E extracta mathematicae Vol. 31, Núm. 2, 227 – 233 (2016)

A Study on Ricci Solitons in
Generalized Complex Space Form

M.M. Praveena, C.S. Bagewadi

Department of Mathematics, Kuvempu University,
Shankaraghatta - 577 451, Shimoga, Karnataka, India

mmpraveenamaths@gmail.com prof bagewadi@yahoo.co.in

Presented by Marcelo Epstein Received June 18, 2016

Abstract: In this paper we obtain the condition for the existence of Ricci solitons in non-flat
generalized complex space form by using Eisenhart problem. Also it is proved that if (g, V, λ)
is Ricci soliton then V is solenoidal if and only if it is shrinking or steady or expanding de-
pending upon the sign of scalar curvature.

Key words: Kähler manifolds, generalized complex space form, parallel second order covari-
ant tensor field, Einstein space, Ricci soliton.

AMS Subject Class. (2010): 53C15, 53C21, 53C35, 53C55, 53C56.

1. Introduction

Ricci flow is an excellent tool in simplifying the structure of the manifolds.
It is defined for Riemannain manifolds of any dimension. It is a process which
deforms the metric of a Riemannian manifold analogous to the diffusion of
heat there by smoothing out the irregularity in the metric. It is given by

∂g(t)

∂t
= −2 Ric(g(t)),

where g is Riemannian metric dependent on time t and Ric(g(t)) is Ricci
tensor.

Let ϕt : M −→ M, t ∈ R be a family of diffeomorphisms and (ϕt : t ∈ R)
is a one parameter family of abelian group called flow. It generates a vector
field Xp given by

Xpf =
df(ϕt(p))

dt
, f ∈ C∞(M).

If Y is a vector field then LXY = limt→0
ϕ∗t Y −Y

t
is known as Lie derivative

of Y with respect to X. Ricci solitons move under the Ricci flow under
ϕt : M −→ M of the initial metric i.e., they are stationary points of the Ricci

227



228 m.m. praveena, c.s. bagewadi

flow in space of metrics. If g0 is a metric on the codomain then g(t) = ϕ
∗
t g0 is

the pullback of g0, is a metric on the domain. Hence if g0 is a solution of the
Ricci flow on the codomain subject to condition LV g0 + 2Ricg0 + 2λg0 = 0 on
the codomain then g(t) is the solution of the Ricci flow on the domain subject
to the condition LV g + 2Ricg + 2λg = 0 on the domain by [12] under suitable
conditions. Here g0 and g(t) are metrics which satisfy Ricci flow.

Thus the equation in general

LV g + 2S + 2λg = 0, (1.1)

is called Ricci soliton. It is said to be shrinking, steady or expanding according
as λ < 0, λ = 0 and λ > 0. Thus Ricci solitons are generalizations of Einstein
manifolds and they are also called as quasi Einstein manifolds by theoretical
physicists.

In 1923, Eisenhart [6] proved that if a positive definite Riemannian man-
ifold (M, g) admits a second order parallel symmetric covariant tensor other
than a constant multiple of the metric tensor then it is reducible. In 1925,
Levy [8] obtained the necessary and sufficient conditions for the existence of
such tensors. Since then, many others investigated the Eisenhart problem
of finding symmetric and skew-symmetric parallel tensors on various spaces
and obtained fruitful results. For instance, by giving a global approach based
on the Ricci identity. Sharma [11] firstly investigated Eisenhart problem on
non-flat real and complex space forms, in 1989.

Using Eisenhart problem Calin and Crasmareanu [4], Bagewadi and In-
galahalli [7, 1], Debnath and Bhattacharyya [5] have studied the existence of
Ricci solitons in f-Kenmotsu manifolds, α-Sasakian, Lorentzian α - Sasakian
and Trans-Sasakian manifolds.

