On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature
This note deals with super quasi-Einstein warped product spaces. Here we establish that if M is a super quasi-Einstein warped product space with nonpositive scalar curvature and compact base, then M is simply a Riemannian product space. Next we give an example of super quasi-Einstein space-time. In...
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irk-123456789-1405812018-07-11T01:23:31Z On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature Pahan, S. Pal, B. Bhattacharyya, A. This note deals with super quasi-Einstein warped product spaces. Here we establish that if M is a super quasi-Einstein warped product space with nonpositive scalar curvature and compact base, then M is simply a Riemannian product space. Next we give an example of super quasi-Einstein space-time. In the last section a warped product is defined on it. 2017 Article On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature / S. Pahan, B. Pal, A. Bhattacharyya // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 353-363. — Бібліогр.: 12 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag13.04.353 Mathematics Subject Classification 2000: 53C20, 53B20 http://dspace.nbuv.gov.ua/handle/123456789/140581 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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This note deals with super quasi-Einstein warped product spaces. Here we establish that if M is a super quasi-Einstein warped product space with nonpositive scalar curvature and compact base, then M is simply a Riemannian product space. Next we give an example of super quasi-Einstein space-time. In the last section a warped product is defined on it. |
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Pahan, S. Pal, B. Bhattacharyya, A. On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature Журнал математической физики, анализа, геометрии |
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Pahan, S. Pal, B. Bhattacharyya, A. |
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On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature |
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On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature |
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On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature |
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On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature |
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On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature |
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on compact super quasi-einstein warped product with nonpositive scalar curvature |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature / S. Pahan, B. Pal, A. Bhattacharyya // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 353-363. — Бібліогр.: 12 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT pahans oncompactsuperquasieinsteinwarpedproductwithnonpositivescalarcurvature AT palb oncompactsuperquasieinsteinwarpedproductwithnonpositivescalarcurvature AT bhattacharyyaa oncompactsuperquasieinsteinwarpedproductwithnonpositivescalarcurvature |
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2025-07-10T10:47:40Z |
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Journal of Mathematical Physics, Analysis, Geometry
2017, vol. 13, No. 4, pp. 353–363
doi:10.15407/mag13.04.353
On Compact Super Quasi-Einstein Warped
Product with Nonpositive Scalar Curvature
Sampa Pahan1, Buddhadev Pal2, and Arindam
Bhattacharyya3
1,3Jadavpur University, Department of Mathematics
Kolkata 700032, India
E-mail: 1sampapahan25@gmail.com
3bhattachar1968@yahoo.co.in
2Banaras Hindu University, Institute of Science, Department of Mathematics
Varanasi 221005, India
E-mail: 2pal.buddha@gmail.com
Received August 24, 2015; revised September 27, 2016
This note deals with super quasi-Einstein warped product spaces. Here
we establish that if M is a super quasi-Einstein warped product space with
nonpositive scalar curvature and compact base, then M is simply a Rie-
mannian product space. Next we give an example of super quasi-Einstein
space-time. In the last section a warped product is defined on it.
Key words: Einstein manifold, super quasi-Einstein manifold, Ricci ten-
sor, Hessian tensor, warped product, warping function.
Mathematical Subject Classification 2010: 53C20, 53B20.
1. Introduction
An n-dimensional (n > 2) Riemannian manifold is Einstein if its Ricci tensor
S of type (0,2) is of the form S = αg, where α is a smooth function, which turns
into S = r
ng, r being the scalar curvature of the manifold. The above equation is
also called the Einstein metric condition [1]. Let (Mn, g), n > 2, be a Riemannian
manifold and US = {x ∈M : S 6= r
ng at x}, then the manifold (Mn, g) is said to
be quasi-Einstein manifold [5, 7] if on US ⊂M we have
S − αg = βA⊗A, (1.1)
The first author is supported by UGC JRF of India 23/06/2013(i)EU-V.
c© Sampa Pahan, Buddhadev Pal, and Arindam Bhattacharyya, 2017
Sampa Pahan, Buddhadev Pal, and Arindam Bhattacharyya
where A is a 1-form on US , and α and β are some functions on US . It is clear that
the 1-form A, as well as the function β, is nonzero at every point on US . From
the above definition, it follows that every Einstein manifold is quasi-Einstein.
