ) + A(Y )H(X , Z , U , V )

+ A(Z)H(Y, X , U , V ) + A(U)H(Y, Z , X , V )

+ A(V )H(Y, Z , U , X ) (7)

where A has the meaning already mentioned in (2). Such a manifold is called a pseudo conhar-
monically symmetric manifold [4] and denoted by (PCHS)n. Since the conformal curvature
tensor vanishes identically for n = 3, we assume that n > 3 throughout the paper. This paper
is organized as follows:

Section 2 deals with some properties of (PCHS)n. Considering special case of conharmonic
curvature tensor of this manifold, some theorems are proved. In section 3, an example is given
for the existence to this manifold.

2. Pseudo Conharmonically Symmetric Manifold

L denotes the symmetric endomorphism of the tangent space at each point of the manifold
corresponding to the Ricci tensor S of type (0,2), that is

g(LX , Y ) = S(X , Y ). (8)

Let ei , (1 ≤ i ≤ n) be an orthonormal basis of the tangent space at any point of the manifold.
From (6), we have

H(X , Y ) =
n
∑

i=1

H(X , ei , ei , Y ) =
n
∑

i=1

H(ei , X , Y, ei)

=−
r

n− 2
g(X , Y ) (9)



F. Zengin, A. Taşcı / Eur. J. Pure Appl. Math, 7 (2014), 246-255 249

and
n
∑

i=1

H(ei , ei , X , Y ) =
n
∑

i=1

H(X , Y, ei , ei) = 0 (10)

where r is the scalar curvature of the manifold. Also, from (6) it follows that [19]

H(X , Y, Z , U) = −H(Y, X , Z , U)

H(X , Y, Z , U) = −H(X , Y, U , Z)

H(X , Y, Z , U) = H(Z , U , X , Y )

H(X , Y, Z , U) +H(X , Z , U , Y ) +H(X , U , Y, Z) = 0. (11)

We assume that our manifold is (PCHS)n. Thus, the relation (7) holds.

Proposition 1 ([19]). In a Riemannian manifold Vn (n> 3), the conharmonic curvature tensor
satisfies the second Bianchi Identity, i.e., the following relation

(∇X H)(Y, Z , U , W ) + (∇U H)(Y, Z , W, X ) + (∇W H)(Y, Z , X , U) = 0

holds if and only if the Ricci tensor is of Codazzi type.

Theorem 1. In a pseudo conharmonically symmetric Riemannian manifold, the conharmonic
curvature tensor satisfies the second Bianchi Identity, i.e.,

(∇X H)(Y, Z , U , W ) + (∇U H)(Y, Z , W, X ) + (∇W H)(Y, Z , X , U) = 0

Proof. Permutating X ,U ,W in (7) and adding these three equations, we obtain

(∇X H)(Y, Z , U , W ) + (∇U H)(Y, Z , W, X ) + (∇W H)(Y, Z , X , U)

=A(X )[2H(Y, Z , U , W ) +H(Y, Z , W, U) +H(Y, Z , W, U)]

+ A(Y )[H(X , Z , U , W ) +H(U , Z , W, X ) +H(W, Z , X , U)]

+ A(Z)[H(Y, X , U , W ) +H(Y, U , W, X ) +H(Y, W, X , U)]

+ A(U)[H(Y, Z , X , W ) + 2H(Y, Z , W, X ) +H(Y, Z , X , W )]

+ A(W )[2H(Y, Z , X , U) +H(Y, Z , U , X ) +H(Y, Z , U , X )]. (12)

Thus, from (11), (12) reduces to

(∇X H)(Y, Z , U , W ) + (∇U H)(Y, Z , W, X ) + (∇W H)(Y, Z , X , U) = 0 (13)

i.e. the conharmonic curvature tensor satisfies the second Bianchi Identity.

Theorem 2. A pseudo conharmonically symmetric manifold with non-zero scalar curvature is of
closed associated 1-form.



