CUBO, A Mathematical Journal

Vol. 24, no. 03, pp. 485–500, December 2022

DOI: 10.56754/0719-0646.2403.0485

Einstein warped product spaces on Lie groups

Buddhadev Pal1

Santosh Kumar1, B

Pankaj Kumar1

1 Department of Mathematics, Institute

of Science, Banaras Hindu University,

Varanasi-221005, India

pal.buddha@gmail.com

thakursantoshbhu@gmail.com B

pankaj.kumar14@bhu.ac.in

ABSTRACT

We consider a compact Lie group with bi-invariant metric,

coming from the Killing form. In this paper, we study Eins-

tein warped product space, M = M1 ×f1 M2 for the cases, (i)
M1 is a Lie group (ii) M2 is a Lie group and (iii) both M1

and M2 are Lie groups. Moreover, we obtain the conditions

for an Einstein warped product of Lie groups to become a

simple product manifold. Then, we characterize the warping

function for generalized Robertson-Walker spacetime, (M =

I ×f1 G2, −dt
2 + f21 g2) whose fiber G2, being semi-simple

compact Lie group of dim G2 > 2, having bi-invariant metric,

coming from the Killing form.

RESUMEN

Consideramos un grupo de Lie compacto con métrica bi-

invariante, que proviene de la forma de Killing. En este

art́ıculo estudiamos espacios productos alabeados de Eins-

tein, M = M1 ×f1 M2 para los casos (i) M1 es un grupo de
Lie (ii) M2 es un grupo de Lie y (iii) ambos M1 y M2 son

grupos de Lie. Más aún, obtenemos condiciones para que

un producto alabeado de Einstein de grupos de Lie sea una

variedad producto simple. Luego, caracterizamos la función

de alabeo para el espacio-tiempo generalizado de Robertson-

Walker, (M = I ×f1 G2, −dt
2 + f21 g2) cuya fibra G2 es un

grupo de Lie compacto semi-simple de dim G2 > 2 con una

métrica bi-invariante, que proviene de la forma de Killing.

Keywords and Phrases: Einstein space, warped product, Lie group, bi-invariant metric, Killing form.

2020 AMS Mathematics Subject Classification: 22E46, 53C21, 53B20.

Accepted: 17 November 2022

Received: 04 June, 2022

©2022 B. Pal et al. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.56754/0719-0646.2403.0485
https://orcid.org/0000-0002-1407-1016
https://orcid.org/0000-0003-0571-9631
https://orcid.org/0000-0001-5778-211X
mailto:pal.buddha@gmail.com
mailto:thakursantoshbhu@gmail.com
mailto:pankaj.kumar14@bhu.ac.in


486 B. Pal, S. Kumar & P. Kumar CUBO
24, 3 (2022)

1 Introduction

R. L. Bishop and B. O’Neill [3], introduced the notion of warped product space to study the

examples of complete Riemannian manifolds of negative sectional curvature. Authors proved that

the completeness of warped space is followed by the completeness of base and fiber spaces. Further,

the results for isometrically immersed warped product manifold into some Riemannian manifold

were considered in [5, 6, 7]. In [9, 19], authors studied the conditions for the warping function to

become a constant by using the relation between the scalar curvatures of a warped manifold with

its base and fiber spaces.

The concept of the warped product has been generalized to the twisted warped product [11, 28],

the doubly warped product and the multiply warped product [25, 32, 33]. A multiply warped

product is a product manifold M = B × M1 × M2 × · · · × Mk, equipped with the metric

g = π∗(gB) + (f1 ◦ π1)2π∗2(g1) + (f1 ◦ π1)
2π∗2(g2) + · · · + (f1 ◦ π1)

2π∗2(gk),

where (B, gB) and (Mi, gi), i ∈ {1, . . . , k}, are pseudo-Riemannian manifolds, fi are smooth func-
tions on (Mi, gi) and πi are projections from M to Mi. In particular, if B = (a, b), k = 1 and

gB = −dt2, then M is known as a generalized Robertson-Walker spacetime [1, 10, 31]. A general-
ized Robertson-Walker spacetime with a fiber of constant scalar curvature is known as a Robertson-

Walker spacetime. The simplest example for Robertson-Walker spacetime is an Einstein static uni-

verse. The product manifold M = M1 × M2 with metric g = (f2 ◦ π2)2π∗1(g1) + (f1 ◦ π1)2π∗2(g2) is
known as a doubly warped product space.