In particular, every Ricci-flat manifold (e.g., Schwarzchild space-time) is quasi-
Einstein. 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 defined by
g(X, ρ) = A(X) for any vector field 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.
M.C. Chaki introduced the super quasi-Einstein manifold in [4], denoted by
S(QE)n, where the Ricci tensor S of type (0, 2), which is not identically zero,
satisfies the condition
S(X,Y ) = αg(X,Y ) + βA(X)A(Y )
+ γ[A(X)B(Y ) +A(Y )B(X)] + δD(X,Y ), (1.2)
where α, β, γ, δ are scalar functions such that β, γ, δ are nonzero and A,B are two
nonzero 1-forms such that g(X,U) = A(X) and g(X,V ) = B(X), U , V being
unit vectors which are orthogonal, i. e., g(U, V ) = 0 and D is a symmetric (0, 2)
tensor with zero trace which satisfies the condition D(X,U) = 0, ∀X ∈ χ(M).
Here α, β, γ, δ are called the associated scalars, and A,B are called the asso-
ciated main and auxiliary 1-forms, respectively, U, V are the main and auxiliary
generators, and D is called the associated tensor of the manifold.
The notion of a warped product generalizes that of a surface of revolution. It
was introduced in [3] for studying manifolds of negative curvature. Let (B, gB)
and (F, gF ) be two Riemannian manifolds with dim B = m > 0, dimF = k > 0
and f : B → (0,∞), f ∈ C∞(B). Consider the product manifold B × F with its
projections π : B × F → B and σ : B × F → F . The warped product B ×f F is
the manifold B×F with the Riemannian structure such that ‖X‖2 = ‖π∗(X)‖2 +
f2(π(p))‖σ∗(X)‖2 for any vector field X on M . Thus we have gM = gB + f2gF
holds on M . Here B is called the base of M and F the fiber. The function
f is called the warping function of the warped product [10]. We will denote
by RicM , RicB, RicF , and Hf the Ricci curvature of M, the lifts to M of the
Ricci curvatures of B and F , and the Hessian of f , respectively. A Riemannian
manifold is said to be super quasi-Einstein if its Ricci tensor is proportional to
the metric, that is,
RicM = αgM (X,Y ) + βA(X)A(Y )
+ γ[A(X)B(Y ) +A(Y )B(X)] + δD(X,Y ). (1.3)
By τM , τB and τF , we will understand the scalar curvatures of M , B and F , that
is, τM = Tr(RicM ), τB = Tr(RicB) and τF = Tr(RicF ). Therefore we have the
followings [10]:
354 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Compact Super Quasi-Einstein Warped Product
Proposition 1.1. The Ricci curvature Ric of the warped product M =
B ×f F with k = dimF satisfies
(1) Ric(X,Y ) = RicB(X,Y )− k
fH
f (X,Y ),
(2) Ric(X,V ) = 0,
(3) Ric(V,W ) = RicF (V,W )− g(V,W )f#, f# = −∆f
f + k−1
f2 |∇f |2
for any horizontal vectors X,Y (that is X,Y ∈ τ(TB)) and any vertical vectors
V,W (that is V,W ∈ τ(TF )), where Hf and ∆f denote the Hessian of f and the
Laplacian of f given by ∆f = − tr(Hf ), respectively.
Proposition 1.2. Let M = B×f F be a warped product manifold. Then the
scalar curvature of M is given by
τM = τB +
τF
f2
+ 2k
∆f
f
− k(k − 1)
|∇f |2
f2
.
From the above Proposition 1.1 we get the following theorem.
Theorem 1.1. Let M = B ×f F be a warped product manifold which is also
a super quasi-Einstein manifold. Then the following conditions hold.
i) When U, V are orthogonal and tangent to the base B, then the Ricci tensors
of B and F satisfy the following conditions:
a) RicB(X,Y ) = αgB(X,Y ) + βgB(X,U)gB(Y,U)
+ γ[gB(X,U)gB(Y, V ) + gB(Y,U)gB(X,V )]
+ δDB(X,Y ) +
k
f
Hf (X,Y ),
b) RicF (X,Y ) = gF (X,Y )
[
αf2 − f∆f + (k − 1)|∇f |2
]
+ δDF (X,Y );
ii) When U, V are orthogonal and tangent to the fibre F , then the Ricci tensors
of B and F satisfy the following conditions:
a) RicB(X,Y ) = αgB(X,Y ) +
k
f
Hf (X,Y ) + δDB(X,Y ),
b) RicF (X,Y ) = gF (X,Y )
[
αf2 − f∆f + (k − 1)|∇f |2
]
+ βf4gF (X,U)gF (Y,U) + γf4[gF (X,U)gF (Y, V )
+ gF (Y,U)gF (X,V )] + δDF (X,Y ).
