PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 98(112) (2015), 227–235 DOI: 10.2298/PIM140527003W
GRADIENT RICCI SOLITONS
ON ALMOST KENMOTSU MANIFOLDS
Yaning Wang, Uday Chand De, and Ximin Liu
Abstract. If the metric of an almost Kenmotsu manifold with conformal
Reeb foliation is a gradient Ricci soliton, then it is an Einstein metric and
the Ricci soliton is expanding. Moreover, let (M
2n+1
, φ, ξ, η, g) be an alm ost
Kenmotsu manifold w ith ξ belonging to the (k, µ)
′
-nullity distr ibution and
h 6= 0. If the metric g of M
2n+1
is a gradient Ricci soliton, then M
2n+1
is
lo cally isometric to the Riemannian product of an (n+1)-dimensional manifold
of constant sectional curvature −4 and a flat n-dimensional manifold, also, the
Ricci soliton is expanding with λ = 4n.
1. Introduction
In 1972, K e nmotsu in [12] introduced a special class of almost contact metric
manifolds, which is known as Kenmotsu manifolds nowadays. Since then, many
authors have investigated Kenmotsu manifolds by using various meaningful geo-
metric conditions. Almost Kenmotsu manifolds were first introduced by Janssens
and Vanhecke in [11], generalizing the class of Ke nmotsu manifolds. Recently,
almost Kenmotsu manifolds were investigated by some authors in [5, 6, 13, 14].
On the other hand, in 1982, Hamilton in [9] introduced the notion of the Ricci
flow to find a canonical metric on a smooth manifold. The Ricc i flow is an evolution
equation for metrics on a Riemannian manifold:
(1.1)
∂
∂
t
g
ij
(t) = −2R
ij
.
A Ricci soliton (see [10]) is a generalization of the Einstein metric (that is, the
Ricci tensor is a constant multiple of the Riemannian metric g) and is defined on
2010 Mathematics Subject Classification: Primary 53C25; Secondary 53D15.
Key words and phrases: al most Kenmotsu manif old, gradient Ricci soliton, η-Einstein con-
dition, nullity distribution.
This work was supported by the National Natural Science Foundation of China (grants No.
11371076 and 11431009) and the Research Foundation of Henan Normal University (grant No.
5101019170126 and 5101019279031).
Communicated by Michael Kunzinger.
227
228 WANG, DE, AND LIU
a Riemannian manifold (M, g) by
(1.2)
1
2
L
V
g + Ric +λg = 0
for some constant λ and a vector field V . Clearly, a Ricci soliton with V zero or
a Killing vector field reduces to an Einstein equation. The Ricci soliton is said to
be shrinking, steady and expanding according as λ is negative, zero and positive
respectively. Compact Ricci solitons are the fixed points of the Ricci flow projected
from the space of metrics onto its quotient modulo diffeomorphisms and scalings,
and often arise as blow-up limits for the Ricci flow. If the vector field V is the
gradient of a potential function −f, then g is called a gradient Ricci soliton and
equation (1.2) becomes
(1.3) ∇∇f = Ric +λg.
Following Perelman [15], we know that a Ricc i soliton on a compact manifold is a
gradient Ricci soliton.
Ricci solitons on contact metric manifods, three-dimensional trans-Sasakian
manifolds and N(k)-quasi-Einstein manifolds were studied by Ghosh [8], Tur an,
De, and Yildiz [18] and Crasmareanu [3], re spe c tively. With regard to the studies
of Ricci solitons on Kenmotsu manifolds, we refer the r e ader to De and Matsuyama
[4] and Ghosh [7], respectively. Moreover, Ricci solitons on f -Kenmotsu manifolds
were studied by Călin and Crasmareanu [2]. As far as we know, ther e are no
studies on Ricci solitons on almost K e nmotsu manifolds. The object of this paper
is to investigate gradient Ricci solitons on almost Kenmotsu manifolds under some
geometric conditions. In fact, we mainly obtain the following results in Section 3.
Theorem 1.1. If the metric of an almost Kenmotsu manifold (M
2n+1
, φ, ξ, η, g)
with conformal Reeb foliation is a gradient Ricci soliton, then one of the following
cases occurs:
case 1: n = 1, M
3
is a three dimensional Kenmotsu manifold of constant
sectional curvature −1 and the gradient Ricci soliton is expanding with λ = 2;
case 2: n > 1, M
2n+1
is an Einstein manifold with the Ricci operator Q =
−2n id and the gradient Ricci soliton is expanding with λ = 2n.
The above theorem is a generalization of Theorem 4.1 of [4] (see Co rollary 3.3
in Section 3). Moreover, gradient Ricci solitons on almost Ke nmotsu manifolds
with ξ belonging to certain nullity distribution and h 6= 0 are classified as follows:
Theorem 1.2. Let (M
2n+1
, φ, ξ, η, g) be an almost Kenmotsu manifold with ξ
belonging to the (k, µ)
′
-nullity distribution and h 6= 0. If the metric g of M
2n+1
is a
gradient Ricci soliton, then M
2n+1
is locally isometric to the Riemannian product
of an (n + 1)-dimensional manifold of constant sectional curvature −4 and a flat
n-dimensional manifold. Moreover, the Ricci soliton is expanding with λ = 4n and
the gradient of the potential function is an eigenvector field of h
′
with eigenvalue −1.