F. Zengin, A. Taşcı / Eur. J. Pure Appl. Math, 7 (2014), 246-255 250

Proof. Contracting on Y and V in (7), we find

(∇X H)(Z , U) =2A(X )H(Z , U) + A(H(U , Z)X ) + A(Z)H(X , U)

+ A(U)H(Z , X ) + A(H(U , Z)X ) (14)

where H is in the form (9). Putting Y = Z = ei in (14), where ei is an orthonormal basis of
tangent space at each point of the manifold and i is summed for 1≤ i ≤ n, we get

(∇X h) = 2A(X )h+ 4A(LX ) (15)

where h = H(LX ) and L denotes the symmetric endomorphism of the tangent space at each
point corresponding to the tensor H(X , Y ). If we take the covariant derivative of (15), we find

∇Y∇X h= 2(∇Y A)(X )h+ 2A(X )(∇Y h) + 4(∇Y A)(LX ) + 4A(∇Y H)(LX ). (16)

Changing X and Y in (16) and subtracting these two equations, we obtain from (6), (11), (14)
and (15), assuming that our manifold admits non-zero scalar curvature then

(∇Y A)(X )− (∇X A)(Y ) = 0. (17)

By the aid of (17), we can say that the associated 1-form of this manifold is closed. Thus, the
proof is completed.

Theorem 3. A Riemannian manifold admits divergence-free conharmonic curvature tensor is of
constant scalar curvature.

Proof. In local coordinates, from the second Bianchi Identity, we have

Rh
i jk,h = Si j,k − Sik, j (18)

and then

Sh
k,h =

1
2

r,k (19)

where r is the scalar curvature and Si j is the Ricci tensor of this manifold. Thus from (6),

Hh
i jk,l = Rh

i jk,l −
1

n− 2
(ghm gi jSmk,l −δh

j Sik,l +δ
h
kSi j,l − ghm gikSmj,l). (20)

Contracting on h and l in (20), we find

Hh
i jk,h = Rh

i jk,h −
1

n− 2
(gi jS

h
k,h − Sik, j + Si j,k + gikSh

j,h). (21)

By using (18) and (19), (21) reduces to

Hh
i jk,h =

n− 3
n− 2

(Si j,k − Sik, j)−
1

2(n− 2)
(gi j r,k − gikr, j). (22)



F. Zengin, A. Taşcı / Eur. J. Pure Appl. Math, 7 (2014), 246-255 251

If we assume that the conharmonic curvature tensor of this manifold is divergence-free then
we find from (22)

(n− 3)(Si j,k − Sik, j)−
1
2
(gi j r,k − gikr, j) = 0. (23)

Multiplying (23) by g i j and putting (19) in (23), we obtain

r,k = 0.

Thus, we get finally that the scalar curvature of this manifold r is constant. This completes
the proof.

Theorem 4. A (PCHS)n admits divergence-free conharmonic curvature tensor is of zero scalar
curvature tensor.

Proof. Assuming that the conharmonic curvature tensor of (PCHS)n is divergence-free,
from (7), we get

2AhHh
i jk + AhHhi jk − A jH ik + AkH i j = 0. (24)

Multiplying (24) by g i j we obtain
2AhHhk = −hAk. (25)

It follows from (6) and (25)
rAk = 0. (26)

Since Ak is not zero for (PCHS)n, we get r must be zero. The proof is completed.

Theorem 5. If (PCHS)n is recurrent then either the scalar curvature of this manifold is zero or
the recurrence vector field and the associated 1-form are related by

λl =
2(n+ 2)

n
Al .

Proof. By taking the covariant derivative of (6), we get

Hhi jk,l = Rhi jk,l −
1

n− 2
(gi jShk,l − gh jSik,l + ghkSi j,l − gikSh j,l). (27)

Comparing (27) with (7), using (6) and assuming our manifold is recurrent, i.e, we have that

Rhi jk,l = λlRhi jk (28)

and
Si j,l = λlSi j , r,l = λl r (29)

are satisfied. Then from (6), (7), (28) and (29), we obtain

(λl − 2Al)Rhi jk − AhRl i jk − AiRhl jk − A jRhilk + AkRhi jl

+
1

n− 2
[Shk(2Al gi j + Ai gl j + A j gil −λl gi j)