Corollary 1.1. Taking the traces of Theorem 1.1, we get the scalar curvature
of M,B and F of two different cases.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 355
Sampa Pahan, Buddhadev Pal, and Arindam Bhattacharyya
i) τM = α(m+k)+β, τB = αm−k∆f
f +β, τF = k
[
αf2 − f∆f + (k − 1)|∇f |2
]
.
ii) τM = α(m+ k) + β, τB = αm− k∆f
f , τF = k
[
αf2 − f∆f + (k − 1)|∇f |2
]
+
βf4.
The proves of Theorem 1.1 and Corollary 1.1 follow similarly to Theorem 2.1
from the paper [12]. We also have the following propositions from [2, 10], where
the expression of Ricci curvature of a warped product space was obtained.
Many authors, like M.C. Chaki [4], C. Özgür [11], etc., have studied su-
per quasi-Einstein manifolds. In [6], D. Dumitru gave a characterization of the
warped product on quasi-Einstein manifold and B. Pal, A. Bhattacharyya studied
a characterization of the warped product on mixed super quasi-Einstein manifold
in [12]. In [9], D. Kim discussed about a compact Einstein warped space with
nonpositive scalar curvature. Motivated by the above papers, in this work we
study super quasi-Einstein warped product spaces with nonpositive scalar cur-
vature. Also, we establish the four-dimensional example of super quasi-Einstein
space-time, and in the last section we give the example of a warped product on
it.
2. Super Quasi-Einstein Warped Product Spaces with
Nonpositive Scalar Curvature
From Proposition 1.1, we get the following result where equation (1.2) be-
comes
Result 2.1. When U , V are orthogonal and tangent to the base B, the warped
product M = B ×f F is a super quasi-Einstein manifold with
RicM (X,Y )=αgM (X,Y )+βA(X)A(Y )+γ[A(X)B(Y )+A(Y )B(X)]+δD(X,Y ),
where D(X,Y ) = g(lX, Y ), l is a symmetric endomorphism if and only if
(2.a) RicB(X,Y ) = αgB(X,Y ) + βgB(X,U)gB(Y,U)
+ γ[gB(X,U)gB(Y, V ) + gB(Y,U)gB(X,V )]
+ δDB(X,Y ) +
k
f
Hf (X,Y ),
(2.b) RicF (X,Y ) = µgF (X,Y ) + δDF (X,Y ),
(2.c) µ =
[
αf2 − f∆f + (k − 1)|∇f |2
]
.
Now, we state a lemma whose detailed proof is given in [9].
356 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Compact Super Quasi-Einstein Warped Product
Lemma 2.1. Let f be a smooth function on a Riemannian manifold B, then
for any vector X, the divergence of the Hessian tensor Hf satisfies
div
(
Hf
)
(X) = Ric(∇f,X)−∆(df)(X), (2.1)
where ∆ = dδ + δd denotes the Laplacian on B acting on differential forms.
Now we prove the following proposition.
Proposition 2.1. Let (Bm, gB) be a compact Riemannian manifold of dimen-
sion m ≥ 2. Suppose that f is a nonconstant smooth function on B satisfying
(2.a) for a constant α ∈ R and a natural number k ∈ N , and if the condition
βgB(X,U)gB(∇f, U) + γ[gB(X,U)gB(∇f, V )
+ gB(∇f, U)gB(X,V )] + gB(lX,∇f) = 0
holds, then f satisfies (2.c) for a constant µ ∈ R. Hence, for a compact Rie-
mannian manifold F with RicF (X,Y ) = µgF (X,Y ) + δDF (X,Y ), we can make
a compact super quasi-Einstein warped product space M = B ×f F with
RicM (X,Y ) = αgM (X,Y ) + βA(X)A(Y )
+ γ[A(X)B(Y ) +A(Y )B(X)] + δD(X,Y ),
where D(X,Y ) = g(lX, Y ), l is a symmetric endomorphism when U , V are
orthogonal and tangent to the base B.
