7_hui.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 3, 2012, 365-372

ISSN 1307-5543 – www.ejpam.com

A Note on Nearly Quasi-Einstein Manifolds

SHYAMAL KUMAR HUI

Nikhil Banga Sikshan Mahavidyalya, Bishnupur, Bankura - 722 122, West Bengal, India

Abstract. The object of the present paper is to study nearly quasi-Einstein manifold. Also we have

studied decomposable Riemannian manifold and it is shown that a decomposable Riemannian mani-

fold is nearly quasi-Einstein if and only if both the decompositions are Einstein.

2010 Mathematics Subject Classifications: 53B30, 53B50, 53C15, 53C25

Key Words and Phrases: quasi-Einstein manifold, nearly quasi-Einstein manifold, Ricci-pseudosymmetric,

scalar curvature, decomposable Riemannian manifold, Killing vector field, projective Killing vector field

1. Introduction

It is well known that a Riemannian manifold (M n, g)(n > 2) is Einstein if its Ricci tensor

S of type (0,2) is of the form S = αg, where α is a constant, which reduces to S = r

n
g, r being

the scalar curvature (constant) of the manifold.

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the

Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For

instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. A non-flat Rieman-

nian manifold (M n, g)(n > 2) is said to be quasi-Einstein manifold [1, 3, 5, 6, 7, 8, 9, 10, 11,

12, 15, 16] if its Ricci tensor S of type (0,2) is not identically zero and satisfies the following:

S(X , Y ) = αg(X , Y ) + βA(X )A(Y ), (1)

where α, β are scalars of which β 6= 0 and A is a nowhere vanishing 1-form defined by

g(X ,ρ) = A(X ) for all X ; ρ being a unit vector field, called the generator of the manifold.

Such an n-dimensional quasi-Einstein manifold is denoted by (QE)n. The scalars α, β are

known as the associated scalars of the manifold. Also the 1-form A is called the associated

1-form of the manifold. From the above definition it follows that every Einstein manifold

is quasi-Einstein. In particular, every Ricci-flat (e.g. Schwarzschild spacetime) manifold is

quasi-Einstein.

Recently the notion of quasi-Einstein manifold have been weakened by De and Gaji [2, 14]

and they introduced the notion of nearly quasi-Einstein manifold with the existence of such

Email address: shyamal_hui�yahoo.
o.in

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S. HUI / Eur. J. Pure Appl. Math, 5 (2012), 365-372 366

notion. A Riemannian manifold (M n, g)(n> 2) is called nearly quasi-Einstein if its Ricci tensor

S is not identically zero and satisfies the condition

S(X , Y ) = αg(X , Y ) + βD(X , Y ), (2)

where α,β are non-zero scalars and D is a symmetric non-zero (0, 2) tensor. The scalars α,

β are known as associated scalars and D is called the associated tensor of the manifold. Such

an n-dimensional manifold is denoted by N(QE)n.

The present paper deals with a study of N(QE)n(n > 2). The paper is organized as fol-

lows. Section 2 is concerned with Ricci-pseudosymmetric N(QE)n and we obtain a N(QE)n
is Ricci-pseudosymmetric if and only if it is D-pseudosymmetric. Section 4 deals with de-

composable Riemannian manifold. It is proved that a decomposable Riemannian manifold

is nearly quasi-Einstein if and only if both the decompositions are Einstein. Section 5 deals

with some global properties of N(QE)n and it is proved that under certain condition such a

manifold does not admit non-zero Killing vector field, non-zero projective Killing vector field

and non-zero conformal Killing vector field. Finally the last section deals with an interesting

example of nearly quasi-Einstein manifold with non-vanishing scalar curvature which is not

quasi-Einstein.

2. Ricci-pseudosymmetry N(QE)n

An n-dimensional Riemannian manifold (M n, g) is called Ricci-pseudosymmetric [4] if the

tensor R · S and Q(g,S) are linearly dependent, where

(R(X , Y ) · S)(Z , U) = −S(R(X , Y )Z , U)− S(Z ,R(X , Y )U), (3)

Q(g,S)(Z , U; X , Y ) = −S((X ∧g Y )Z , U)− S(Z , (X ∧g Y )U). (4)

Thus the condition of Ricci-pseudosymmetry is

(R(X , Y ) · S)(Z , U) = LSQ(g,S)(Z , U; X , Y ) (5)

holding on the set US = {x ∈ M : S 6= r

n
g at x}, where LS is some function on US. If

R · S = 0 then M is called Ricci-semisymmetric. Every Ricci-semisymmetric manifold is

Ricci-pseudosymmetric but the converse is not true [4]. In [2] De and Gaji studied Ricci-

semisymmetric N(QE)n.

Now we prove the following:

Theorem 1. A nearly quasi-Einstein manifold is Ricci-pseudosymmetric if and only if it is D-

pseudosymmetric.