A pseudo-Riemannian manifold M with metric g is an Einstein manifold provided Ric = cg, where

Ric is a Ricci curvature and c is some real constant. The Einstein metric g is of much interest,

both in geometry and physics. A warped product with a constant warping function is considered

as simply Riemannian product. In [2, p. 265], A. L. Besse proposed the question, “Does there

exist a compact Einstein warped product with non-constant warping function?”. Some answers

to the question were given in [16, 30]. If M is an Einstein warped product space of nonpositive

scalar curvature with a compact base manifold, then the warped product space is reduced to a

simply Riemannian product [16]. In [24, 26], authors studied Einstein warped product space by

using quarter and semi symmetric connections. The triviality results for Einstein warped product

space with non-compact base manifold were studied in [30].

In 1976, Milnor investigated the curvature properties of left-invariant metrics in Lie groups [20].

Most of the Lie groups carry the more than one left-invariant metric, because in [18], authors

showed that for a non-Abelian Lie group with a unique left-invariant metric up to homothety,

the group is either the hyperbolic space Hn, or Rn−3 × H3, where H3 is a Heisenberg group.
The Heisenberg group H3 has a unique Riemannian metric up to homothety, whereas it has three



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Einstein warped product spaces on Lie groups 487

metrics in the Lorentzian case [29]. Classifications for four-dimensional nilpotent Lie groups were

considered in [4, 17]. The class of Lie groups obtaining a bi-invariant metric is smaller than that

of Lie groups with a left-invariant metric. In [14, 15], authors study the warped product Einstein

metrics on spaces of constant scalar curvature and homogeneous spaces. The classifications of

warped product Einstein metric were studied in [13]. In [8, 22], the authors study the general

helices and slant helices in three dimensional Lie group equipped with a bi-invariant metric.

In our paper, we discuss the few possible answers to the question “Does there exist a compact

Einstein warped product with non-constant warping function?” for a compact Einstein warped

product of Lie groups. We know that every compact Lie group has a bi-invariant metric and

bi-invariant metric is much easier to handle than the left invariant metric. That is why, we use

the bi-invariant metric in our paper. Now the results of the left invariant metric are still open

to study. Section 2, of this paper includes some of the basic results. The central part of our

paper is section 3, where we prove our main results for a warped product having either base

manifold or fiber manifold is a compact Lie group with bi-invariant metric, coming from the

Killing form. We show that an Einstein warped product space of nonnegative scalar curvature

with a one-dimensional base manifold (Riemannian manifold) and fiber being a compact Lie group

with bi-invariant metric, coming from the Killing form does not exist. Also, the characteristic of

warping function in generalized Robertson-Walker spacetime is studied in Theorem 3.9. Finally,

we give examples of warped products, obtained using a semi-simple compact Lie group taking

bi-invariant metric from the Killing form.

2 Preliminaries

A Lie group G1 is a smooth manifold with a group structure such that the multiplicative and

inverse maps are smooth. To study the geometry of G1, it becomes necessary to associate a left

invariant metric with it. A metric in which left multiplication behaves as an isometry is known

as a left invariant metric, and for a metric in which right multiplication behaves as an isometry is

known as a right invariant metric. Left multiplication and right multiplication on G1, are defined

as La1 : G1 7→ G1, La1x1 = a1x1 and Ra1 : G1 7→ G1, Ra1x1 = x1a1, for all a1, x1 ∈ G1. Let g1 be
the Lie algebra of G1, then an adjoint representation, Ad : G1 7→ g1, of a Lie group G1 is a map
such that Ada1 : g1 7→ g1 is linear isomorphism given by Ada1 = d(Ra−11 ◦ La1)e1 for all a1 ∈ G1.
An inner product g1 on g1 is said to be Ad-invariant if

g1(Ada1X1, Ada1Y1) = g1(X1, Y1),

for all a1 ∈ G1 and X1, Y1 ∈ g1.

A metric g1, which is both left invariant and right invariant is said to be a bi-invariant metric.



488 B. Pal, S. Kumar & P. Kumar CUBO
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The metric g1 is bi-invariant if and only if

g1([S1, K1], T1) = g1(K1, [T1, S1]) = g1(S1, [K1, T1]),

for all S1, K1, T1 ∈ g1. Also, using the Koszul formula and above equation, we obtain

∇S1K1 =
1

2
[S1, K1], ∀ S1, K1 ∈ g1.