Proof. By taking the trace of both sides of (2.a), we have
S = αm− k∆f
f
+ β, (2.2)
where S denotes the scalar curvature of B given by tr(Ric). Note that the second
Bianchi identity implies (see [10])
dS = 2 div(Ric). (2.3)
From equations (2.2) and (2.3), we obtain
div Ric(X) =
k
2f2
{∆fdf − fd(∆f)(X)}. (2.4)
On the other hand, by the definition, we have
div
(
1
f
Hf
)
(X) =
∑
i
(
DEi
(
1
f
Hf
))
(Ei, X)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 357
Sampa Pahan, Buddhadev Pal, and Arindam Bhattacharyya
= − 1
f2
Hf (∇f,X) +
1
f
divHf (X)
for any vector field X and an orthonormal frame E1, E2, . . . , Em of B. Since
Hf (∇f,X) = (DX df)(∇f) = 1
2 d
(
|∇f |2
)
(X), the last equation becomes
div
(
1
f
Hf
)
(X) = − 1
2f2
d
(
|∇f |2
)
(X) +
1
f
divHf (X)
for a vector field X on B. Hence, from (2.a) and (2.1), it follows that
div(
1
f
Hf )(X) =
1
2f2
{
(k − 1)d
(
|∇f |2
)
− 2f d(∆f) + 2αf df
}
+
1
f
βgB(X,U)gB(∇f, U)
+
1
f
γ[gB(X,U)gB(∇f, V ) + gB(∇f, U)gB(X,V )]
+
1
f
δDB(X,∇f). (2.5)
But, (2.a) gives div RicB = div( kfH
f ) + divDB. Therefore, (2.4) and (2.5) imply
that d
(
−f∆f + (k − 1)|∇f |2 + αf2
)
= 0, that is, −f∆f + (k− 1)|∇f |2 +αf2 =
µ for some constant µ. Thus the first part of the proposition is proved. For a
compact Riemannian manifold (F, gF ) of dimension k with RicF = µgF + δDF ,
we can construct a compact super quasi-Einstein warped product M = B ×f F
by the sufficiencies of Result 2.1.
In a similar way, we get the following result and proposition when U , V are
orthogonal and tangent to the fiber F .
Result 2.2. When U , V are orthogonal and tangent to the fiber F , the warped
product M = B ×f F is a super quasi-Einstein manifold with RicM (X,Y ) =
αgM (X,Y ) + βA(X)A(Y ) + γ[A(X)B(Y ) + A(Y )B(X)] + δD(X,Y ), where
D(X,Y ) = g(lX, Y ), l is a symmetric endomorphism. if and only if
(2.d) RicB(X,Y ) = αgB(X,Y ) +
k
f
Hf (X,Y ) + δDB(X,Y ),
(2.e) RicF (X,Y ) = gF (X,Y )
[
αf2 − f∆f + (k − 1)|∇f |2
]
+ βf4gF (X,U)gF (X,U)
+ γf4[gF (X,U)gF (Y, V ) + gF (Y, U)gF (X,V )] + δDF (X,Y ),
(2.f) µ =
[
αf2 − f∆f + (k − 1)|∇f |2
]
.
358 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Compact Super Quasi-Einstein Warped Product
Proposition 2.2. Let (Bm, gB) be a compact Riemannian manifold of dimen-
sion m ≥ 2. Suppose that f is a nonconstant smooth function on B satisfying
(2.d) for a constant α ∈ R and a natural number k ∈ N , and if the condition
δgB(lX,∇f) = 0 holds, then f satisfies (2.f) for a constant µ ∈ R. Hence, for a
compact super quasi-Einstein manifold F with
RicF (X,Y ) = gF (X,Y )[αf2 − f∆f + (k − 1)|∇f |2 + βf4gF (X,U)gF (Y, U)
+ γf4[gF (X,U)gF (Y, V ) + gF (Y,U)gF (X,V )] + δDF (X,Y ),
we can make a compact super quasi-Einstein warped product space M = B ×f F
with
RicM (X,Y ) = αgM (X,Y ) + βA(X)A(Y )
+ γ[A(X)B(Y ) +A(Y )B(X)] + δD(X,Y ),
where D(X,Y ) = g(lX, Y ), l is a symmetric endomorphism when U , V are
orthogonal and tangent to the fiber F.