Proof. We now consider a Ricci-pseudosymmetric N(QE)n. Then from (3)–(5), we can

write

S(R(X , Y )Z , U) + S(Z ,R(X , Y )U) = LS{S(X , U)g(Y, Z) (6)

−S(Y, U)g(X , Z)+ S(X , Z)g(Y, U)− S(Y, Z)g(X , U)}.



S. HUI / Eur. J. Pure Appl. Math, 5 (2012), 365-372 367

Using (2) in (6), we get

D(R(X , Y )Z , U) + D(Z ,R(X , Y )U) = LS{D(X , U)g(Y, Z) (7)

−D(Y, U)g(X , Z)+ D(X , Z)g(Y, U)− D(Y, Z)g(X , U)},

which implies that the manifold is D-pseudosymmetric.

Conversely, if the manifold is D-pseudosymmetric, then (7) holds. By virue of (2), it fol-

lows from (7), we get the relation(6) and consequently, the manifold is Ricci-pseudosymmetric.

Corollary 1. A nearly quasi-Einstein manifold is Ricci-semisymmetric if and only if it is D-

semisymmetric [2].

3. Decomposable Riemannian manifold

A non-flat Riemannian manifold (M n, g) is said to be decomposable [19] if it can be

expressed as M
p

1
×M

n−p

2
for 2≤ p ≤ n− 2, that is, in some coordinate neighbourhood of the

Riemannian manifold (M n, g), the metric can be expressed as

ds2 = gi jd x id x j = g̃abd x ad x b+
∗
g
αβ d xαd xβ , (8)

where g̃ab are functions of x1, x2, · · · , x p(p < n) denoted by x̃ and
∗
g
αβ are functions of

x p+1, x p+2, · · · , xn denoted by
∗
x ; a, b, c, · · · run from 1 to p and α,β ,γ, · · · run from p+ 1 to

n. The two parts of (8) are the metrics of M
p

1 (p ≥ 2) and M
n−p

2 (n− p ≥ 2) which are called

the decomposition of the manifold M n = M
p

1 ×M
n−p

2 (2≤ p ≤ n− 2).

Let (M n, g) be a Riemannian manifold such that M
p

1 × M
n−p

2 for 2 ≤ p ≤ n − 2. Here

throughout this section each object denoted by a “tilde” is assumed to be from M1 and each

object denoted by a “star” is assumed to be from M2.

Let X̃ , Ỹ , Z̃ , Ũ , Ṽ ∈ χ(M1) and
∗
X ,
∗
Y ,
∗
Z ,
∗
U ,
∗
V∈ χ(M2), then we have the following

relations:

R(
∗
X , Ỹ , Z̃ , Ũ) = 0= R(X̃ ,

∗
Y , Z̃ ,

∗
U) = R(X̃ ,

∗
Y ,
∗
Z ,
∗
U),

(∇ ∗
X
R)(Ỹ , Z̃ , Ũ , Ṽ ) = 0= (∇X̃ R)(Ỹ ,

∗
Z , Ũ ,

∗
V ) = (∇ ∗

X
R)(Ỹ ,

∗
Z , Ũ ,

∗
V ),

R(X̃ , Ỹ , Z̃ , Ũ) = R̃(X̃ , Ỹ , Z̃ , Ũ); R(
∗
X ,
∗
Y ,
∗
Z ,
∗
U) =

∗
R (
∗
X ,
∗
Y ,
∗
Z ,
∗
U),

S(X̃ , Ỹ ) = S̃(X̃ , Ỹ ); S(
∗
X ,
∗
Y ) =

∗
S (
∗
X ,
∗
Y ),

(∇X̃ S)(Ỹ , Z̃) = (∇̃X̃ S)(Ỹ , Z̃); (∇ ∗
X
S)(
∗
Y ,
∗
Z) = (

∗
∇ ∗

X
S)(
∗
Y ,
∗
Z),

and r = r̃+
∗
r,

where r, r̃, and
∗
r are the scalar curvature of M , M1, M2 respectively.

In [19] Yano and Kon find a necessary and sufficient condition that both the decomposi-

tions of a decomposable Riemannian manifold are Einstein and they obtained that



S. HUI / Eur. J. Pure Appl. Math, 5 (2012), 365-372 368

Theorem 2. In a decomposable Riemannian manifold M n = M
p

1 × M
n−p

2 (2 ≤ p ≤ n− 2), a

necessary and sufficient condition that the two decompositions are both Einstein is that the Ricci

tensor of the manifold has the form

S(X , Y ) = ag(X , Y ) + bF(X , Y ), (9)

a and b being necessarily constant and F is a (0, 2) type metric tensor such that

F(X , Y ) = g̃(X̃ , Ỹ )+
∗
g (
∗
X ,
∗
Y ). (10)

By virtue of Theorem 2, we can state the following:

Theorem 3. A decomposable Riemannian manifold M n = M
p

1 ×M
n−p

2 (2≤ p ≤ n− 2) is nearly

quasi-Einstein if and only if both the decompositions are Einstein.