In 1989 the author Olszak [9] has worked on existence of generalized com-
plex space form. The authors Parveena and Bagewadi [2, 10] extended the
study to some curvature tensors on generalized complex space form. Moti-
vated by these ideas, in this paper, we made an attempt to study Ricci solitons
of generalized complex space form by using Eisenhart problem.

2. Preliminaries

A Kähler manifold is an n(even)-dimensional manifold, with a complex
structure J and a positive-definite metric g which satisfies the following con-
ditions;

J2(X) = −X, g(JX, JY ) = g(X, Y ) and (∇XJ)(Y ) = 0, (2.1)



a study on ricci solitons in generalized complex space form 229

where ∇ means covariant derivative according to the Levi-Civita connection.
The formulae [3]

R(X, Y ) = R(JX, JY ), (2.2)

S(X, Y ) = S(JX, JY ), (2.3)

S(X, JY ) + S(JX, Y ) = 0, (2.4)

are well known for a Kähler manifold.

Definition 2.1. A Kähler manifold with constant holomorphic sectional
curvature c is said to be a complex space form and its curvature tensor is
given by

R(X, Y )Z =
c

4

[
g(Y, Z)X − g(X, Z)Y + g(X, JZ)JY

− g(Y, JZ)JX + 2g(X, JY )JZ
]
.

The models now are Cn, CP n and CHn, depending on c = 0, c > 0 or c < 0.

Definition 2.2. An almost Hermition manifold M is called a generalized
complex space form M(f1, f2) if its Riemannian curvature tensor R satisfies,

R(X, Y )Z = f1{g(Y, Z)X − g(X, Z)Y } + f2{g(X, JZ)JY
− g(Y, JZ)JX + 2g(X, JY )JZ}. (2.5)

3. Parallel symmetric second order covariant tensor and Ricci
soliton in a non-flat generalized complex space form

Let h be a (0, 2)-tensor which is parallel with respect to ∇ that is ∇h = 0.
Applying the Ricci identity [11]

∇2h(X, Y ; Z, W) − ∇2h(X, Y ; W, Z) = 0. (3.1)

We obtain the relation [11]:

h(R(X, Y )Z, W) + h(Z, R(X, Y )W) = 0. (3.2)

Using equation (2.5) in (3.2) and putting X = W = ei, 1 ≤ i ≤ n after
simplification, we get

f1{g(Y, Z)(tr.H) − h(Y, Z)} + f2{h(JY, JZ) − g(Y, JZ)(tr.HJ)
+ 2h(JZ, JY )} − {(n − 1)f1 − 3f2}h(Z, Y ) = 0, (3.3)



230 m.m. praveena, c.s. bagewadi

where H is a (1, 1) tensor metrically equivalent to h. Symmetrization and
anti-symmetrization of (3.3) yield:

[nf1 − 3f2]
f1

h(Z, Y ) −
3f2
f1

h(JY, JZ) = (tr.H)g(Y, Z), (3.4)

[(n − 2)f1 − 3f2]
f2

h(Y, Z) + h(JZ, JY ) = g(Y, JZ)(tr.HJ). (3.5)

Replacing Y, Z by JY, JZ respectively in (3.4) and adding the resultant equa-
tion from (3.4), provide we obtain:

hs(Y, Z) = β.(tr.H)g(Y, Z), (3.6)

where

β =
f1

nf1 − 6f2
.

Replacing Y, Z by JY, JZ respectively in (3.5) and adding the resultant equa-
tion from (3.5), provide we obtain:

ha(Y, Z) =
f2

[(n − 2)f1 − Hf2]
(tr.HJ)g(Y, JZ). (3.7)

By summing up (3.6) and (3.7) we obtain the expression:

h = {β.(tr.H)g + ρ(tr.HJ)Ω}, (3.8)

where

ρ =
f2

[(n − 2)f1 − Hf2]
.

Hence we can state the following.

Theorem 3.1. A second order parallel tensor in a non-flat generalized
complex space form is a linear combination (with constant coefficients) of the
underlying Kaehlerian metric and Kaehlerian 2-form.