Corresponding to bi-invariant metric g1 on m1- dimensional Lie group G1, the Riemann curvature

tensor R, and the Ricci tensor Ric, are given by

R(X1, Y1)Z1 =
1

4

[
[X1, Y1], Z1

]
,

Ric(X1, Y1) =
1

4
g1
(
[X1, Ei], [Y1, Ei]

)
,

where {E1, . . . , Em1}, is an orthonormal frame for g1. From [12, p. 622], we get the existence of
bi-invariant metric on Lie group.

Proposition 2.1. Let G1 be a Lie group with Lie algebra g1 and metric g1, then g1 induces a

bi-invariant metric if and only if Ad(G1) is compact. In other words, every compact Lie group has

a bi-invariant metric.

Also, for a connected Lie group G1, the metric g1 induce a bi-invariant metric if and only if

Ada1 : g1 7→ g1, is skew adjoint for all a1 ∈ G1, which means

g1(Ada1X1, Y1) = −g1(X1, Ada1Y1), ∀ X1, Y1 ∈ g1.

Definition 2.2 ([2, 23]). The Killing form B : g × g 7→ R is a symmetric B(X1, Y1) = B(Y1, X1),
Ad(G1)-invariant B([X1, Y1], Z1) = B(X1, [Y1, Z1]) and bilinear form, defined by

B(X1, Y1) = tr(ad(X1) ◦ ad(Y1)),

where ad(X1) : g1 7→ g1 is a map, sending each Z1 to [X1, Z1], for all X1, Y1, Z1 ∈ g1.

A Killing form on a Lie group G1 is nondegenerate if and only if G1 is semisimple. In case of

compact semisimple Lie group, the Killing form is always negative definite. From [23, p. 304–306],

we have



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Einstein warped product spaces on Lie groups 489

Corollary 2.3. Let G1 be a semisimple compact Lie group with bi-invariant metric g1, then

(a.1) For nondegenerate plane spanned by S and K in g1, the sectional curvature is given by

K =
1

4

( g1([S, K], [S, K])
g1(S, S)g1(K, K) − g1(S, K)g1(S, K)

)
.

(a.2) If the metric g1 is induced from the Killing form, then G1 is an Einstein (Ric1 = −14g1) and
the scalar curvature (τ), is given by

τ =
1

4
dim(G).

It is clear from (a.1), that if g1 is a Riemannian metric then K ≥ 0 and K = 0, if G1 is an Abelian
group.

Let (M1, g1) and (M2, g2) be two pseudo-Riemannian manifolds of dimensions m1, m2 and f1

be a positive smooth function on M1. Then for natural projections π1 : M1 × M2 → M1 and
π2 : M1 × M2 → M2, the warped product (M = M1 ×f1 M2, g) is a product manifold M1 × M2
with the metric

g = π∗1(g1) + (f1 ◦ π1)
2π∗2(g2),

where ∗ representing the pull-back operator and f1 is a warping function on M. Whereas M1 and

M2 are known as the base, and the fiber of (M, g), respectively. Let Ric, Ric1 and Ric2 are Ricci

tensors on M, M1 and M2, respectively. Then from [23, p. 211], we have

Proposition 2.4. Let M = M1 ×f1 M2 be a warped product space, then Ricci tensors on M, M1
and M2, satisfies

Ric = Ric1 −
m2
f1

Hf1 + Ric2 − f♯g2, (2.1)

where f♯ = −f1∆f1+(m2−1)g1(grad f1, grad f1). Here grad f1, Hf1 and ∆f1 denote the gradient
of f1, the Hessian of f1 and the Laplacian of f1, defined as ∆f1 = −trHf1.

Corollary 2.5. The warped product M = M1 ×f1 M2 is an Einstein with Ric = λg if and only if

(a.3) Ric1 = λg1 +
m2
f1

Hf1,

(a.4) (M2, g2) is an Einstein, such that Ric2 = νg2, where ν = f
♯ + λf21 .

3 Main results

Proposition 3.1. Let (M2, g2) be a pseudo-Riemannian manifold and (G1, g1) be a semi-simple

compact Lie group whose bi-invariant metric coming from the Killing form. Then warped product



490 B. Pal, S. Kumar & P. Kumar CUBO
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manifold (M = G1 ×f1 M2, g), is an Einstein manifold (Ric = λg) if and only if

(a.5) Hf1 = −
(1 + 4λ)f1

4m2
g1,

(a.6) (M2, g2) is an Einstein with Ric2 = νg2, where

ν = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 .