The paper is o rganized as follows. In Section 2, we reca ll some well known basic
formulas and properties of almost Kenmotsu manifolds. In Section 3, by applying
some results proved by Pastore a nd Saltarelli, we completely classify the gradi-
ent Ricci solitons on almost Kenmotsu manifolds with conformal Reeb foliation.
GRADIENT RICCI SOLITONS ON ALMOS T KENMOTSU M ANIFOLDS 229
Finally, by using some results obtained by Dileo and Pastore, we a lso obtain the
classification of gr adient Ricci solitons on almost Kenmtsu manifolds with h 6= 0
whose Reeb vector field ξ belongs to the (k, µ)
′
-nullity distribution.
2. Almost Kenmotsu manifolds
Following [5, 6, 12], we first recall s ome basic notions and properties of almost
Kenmotsu manifolds. An almost contact structure (see Blair [1]) on a (2n + 1)-
dimensional smooth ma nifold M
2n+1
is a triplet (φ, ξ, η), where φ is a (1, 1)-type
tensor field, ξ a global vector field (which is called the characteristic vector field)
and η a 1-form, such that
(2.1) φ
2
= −id +η ⊗ ξ, η(ξ) = 1,
where id denotes the identity mapping. This implies that φ(ξ) = 0, η ◦ φ = 0 and
rank(φ) = 2n. A Riemannian metric g on M
2n+1
is said to be compatible with the
almost contact structure (φ, ξ, η) if g(φX, φY ) = g(X, Y ) −η(X)η(Y )for any vector
fields X, Y ∈ Γ(T M ), where Γ(T M ) denotes the Lie algebra of all differentiable
vector fields on M
2n+1
.
An a lmost contact structure endowed with a compatible Riemannian metric is
said to be an almost contact metric structure. The fundamental 2-form Φ of an
almost co ntact metric structure is defined by Φ(X, Y ) = g(X, φY ) for any vector
fields X and Y on M
2n+1
. We may define on the product manifold M
2n+1
×R an
almost complex structure J by
J
X, f
d
dt
=
φX − f ξ, η(X)
d
dt
,
where X denotes a vector field tangent to M
2n+1
, t is the coordinate of R and f is
a C
∞
-function on M
2n+1
× R. From Blair [1], the normality of an almost contact
structure is expressed by the vanishing of the tensor N
φ
= [φ, φ] + 2dη ⊗ ξ, where
[φ, φ] is the Nijenhuis tensor of φ. An almost Kenmotsu manifold is defined as an
almost contact metric manifold such that dη = 0 and dΦ = 2η∧Φ. A normal almost
Kenmotsu manifold is said to be a Kenmotsu manifold (see [11]). It is known [12]
that a Kenmotsu manifold M
2n+1
is locally a warped product I ×
f
M
2n
(where
M
2n
is a Kählerian manifold, I is an ope n interval with coordinate t and f = ce
t
for some positive constant c).
Now let (M
2n+1
, φ, ξ, η, g) be an almost Ke nmotsu manifold. We consider two
tensor fields l = R(·, ξ)ξ and h =
1
2
L
ξ
φ on M
2n+1
, where R denotes the curvature
tensor and L is the Lie differentiation. Following [13], the two (1, 1)-type tensor
fields l and h are symmetric and satisfy
(2.2) hξ = 0, lξ = 0, tr h = 0, tr(hφ) = 0, hφ + φh = 0.
We also have the following formulas pre sented in [5, 6, 13]:
∇
X
ξ = −φ
2
X −φhX (⇒ ∇
ξ
ξ = 0),(2.3)
φlφ − l = 2(h
2
− φ
2
),(2.4)
tr(l) = S(ξ, ξ) = g(Qξ, ξ) = −2n − tr h
2
,(2.5)
230 WANG, DE, AND LIU
R(X, Y )ξ = η(X)(Y − φhY ) − η(Y )(X − φhX) + (∇
Y
φh)X − (∇
X
φh)Y,(2.6)
∇
ξ
h = −φ − 2h − φh
2
− φl(2.7)
for any X, Y ∈ Γ(T M ), where S, Q and ∇ denote the Ricci c urvature tensor, the
Ricci operator with respect to g and the Levi-Civita connection of g, respectively.
On an almost contact metric manifold M , if the Ricci operator satisfies
Q = α id +βη ⊗ ξ,
where both α and β are smooth functions, then M is called an η-Einstein manifold.
Clearly, an η-Einstein manifold with β = 0 and α a constant is an Einstein manifold.
An η-Einstein manifold is said to be proper η-Einstein if β 6= 0.