Corollary 3.1. The only symmetric (anti-symmetric) parallel tensor of
type (0, 2) in a non-flat generalized complex space form is the Kaehlerian
metric (Kaehlerian 2-form) up to a constant multiple.

Corollary 3.2. A locally Ricci symmetric (∇S = 0) non-flat generalized
complex space form is an Einstein manifold.



a study on ricci solitons in generalized complex space form 231

Proof. If H = S in (3.8) then tr.H = r and tr.HJ = 0 by virtue of (2.4).
Equation (3.8) can be written as

S(Y, Z) = βr g(Y, Z). (3.9)

Remark 3.1. The following statements for non-flat generalized complex
space form are equivalent.

1. Einstein

2. locally Ricci symmetric

3. Ricci semi-symmetric that is R · S = 0 if f1 ̸= 0.

Proof. The statements (1) → (2) → (3) are trivial. Now, we prove the
statement (3) → (1) is true.
Here R · S = 0 means

(R(X, Y ) · S(U, W)) = 0.

Which implies

S(R(X, Y )U, W) + S(U, R(X, Y )W) = 0. (3.10)

Using equations (2.5) in (3.10) and putting Y = U = ei, where {ei} is an
orthonormal basis of the tangent space at each point of the manifold and
taking summation over i (1 ≤ i ≤ n) we get after simplification that

f1{nS(X, W) − rg(X, W)} = 0. (3.11)

If f1 ̸= 0, then (3.11) reduced to

S(X, W) =
r

n
g(X, W). (3.12)

Therefore, we conclude the following.

Lemma 3.1. A Ricci semi-symmetric non-flat generalized complex space
form is an Einstein manifold if f1 ̸= 0.



232 m.m. praveena, c.s. bagewadi

Corollary 3.3. Suppose that on a non-flat generalized complex space
form, the (0, 2) type field LV g + 2S is parallel where V is a given vector field.
Then (g, V ) yield a Ricci soliton if JV is solenoidal. In particular, if the given
non-flat generalized complex space form is Ricci semi-symmetric with LV g
parallel, we have same conclusion.

Proof. From Theorem (3.1) and corollary (3.2), we have λ = −βr as seen
below:

(LV g + 2S)(Y, Z) =
[
β tr(LV g + 2S)g(Y, Z)

+ ρ.tr((LV g + 2S)J)Ω(Y, Z)
]

=
[
2β(div V + r)g(Y, Z) + ρ[2(div JV )Ω(Y, Z)

+ 2(tr.SJ)Ω(Y, Z)
]
,

(3.13)

by virtue of (2.4) the above equation becomes

(LV g + 2S)(Y, Z) =
[
2β(div V + r)g(Y, Z) + 2ρ(div JV )Ω(Y, Z)

]
. (3.14)

By definition (g, V, λ) yields Ricci soliton. If div JV = 0 then div V = 0
becouse JV = iV i.e.,

(LV g + 2S)(Y, Z) = 2βr g(Y, Z) = −2λg(Y, Z). (3.15)

Therefore λ = −βr.

Corollary 3.4. Let (g, V, λ) be a Ricci soliton in a non-flat generalized
complex space form. Then V is solenoidal if and only if it is shrinking or
steady or expanding depending upon the sign of scalar curvature.

Proof. Using equation (3.12) in (1.1) we get

(LV g)(Y, Z) + 2
r

n
g(Y, Z) + 2λg(Y, Z) = 0. (3.16)

Putting Y = Z = ei where {ei} is an orthonormal basis of the tangent space
at each point of the manifold and taking summation over i (1 ≤ i ≤ n), we
get

(LV g)(ei, ei) + 2
r

n
g(ei, ei) + 2λg(ei, ei) = 0. (3.17)

The above equation implies

div V + r + λn = 0. (3.18)



a study on ricci solitons in generalized complex space form 233

If V is solenoidal then div V = 0. Therefore the equation (3.18) can be
reduced to

λ =
−r
n

.

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