Proof. Let (M = G1 ×f1 M2, g) be an Einstein manifold (Ric = λg), where (M2, g2) is a pseudo-
Riemannian manifold and (G1, g1) is a semi-simple compact Lie group taking bi-invariant metric

from the Killing form. Then from (2.1), we have

λg1 + f
2
1 λg2 = Ric1 −

m2
f1

Hf1 + Ric2 − f♯g2, (3.1)

where λ is some constant and f♯ = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1). Now, by restricting
the argument (horizontal and vertical vectors) on G1, M2, and taking Ric1 = −14g1 in (3.1), we
get 


λg1 = −

1

4
g1 −

m2
f1

Hf1,

f21 λg2 = Ric2 − f♯g2.
(3.2)

Conversely, assume that (M = G1 ×f1 M2, g) be a warped product with conditions (a.5) and (a.6).
Then from (2.1), we get

Ric = λg1 +
m2
f1

Hf1 −
m2
f1

Hf1 + νg2 − f♯g2. (3.3)

Since ν = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 , so from (3.3), we have

Ric = λ(g1 + f
2
1 g2) = λg. (3.4)

Proposition 3.2. Let (M1, g1) be a pseudo-Riemannian manifold and (G2, g2) be a semi-simple

compact Lie group whose bi-invariant metric coming from the Killing form. Then warped product

manifold (M = M1 ×f1 G2, g), is an Einstein manifold (Ric = λg) if and only if

(a.7) Ric1 = λg1 +
m2
f1

Hf1,

(a.8) (M2, g2) is an Einstein with Ric2 = νg2, where

ν = −
1

4
= −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 .



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Einstein warped product spaces on Lie groups 491

Proof. Since (G2, g2) is a semi-simple compact Lie group taking bi-invariant metric from the Killing

form, so using Ric2 = −14g2 in (a.6), we have

Ric2 = −
1

4
g2 = νg2 = (f

♯ + λf21 )g2.

Lemma 3.3 ([16]). Let f1 be a smooth function on semi-Riemannian manifold M1, then the

divergence of Hessian tensor satisfies

div(Hf1)(X1) = Ric1(grad f1, X1) − d(∆f1)(X1), (3.5)

for all X1 ∈ ΓTM1.

Theorem 3.4. Let (G1, g1) be a semi-simple compact Lie group of dimension m1 > 2 and whose

bi-invariant metric coming from the Killing form. If 4m2H
f1 + (1 + 4λ)f1g1 = 0, where λ ∈ R,

m2 ∈ N and f1 is a non constant smooth function on G1, then f1 satisfy the condition

ν = −f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 ,

where ν ∈ R.

Proof. The trace of (a.5), provide us

m2
f1

∆f1 +
(1 + 4λ)m1

4
= 0. (3.6)

On differentiating (3.6), we get

m2
f21

(∆f1df1 − f1d(∆f1)) = 0. (3.7)

By the definition of divergence and Hessian for any vector field X1 and g1-orthonormal frame

{E1, . . . , Em1} on G1, we have

div

(
1

f1
Hf1

)
(X1) =

∑
i

ϵi

(
DEi(

1

f1
Hf1)

)
(Ei, X1)

= −
1

f21
Hf1(grad f1, X1) +

1

f1
div(Hf1)(X1), (3.8)

where ϵi = g1(Ei, Ei). Using the fact that Ric1 = −14g1, in equation (3.5), the divergence of
Hessian becomes

div(Hf1)(X1) = −
1

4
g1(grad f1, X1) − d(∆f1)(X1). (3.9)



492 B. Pal, S. Kumar & P. Kumar CUBO
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Also, from (a.5) and Hf1(grad f1, X1) = (DX1df1)(grad f1) =
1
2
d(g1(grad f1, grad f1)), we have

−
1

4
g1(grad f1, X1) =

m2
2f1

d(g1(grad f1, grad f1))(X1) + λdf1(X1). (3.10)

In view of equations (3.9) and (3.10), the equation (3.8) becomes

div

(
1

f1
Hf1

)
=

1

2f21

(
(m2 − 1)d(g1(grad f1, grad f1)) + 2λ1f1df1(X1) − 2f1d(∆f1)