F. Zengin, A. Taşcı / Eur. J. Pure Appl. Math, 7 (2014), 246-255 252

− Sik(2Al gh j + Ah gl j + A j ghl −λl gh j)

+ Si j(2Al ghk + Ah glk + Ak ghl −λl ghk)

− Sh j(2Al gik + Ai glk + Ak gil −λl gik)

+ Slk(Ah gi j − Ai gh j)− Sl j(Ah gik − Ai ghk)

+ Sil(A j ghk − Ak gh j)− Shl(A j gik − Ak gi j)]

=0 (30)

Multiplying (30) by ghk and g i j , we get

r(−n(λl − 2Al) + 4Al) = 0. (31)

It follows from (31) that either r is zero or λl =
2(n+2)

n Al . Thus, this completes the proof.

3. An Example of (PCHS)n

In this section we will give an example for (PCHS)n satisfying the conditions (6) and (7).
We define a Riemannian metric on Rn (n≥ 4) by the formula,[18]

ds2 = ϕ(d x1)2 + kαβd xαd xβ + 2d x1d xn (32)

where [kαβ] is a symmetric and non-singular matrix consisting of constant and ϕ is a function
of x1, x2, . . . , xn−1 and independent of xn. Let each Latin index runs over 1, 2, . . . , n and each
Greek index runs over 2,3, . . . , (n− 1).
In the metric considered, the only non-vanishing components of Christoffel symbols, the cur-
vature tensor and the Ricci tensor are, according to [18]

Γβ11 = −
1
2

kαβϕ,α, Γn
11 =

1
2
ϕ,1 Γn

1α =
1
2
ϕ,α

R1αβ1 =
1
2
ϕ,αβ , S11 =

1
2

kαβϕ,αβ (33)

where “,” denotes the partial differentiation with respect to the coordinates and kαβ are the
elements of the matrix inverse to [kαβ].
We consider kαβ as the kronecker symbol δαβ and ϕ as, [8]

ϕ = (Mαβ +δαβ)x
αxβ e(x

1)2 (34)

where Mαβ are constant and satisfy the relations

Mαβ = 0 for α 6= β
Mαβ 6= 0 for α= β

n−1
∑

α=2

Mαα = 0. (35)



F. Zengin, A. Taşcı / Eur. J. Pure Appl. Math, 7 (2014), 246-255 253

In this case, we have the following relations

ϕ,αβ =2(Mαβ +δαβ)e
(x1)2

δαβδ
αβ =n− 2

δαβMαβ =ΣMαα = 0. (36)

Thus, from (34) and (36), we have

δαβϕ,αβ = 2(n− 2)e(x
1)2 . (37)

By using (33), we find the only non-zero components for Rhi jk and Si j as

R1αα1 =
1
2
ϕ,αα = (1+Mαα)e

(x1)2

S11 =
1
2
ϕ,αβδ

αβ = (n− 2)e(x
1)2 . (38)

Hence, the only non-zero components of the conharmonic curvature tensor Hhi jk are

H1αα1 =R1αα1 −
1

n− 2
(gααS11)

=(1+Mαα)e
(x1)2 −

1
n− 2

(n− 2)e(x
1)2

=Mααe(x
1)2 (39)

which never vanish. In this case, from (39), the only non-zero components of the derivative
of Hhi jk are found as

H1αα1,1 =2x1Mααe(x
1)2

=2x1H1αα1. (40)

Let us consider the associated 1-form as

Ai(x) =

¨

x1

2 , for i = 1

0, otherwise
(41)

at any point x ∈ Rn.
To verify the relation (7) it is sufficient to prove that the equation

H1αα1,1 = 4A1H1αα1. (42)

By the aid of (40) and (41), we can easily see that (42) is satisfied. The other components of
each term of (7) vanish identically and the relation (7) holds trivially.
Under our assumptions (32), (34) and (35), this manifold is a (PCHS)n.



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ACKNOWLEDGEMENTS The authors wish to express their sincere thanks and gratitude to
the referee for his valuable suggestions towards the improvement of the paper.

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