Proof. By taking the trace of both sides of (2.d), we have
S = αm− k∆f
f
, (2.6)
where S denotes the scalar curvature of B given by tr(Ric). From equations (2.6)
and (2.3), we obtain
div Ric(X) =
k
2f2
{∆f df − f d(∆f)(X)}. (2.7)
Hence, from (2.d) and (2.1), it follows that
div
(
1
f
Hf
)
(X) =
1
2f2
{(k − 1) d
(
|∇f |2
)
− 2f d(∆f) + 2λf df}+
1
f
δDB(X,∇f). (2.8)
But, (2.d) gives div RicB = div
(
k
fH
f
)
+divDB. Therefore, (2.7) and (2.8) imply
that d(−f∆f + (k − 1)|∇f |2 + λf2) = 0, that is, −f∆f + (k − 1)|∇f |2 + αf2 =
µ for some constant µ. Thus the first part of Proposition 2.2 is proved. For a
compact Riemannian manifold (F, gF ) of dimension k with
RicF (X,Y ) = gF (X,Y )
[
αf2 − f∆f + (k − 1)|∇f |2
]
+ βf4gF (X,U)gF (X,U)
+ γf4[gF (X,U)gF (Y, V ) + gF (Y,U)gF (X,V )] + δDF (X,Y ),
we can construct a compact super quasi-Einstein warped product M = B ×f F
by the sufficiencies of Result 2.2.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 359
Sampa Pahan, Buddhadev Pal, and Arindam Bhattacharyya
Now we prove the following theorem.
Theorem 2.1. Let M = B ×f F be a compact super quasi-Einstein warped
space. If M has nonpositive scalar curvature, then the warped product becomes a
Riemannian product.
Proof. Equations (2.c) and (2.f) become
div(f∆f) + (k − 2)|∇f |2 + αf2 = µ. (2.9)
By integrating (2.9) over B, we get
µ =
k − 2
V (B)
∫
B
|∇f |2 +
α
V (B)
∫
B
f2, (2.10)
where V (B) denotes the volume of B.
1. Suppose k ≥ 3. Let p be a maximum point of f on B. Then we have f(p) >
0,∇f(p) = 0 and ∆f(p) ≥ 0. Hence, from (2.c), (2.f) and (2.10), we obtain the
following:
0 ≤ f(p)∆f(p) = αf2(p)− µ
=
2− k
V (B)
∫
B
|∇f |2 +
α
V (B)
∫
B
(
f2(p)− f2
)
≤ 0. (2.11)
If α < 0, then f is constant.
2. Suppose k = 1, 2. Let p be a minimum point of f on B. Then we have
f(q) > 0,∇f(q) = 0 and ∆f(p) ≤ 0. Hence, from (2.c), (2.f) and (2.10), we
obtain the following:
0 ≥ f(q)∆f(q) = αf2(q)− µ
=
2− k
V (B)
∫
B
|∇f |2 +
α
V (B)
∫
B
(
f2(q)− f2
)
≥ 0. (2.12)
If k = 1 and α < 0, then from (2.12), f is constant. If k = 2 and α = 0, (2.9)
and (2.10) imply that f is harmonic on B, then f is constant. This completes
the proof of the theorem.