)
. (3.11)

But the divergence of (a.5), implies that div
(

1
f1
Hf1

)
= 0. Hence from (3.11), we get

(m2 − 1)d(g1(grad f1, grad f1)) + 2λ1f1df1 − 2f1d(∆f1) = 0. (3.12)

Therefore from equations (3.7) and (3.12), we obtain

d
(
(m2 − 1)(g1(grad f1, grad f1)) + λ1f21 − f1(∆f1)

)
= d(ν) = 0. (3.13)

Hence from equation (3.13), we can conclude that for a compact Einstein manifold (M2, g2) with

dimension m2 and Ric2 = νg2, the construction of an Einstein warped manifold M = G1 ×f1 M2
is possible.

Corollary 3.5. Let M = G1×f1 M2 be an Einstein warped product space with semi-simple compact
Lie group G1 of dimension m1 > 2 and whose bi-invariant metric coming from the Killing form.

Then M reduces to a simply Riemannian product.

Proof. Rearranging the equation (3.6), we have

∆f1 =
(1 + 4λ)m1

4m2
f1. (3.14)

As λ is a constant, so for λ ≤ −1
4
, equation (3.14) implies that ∆f1 ≤ 0, hence f1 is constant.

Similarly if λ ≥ −1
4
, then ∆f1 ≥ 0. Since according to the weak maximum principle, if f1 is

subharmonic or superharmonic i.e. (∆f1 ≥ 0 or ∆f1 ≤ 0), then f1 is constant [27, p. 75]. Hence
M is a simply Riemannian product.

In our next result, we prove that if fiber space of warped space is also a semi-simple compact Lie

group of dimension m2 > 2 and inherits the bi-invariant metric from the Killing form, then the

only possible values for f1 are ±1.

Corollary 3.6. Let G1 and G2 be semi-simple compact Lie groups of dimensions m1, m2 > 2 and

bi-invariant metric tensors coming from their respective Killing forms. Then M = G1 ×f1 G2 is
an Einstein if and only if f1 = ±1.



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Einstein warped product spaces on Lie groups 493

Proof. Let M = G1 ×f1 G2 be an Einstein, then from Proposition 3.1, Corollary 3.5 and using the
fact that Ric2 = −14g2, we obtain, ν = λ = −

1
4
. Therefore f21 = 1.

Now conversely assume that f1 = ±1, then Ric = Ric1 + Ric2 = −14(g1 + g2) = −
1
4
g, hence

M = G1 × G2 is an Einstein.

Next, we consider those warped product spaces whose base is any pseudo-Riemannian manifold

and fiber space is a semi-simple compact Lie group of dimension m2 > 2, taking bi-invariant metric

from the Killing form.

Theorem 3.7. Let M = M1 ×f1 G2 be an Einstein warped product space with fiber G2 as a
semi-simple compact Lie group of dimension m2 > 2 and having bi-invariant metric coming from

the Killing form. If M has negative scalar curvature, then the warped product becomes a simply

Riemannian product.

Proof. Let M = M1 ×f1 G2 be an Einstein warped product space with fiber G2 as a semi-simple
compact Lie group of dimension m2 > 2, having bi-invariant metric is coming from the Killing

form. Then from (a.4), we can say that

−f1∆f1 + (m2 − 1)g1(grad f1, grad f1) + λf21 = −
1

4
. (3.15)

Since M is an Einstein, therefore the trace of Ric = λg, implies that

τ = λ(m1 + m2), (3.16)

where τ is a scalar curvature of M. Now assume that p1 and p2 are maximum and minimum points

of f1 on M1. Therefore grad f1(p1) = grad f1(p2) = 0, ∆f1(p1) ≥ 0 and ∆f1(p2) ≤ 0. From (3.16)
it is clear that τ ≤ 0, implies λ ≤ 0, therefore

f1(p1)
2 ≥ f1(p2)2 =⇒ λf1(p1)2 ≤ λf1(p2)2 =⇒ λf1(p1)2 +

1

4
≤ λf1(p2)2 +

1

4
, (3.17)

where ν is some constant. Since ∆f1(p2)f(p2) ≤ 0 and ∆f1(p1)f(p1) ≥ 0, therefore from (3.15),
λf1(p2)

2 + 1
4
≤ 0 and λf1(p1)2 + 14 ≥ 0, gives us

λf1(p2)
2 +

1

4
≤ λf1(p1)2 +

1

4
. (3.18)

Comparing equations (3.17) and (3.18), we have f1(p1) = f1(p2) for λ < 0.