4. Some global properties of N(QE)n

This section is concerned with a compact, orientable N(QE)n(n > 2) without boundary

with α, β as associated scalars and D as the structure tensor. Then we prove the following:

Theorem 4. If in a compact, orientable N(QE)n(n> 2) without boundary, the associated scalars

and the structure tensor are such that α < 0 and βD(X , X ) < 0, then there exists no non-zero

Killing vector field in this manifold.

Proof. It is known that [17] for a vector field X in a Riemannian manifold M , the following

relation holds
∫

M

h

S(X , X )− |∇X |2− (div X )2
i

dv ≤ 0, (11)

where “dv” denotes the volume element of M . If X is a Killing vector field, then div X = 0

[18]. Hence (11) takes the following form

∫

M

h

S(X , X )− |∇X |2
i

dv = 0. (12)

Let us consider α < 0 and βD(X , X )< 0. Hence by virtue of (2) we have

∫

M=N(QE)n

h

α|X |2+ βD(X , X )− |∇X |2
i

dv

≥

∫

M

h

S(X , X )− |∇X |2
i

dv,

which yields by virtue of (12) that

∫

M

h

α|X |2+ βD(X , X )− |∇X |2
i

dv ≥ 0.



S. HUI / Eur. J. Pure Appl. Math, 5 (2012), 365-372 369

If α < 0 and βD(X , X )< 0, then the last relation reduces to
∫

M

h

α|X |2+ βD(X , X )− |∇X |2
i

dv = 0.

Hence X = 0. This proves the theorem.

Definition 1. [18] A vector field X in a Riemannian manifold (M n, g) (n > 2) is said to be

projective Killing vector field if it satisfies

($X g)(Y, Z) =ω(Y )Z +ω(Z)Y

for any vector fields Y and Z,ω being a certain 1-form and $ is the operator of Lie differentiation.

Theorem 5. If in a compact, orientable N(QE)n(n> 2) without boundary, the associated scalars

and the structure tensor are such that α ≤ 0 and βD(X , X ) ≤ 0, then a projective Killing vector

field has vanishing covariant derivative, and if α < 0 and βD(X , X ) < 0, then there exists no

non-zero projective Killing vector field in this manifold.

Proof. We know that [17] for a vector field X in a Riemannian manifold M , the following

relation holds
∫

M

h

S(X , X )−
1

4
|dξ|2−

n− 1

2(n+ 1)
(div X )2
i

dv = 0, (13)

where ξ is an 1-form corresponding to the vector field X . We now assume α ≤ 0 and

βD(X , X ) ≤ 0. Therefore (13) yields S(X , X ) ≤ 0 and hence from (13) we obtain dξ = 0

and div X = 0. This implies that X is harmonic as well as a Killing vector field. Consequently

its covariant derivative vanishes. This proves the theorem.

Definition 2. [18] A vector field X in a Riemannian manifold (M n, g) (n > 2) is said to be

conformal Killing vector field if it satisfies

$X g = 2ρg

for any vector field X , where ρ is given by ρ = − 1

n
(div X ) and $ is the operator of Lie differenti-

ation.

Theorem 6. If in a compact, orientable N(QE)n(n> 2) without boundary, the associated scalars

and the structure tensor are such that α < 0 and βD(X , X ) < 0, then there exists no non-zero

conformal Killing vector field in this manifold.

Proof. It is known from [17] that for a vector field X in a Riemannian manifold M , the

following relation holds
∫

M

h

S(X , X )− |∇X |2−
n− 2

n
(div X )2
i

dv = 0, (14)

where dv denotes the volume element of M . Now we assume that the associated scalars and

the structure tensor are such that α < 0 and βD(X , X ) < 0. Then proceeding similarly as

before we obtain

∇X = 0, div X = 0.

This proves the theorem.



REFERENCES 370

5. Example of N(QE)n

We define a Riemannian metric g on the n-dimensional real number space Rn by the

formula

ds2 = ekx1

[(d x1)2 + sin2 x3(d x2)2 + (d x3)2] + f (x4)(d x4)2 +

n
∑

l=5

(d x l)2, (15)

where x1 is non-zero finite, 0 < x3 < π

2
, k is a non-zero finite real number excepting ±2 and

f is a positive smooth function of x4 only. Then the only non-vanishing components of the

Christoffel symbols, the curvature tensor, the Ricci tensors are given by

Γ1
11 =

k

2
= Γ2

12 = Γ
3
13 = −Γ

1
33,Γ1

22 = −
k

2
sin2 x3,

Γ3
22 = − sin x3 cos x3,Γ2

23 = cot x3,Γ4
44 =

1

2

f ′(x4)

f (x4)
,

R2332 =
�k2

4
− 1
�

ekx1

sin2 x3,S22 =
�k2

4
− 1
�

sin2 x3,S33 =
�k2

4
− 1
�

.