3. Example of 4-Dimensional Super Quasi-Einstein Space-Time
Here we construct a nontrivial concrete example of a super quasi-Einstein
space-time. Let us consider a Lorentzian metric g on M4 by
ds2 = gijdx
idxj = −k
r
(dt)2 +
1
c
r − 4
(dr)2 + r2(dθ)2 + (r sin θ)2(dφ)2,
360 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Compact Super Quasi-Einstein Warped Product
where i, j = 1, 2, 3, 4 and k, c are constant. Then the only nonvanishing compo-
nents of Christofell symbols, the curvature tensors, and the Ricci tensors are:
Γ2
33 = 4r − c, Γ1
12 = − 1
2r
, Γ2
22 =
c
2r(c− 4r)
, Γ3
32 = Γ4
42 =
1
r
,
Γ2
33 = 4r − c, Γ4
43 = cot θ, Γ2
44 = (4r − c)(sin θ)2, Γ3
44 = −sin 2θ
2
(3.1)
R1221 = − k(c− 3r)
r3(c− 4r)
, R1331 =
k(c− 4r)
2r2
, R1441 =
k(c− 4r)(sin θ)2
2r2
,
R2332 =
c
2(4r − c)
, R2442 =
c(sin θ)2
2(4r − c)
, R3443 = r(c− 5r)(sin θ)2,
R11 = − k
r3
, R22 = − 3
r(c− 4r)
, R33 = −3, R44 = −3(sin θ)2. (3.2)
From the above, it can be said that M4 is a Lorentzian manifold of the nonva-
nishing scalar curvature and the scalar curvature r1 = − 8
r2 . We shall now show
that this manifold is S(QE)4.
Let us consider the associated scalars α, β, γ and δ and the associated tensor
D as follows:
α = − 3
r2
, β = −1
r
, γ =
1
r
, δ =
1
r2
, (3.3)
and
D11 = 0, D22 =
1
r
, D33 =
1
r
, D44 = −2
r
,
D12 =
2
√
k
r
, D21 =
2
√
k
r
, D13 =
2
√
k
r
, D31 =
2
√
k
r
,
D14 =
√
k
r
, D41 =
√
k
r
, D23 =
√
k
r
, D32 =
√
k
r
,
D24 =
1
2r
, D42 =
1
2r
, D34 =
1
2r
, D43 =
1
2r
, (3.4)
and the 1-forms are given by
Ai(x) =
2
√
k
r for i = 1
1
r for i = 2, 3
−1
r for i = 4
and Bi(x) =
{
− 3
2r for i = 4
0 otherwise.
Then we have
i) R11 = αg11 + βA1A1 + γ[A1B1 +A1B1] + δD11,
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 361
Sampa Pahan, Buddhadev Pal, and Arindam Bhattacharyya
ii) R22 = αg22 + βA2A2 + γ[A2B2 +A2B2] + δD22,
iii) R33 = αg33 + βA3A3 + γ[A3B3 +A3B3] + δD44,
iv) R44 = αg44 + βA4A4 + γ[A4B4 +A4B4] + δD44.
Since all the cases other than (i)–(iv) are trivial, we can say that
Rij = αgij + βAiAj + γ[AiBj +AjBi] + δDij , i, j = 1, 2, 3, 4.
Example 3.1. Let (M4, g) be a Lorentzian manifold endowed with the metric
given by
ds2 = gijdx
idxj = −k
r
(dt)2 +
1
c
r − 4
(dr)2 + r2(dθ)2 + (r sin θ)2(dφ)2,
where i, j = 1, 2, 3, 4 and k, c are constant. Then (M4, g) is an S(QE)4 space-time
with nonvanishing and nonconstant scalar curvature.
4. Example of Warped Product on Super Quasi-Einstein
Space-Time
Here we consider the example (3.1), a 4-dimensional example of super quasi-
Einstein space-time endowed with the Lorentzian metric given by
ds2 = gijdx
idxj = −k
r
(dt)2 +
1
c
r − 4
(dr)2 + r2(dθ)2 + (r sin θ)2(dφ)2,
where i, j = 1, 2, 3, 4 and k, c are constant. Now we have already proved that it
is a super quasi-Einstein space-time with nonzero and constant scalar curvature.
Therefore the above space-time of the form R×f ( c4 ,∞)×S2, where S2 is the
2-dimensional Euclidean sphere, the warping function f : R→ (0,∞) is given by
f(t) = 1√
c
r
−4
, r < c
4 . Here R is the base B, and F = ( c4 ,∞) × S2 is the fiber.
Therefore the metric ds2
M = ds2
B + f2ds2
F , that is,
ds2 = gijdx
idxj =
−k
r
(dt)2 +
1
c
r − 4
[
(dr)2 + (cr − 4r2)((dθ)2 + sin2 θ(dφ)2)
]
,
is the example of a warped product on S(QE)4 space-time.
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