Theorem 3.8. Let M = I ×f1 G2 be an Einstein warped product space with the metric g =
dt2 + f21 (t)g2, where I is an open interval in R and G2 is a semi-simple compact Lie group of

dimension m2 > 2 and having bi-invariant metric coming from the Killing form. If M has non



494 B. Pal, S. Kumar & P. Kumar CUBO
24, 3 (2022)

negative scalar curvature, then there does not exist any such f1, so that M = I ×f1 G2 is an
Einstein warped product space.

Proof. Let M = I ×f1 G2 have positive scalar curvature (λ > 0). Then taking f1 = e
u
2 , the Hessian

of f1,

Hf1 =
u′′

2
e

u
2 +

(u′)2

4
e

u
2 .

Using the above equation in (a.7), we have

u′′

2
+

(u′)2

4
= −

λ

m2
. (3.19)

Also, from (a.8), we get

(u′′
2

+
(u′)2

4

)
+ (m2 − 1)

(u′)2

4
+ λ = −

1

4
e−u. (3.20)

Thus from (3.19) and (3.20), we obtain

(u′)2 = −
( 1
m2 − 1

e−u +
4

m2
λ
)
. (3.21)

The possible solutions for (3.21) (with the help of Maple), are



u = − ln

(
− 4λ(m2−1)

m2

)
,

u = − ln
(
− 4(m2−1)

m2
λ
(
1 + tan2

(
−

√
λ
m2

t + c
√

λ
m2

)) )
,

(3.22)

where c is some constant. It is clear from (3.22) that the function u is not well defined. Furthermore,

as u is a real valued function, therefore (u′)2 ≥ 0 and −
(
e−u 1

m2−1
+ 4

m2
λ
)
< 0, for any point on

I. Therefore from equation (3.21), we can conclude that there does not exist any real solution for

the equation.

For λ = 0, (a.7) and (a.8), imply that f′′1 = 0 and f1f
′′
1 + (m2 − 1)(f′1)2 = −

1
4
, respectively. Hence

f1 = at + b =⇒ (m2 − 1)(a)2 = −
1

4
, (3.23)

where a and b are some real constants. Thus from (3.22) and (3.23), we can say that there does

not exist any such f1 such that M = I ×f1 G2 be an Einstein warped product space of non negative
scalar curvature.

Next, we find the characteristic of warping function in generalized Robertson-Walker spacetime,

whose fiber is semi-simple and compact Lie group of dimension m2 > 2.



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Einstein warped product spaces on Lie groups 495

Theorem 3.9. Let M = I ×f1 G2 be an Einstein warped product space with the metric g =
−dt2 + f21 (t)g2, where I is an open interval in R and G2 is a semi-simple compact Lie group of
dimension m2 > 2 and having bi-invariant metric coming from the Killing form. Then

(i) If M is Ricci flat, then there exists a non-constant function f1 on I such that f1 =
1

2
√
m2−1

t+

b, where b is some constant.

(ii) If M has positive scalar curvature (τ > 0) or negative scalar curvature (τ < 0), then there

does not exist any such f1, so that M = I ×f1 G2 be an Einstein warped product space.

Proof. Let M = I ×f1 G2 be an Einstein warped product space with the metric g = −dt2 +f21 (t)g2,
then from Proposition 3.2, we get

f′′1 =
λf1
m2

, and f1f
′′
1 − (m2 − 1)(f

′
1)

2 + λf21 = −
1

4
. (3.24)

From these two differential equations, we obtain

(f′1)
2 −

λ(1 + m2)

m2(m2 − 1)
f21 =

1

4(m2 − 1)
. (3.25)

As λ is constant, therefore to obtain the solutions for differential equation (3.25), we have to

consider all possible values of λ.

(i) If λ = 0, then from (3.25), we obtain

f1 =
1

2
√
m2 − 1

t + b, (3.26)

where b is some constant. Since f1 is also satisfying (3.24), hence in the Ricci flat manifold

case, it is possible to find a non-constant function on I.