3. Proofs of main results
We assume that the metric g of an almost Kenmotsu manifold (M
2n+1
, φ, ξ, η, g)
is a gradient s oliton; then equation (1.3) becomes
(3.1) ∇
Y
Df = QY + λY
for any Y ∈ Γ(T M ), where D denotes the gradient operator of g. It fo llows from
(3.1) that
(3.2) R(X, Y )Df = (∇
X
Q)Y − (∇
Y
Q)X
for any X, Y ∈ Γ(T M). Befor e giving the detailed proof of our Theorem 1.1, we
first present the following res ult which is directly deduced from Theorem 5 .1 and
Remark 5.1 of [14]. Throughout this paper, we denote by D the distribution defined
by D = ker η.
Lemma 3.1. Let (M
2n+1
, φ, ξ, η, g) be an η-Einstein almost Kenmotsu manifold
with h = 0, then one of the following cases occurs:
case 1: n = 1, the Ricci operator of M
3
is Q = −(β + 2 ) id +βη ⊗ ξ and
ξ(β) = −2β;
case 2: n > 1, β = 0, the Ricci operator is Q = −2n id and the integral subman-
ifolds of the distribution D are Einstein almost Kählerian Ricci-flat hypersurfaces;
case 3: n > 1, β is not a constant, X(β) = 0 for any X ∈ D and ξ(β) = −2β.
Hence, the Ricci operator is Q = −2n id +βφ
2
, where β is locally given by β = ce
−2t
for some constant c 6= 0.
From Lemma 3.1 we see that for an η-Einstein almost Kenmotsu manifold
M
2n+1
with h = 0, if either α or β is a constant then M
2n+1
is an Einstein
manifold with Q = −2n id.
Proof of Theorem 1.1. From Pastore and Saltarelli [14], we see that the
Reeb foliation of an almost Kenmo tsu manifold is conformal if and only if h = 0.
Using h = 0 in (2.3) and (2.6) we have
(3.3) ∇
X
ξ = X − η(X)ξ
for any Y ∈ Γ(T M ) and
(3.4) R(X, Y )ξ = η(X)Y − η(Y )X
GRADIENT RICCI SOLITONS ON ALMOS T KENMOTSU M ANIFOLDS 231
for any X, Y ∈ Γ(T M ), re spe c tively. Replacing X by ξ in (3.4) gives an equation;
then taking the inner product o f the resulting equation with Df and making use
of the curvature properties, we obtain
(3.5) g(R(ξ, Y )Df, ξ) = −Y (f) + η(Y )ξ(f )
for a ny Y ∈ Γ(T M ). On the other hand, by a straightforward calculation, we may
obtain from (3.2) that
(3.6) g(R(ξ, Y )Df, ξ) = 0
for a ny Y ∈ Γ(T M ), where Qξ = −2nξ (which is deduced from equation (3.4)) and
(3.3) are used. Clearly, from (3.5) and (3.6) we have
Df = ξ(f)ξ.
Substituting the above equation into (3.1) gives
(3.7) QY = (ξ(f) − λ)Y + (Y (ξ(f )) − ξ(f)η(Y ))ξ
for any Y ∈ Γ(T M). Then taking the inner product of relation (3.7) with ξ and
making use of Qξ = −2nξ we obtain
(3.8) Y (ξ(f)) = (λ − 2n)η(Y )
for any Y ∈ Γ(T M ). Finally, putting (3.8) into (3.7) we get
(3.9) Q = (ξ(f ) − λ) id +(λ − ξ(f ) − 2n)η ⊗ ξ
This means that M
2n+1
is an η-Eins tein manifold. In this context, it follows from
Lemma 3.1 that ξ(β) = −2β for n > 1, applying this equation on (3.9) we have
ξ(ξ(f)) = 2(λ −2n −ξ(f )). On the other hand, re placing Y by ξ in equation (3.8)
implies that ξ(ξ(f)) = λ − 2n. Consequently, it follows that ξ(f ) =
λ−2n
2
being a
constant. This means that case 3 of Lemma 3.1 can not occur. Taking into account
equation (3.8) we obtain λ = 2n and hence ξ(f ) = 0, then we see from (3.9) that
Q = −2 n id. It is well known that the curvature tensor of a three dimensional
Riemannian manifold (M
3
, g) is given by
R(X, Y )Z = g(Y, Z)QX − g(X, Z)QY + g(QY, Z)X
− g(QX, Z)Y −
r
2
g(Y, Z)X − g(X, Z)Y
for any X, Y, Z ∈ Γ(T M ), where r denotes the scalar curvature of M
3
. It is also
well known that an almost Kenmotsu manifold of dimension 3 with h = 0 is a
Kenmotsu manifold (see [5]). Then putting Q = −2 id in the above equation
proves case 1 of Theorem 1.1. The proof of case 2 of Theore m 1.1 follows from the
above arguments.
Remark 3.1. We obs e rve from [6, 14] that for an almost Kenmotsu manifold,
the following four conditions a re equivalent: (1) the Reeb foliation is conformal; (2)
ξ belong s to the k-nullity distribution; (3) ξ belongs to the (k, µ)-nullity distribu-
tion; (4) the tensor field h vanishes. Therefore, for an almost Kenmotsu ma nifold
whose Riemanian metric is a g radient Ricci soliton, under one of the above four
conditions, the conclusion of Theorem 1.1 still holds.