Here the scalar curvature of the manifold is r = 2( k2

4
− 1)e−kx1

6= 0. Therefore Rn with the

considered metric is a Rimennian manifold (M n, g) of non-vanishing scalar curvature. We

shall now show that this M n is a nearly quasi-Einstein manifold, i.e., it satisfies (2).

Let us now consider the associated scalars and the components of the structure tensor of

D as follows:

α=
1

2

�k2

4
− 1
�

e−kx1

, β =
1

2

�k2

4
− 1
�

, (16)

and

Di j(x) =







sin2 x3 for i, j = 2,2,

1 for i, j = 3,3,

0 otherwise

(17)

at any point x ∈ M .

Then it can be easily shown that the manifold under consideration is nearly quasi-Einstein

manifold. Hence we can state the following:

Theorem 7. Let (M n, g) be a Riemannian manifold endowed with the metric given in (15).

Then (M n, g) is a nearly quasi-Einstein manifold with non-vanishing scalar curvature, which is

not quasi-Einstein.

References

[1] M. C. Chaki and R. K. Maity. On quasi-Einstein manifolds, Publ. Math. Debrecen 57,

297–306. 2000.



REFERENCES 371

[2] U. C. De and A. K. Gaji. On nearly quasi-Einstein manifolds, Novi Sad J. Math., 38(2),

115–121. 2008

[3] F. Defever, R. Deszcz, M. Hotloś, M. Kucharski and Z. Sentürk. Generalisations of

Robertson-Walker spaces, Annales Univ. Sci. Budapest. Eötvös Sect. Math., 43, 13–24.

2000.

[4] R. Deszcz. On pseudosymmetric spaces, Bull. Belg. Math. Soc. Ser A, 44, 1–34. 1992.

[5] R. Deszcz, F. Dillen, L. Verstraelen and L. Vrancken. Quasi-Einstein totally real submani-

folds of S6(1), Tohoku Math. J., 51, 461–478. 1999.

[6] R. Deszcz and M. Glogowska. Some examples of nonsemisymmetric Ricci-semisymmetric

hypersurfaces, Colloq. Math., 94, 87–101. 2000.

[7] R. Deszcz, M. Glogowska, M. Hotloś and Z. Sentürk, Z. On certain quasi-Einstein semi-

symmetric hypersurfaces, Annales Univ. Sci. Budapest. Eötvös Sect. Math., 41, 151–164.

1998.

[8] R. Deszcz and M. Hotloś. On some pseudosymmetry type curvature condition, Tsukuba J.

Math., 27, 13–30. 2003.

[9] R. Deszcz and M. Hotloś On hypersurfaces with type number two in spaces of constant

curvature, Annales Univ. Sci. Budapest. Eötvös Sect. Math., 46, 19–34. 2003.

[10] R. Deszcz and M. Hotloś and Z. Sentürk. Quasi-Einstein hypersurfaces in semi-Riemannian

space forms, Colloq. Math., 81, 81–97. 2001.

[11] R. Deszcz and M. Hotloś and Z. Sentürk. On curvature properties of quasi-Einstein hyper-

surfaces in semi-Euclidean spaces, Soochow J. Math., 27(4), 375–389. 2001.

[12] R. Deszcz, P. Verheyen and L. Verstraelen. On some generalized Einstein metric conditions,

Publ. Inst. Math. (Beograd), 60:74, 108–120. 1996.

[13] D. Ferus. A remark on Codazzi tensors on constant curvature space, Lecture Notes Math.,

838, Global Differential Geometry and Global Analysis, Springer-Verlag, New York,

1981.

[14] A. K. Gaji and U. C. De. On the existence of nearly quasi-Einstein manifolds, Novi Sad J.

Math., 39(2), 111–117. 2009.

[15] A. A. Shaikh, Y. H. Kim and S. K. Hui. On Lorentzian quasi-Einstein manifolds, J. Korean

Math. Society, 48(4), 669–689. 2011.

[16] A. A. Shaikh, D. W. Yoon and S. K. Hui. On quasi-Einstein spacetimes, Tsukuba J. Math.,

33(2), 305–326. 2009.

[17] Y. Watanabe. Integral inequalities in compact orientable manifolds, Riemannian or Kahle-

rian, Kodai Math. Sem. Rep., 20, 261–271. 1968.



REFERENCES 372

[18] K. Yano. Integral formulas in Riemannian Geometry, Marcel Dekker, New York, 1970.

[19] K. Yano and M. Kon. Structure on manifolds, World Scientific Publ., Singapore, 1984.