(ii) (a) Let M be an Einstein manifold with positive scalar curvature λ > 0, then from (3.25),

the possible solutions are



f1 = ±

√
−m2

4λ(1+m2)
,

f1 =

√
m2(m2−1)

2
√

λ(1+m2)

(
− 1

4(m2−1)
e

√
λ(1+m2)

m2(m2−1)
(c1−t) + e

√
λ(1+m2)

m2(m2−1)
(t−c1)

)
,

f1 =

√
m2(m2−1)

2
√

λ(1+m2)

(
− 1

4(m2−1)
e

√
λ(1+m2)

m2(m2−1)
(t−c1) + e

√
λ(1+m2)

m2(m2−1)
(c1−t)

)
,

(3.27)

where c1 is some constant. As m2 > 2, so f1 = ±
√

−m2
4λ(1+m2)

/∈ R, hence constant
solution of f1 is not possible. From second and third part of (3.27), we have

f′′1 =
λ(1 + m2)

m2(m2 − 1)
f1. (3.28)



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Equation (3.28), showing that second and third part of (3.27), is not satisfying (3.24).

Hence there does not exist such type of f1 which satisfies the equation (3.24) for λ > 0.

(b) Let M be an Einstein manifold with negative scalar curvature λ < 0, then (3.25),

reduced to

(f′1)
2 +

a(1 + m2)

m2(m2 − 1)
f21 =

1

4(m2 − 1)
, (3.29)

where λ = −a and a is some positive real number. The solutions for differential equation
(3.29), are 


f1 = ±

√
m2

4a(1+m2)
,

f1 = ±
√

m2
4a(1+m2)

sin
(√

a(1+m2)
m2(m2−1)

(−t + c1)
)
.

(3.30)

Since solutions obtained in (3.30), are not satisfying the equation (3.24), hence there is

no solution for (3.24).

Examples for warped product of Lie groups

The Lie groups SU(n), n ≥ 2, and SO(n), n ≥ 3 are examples of semi-simple compact Lie groups.
The Lie algebra su(n) of SU(n), set of n × n skew hermitian matrices with zero trace. For n = 2,
the elements of su(2),

X1 =


 a1ι a2 + a3ι
−a2 + a3ι −a1ι


 , a1, a2, a3 ∈ R.

Similarly, If Y1 ∈ su(2), then

Y1 =


 b1ι b2 + b3ι
−b2 + b3ι −b1ι


 , b1, b2, b3 ∈ R.

The basis E1, E2 and E3, for su(2), can be chosen as

E1 =


 0 1
−1 0


 , E2 =


0 ι
ι 0


 , E3 =


ι 0
0 −ι


 .

Hence AdX1 and AdX2, are obtained as



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Einstein warped product spaces on Lie groups 497

AdX1 =




0 −2a1 2a3
2a1 0 −2a2
−2a3 2a2 0


 , AdX2 =




0 −2b1 2b3
2b1 0 −2b2
−2b3 2b2 0


 .

Thus, the Killing form B(X1, Y1) on su(2), will be

B(X1, Y1) = tr(AdX1 ◦ AdY1) = −8a1b1 − 8a2b2 − 8a3b3 = 4tr(X1Y1).

So, we can made the following examples from all the above discussions.

1. The warped product manifold M = SU(2)×f1M2, with metric g = B+f21 g2, where (M2, g2) is
any pseudo-Riemannian manifold and non constant function f1 on SU(2), is not an Einstein.

2. The product manifold M = SU(2)×SO(2), with metric g = B1+B2, is an Einstein manifold,
where B1 and B2 are Killing forms on su(2) and so(2), respectively.

Conclusion 3.10. In [21], Mustafa proved that for every compact manifold G1 there exist a metric

on it such that non trivial Einstein warped products with base G1 cannot be constructed. In our

paper, from Corollary 3.5, we can say that bi-invariant metric generated by the Killing form on

semi-simple compact Lie group G1 is one in which we cannot construct non trivial Einstein warped

product with base G1.

Data availability statement

No new data were created and analyzed in this study.

Acknowledgment

The authors would like to express their thanks and gratitude to the referees for their valuable

suggestions to improve the paper. Also, Santosh Kumar would like to thank the UGC JRF of

India for their financial support, Ref. No. 1068/ (CSIR - UGC NET JUNE 2019).



498 B. Pal, S. Kumar & P. Kumar CUBO
24, 3 (2022)

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	Introduction
	Preliminaries
	Main results