Filomat 32:1 (2018), 179–186
https://doi.org/10.2298/FIL1801179D
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Ricci Semisymmetric Almost Kenmotsu Manifolds
with Nullity Distributions
Sharief Deshmukh
a
, Uday Chand De
b
, Peibiao Zhao
c
a
Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh-11451, Saudi Arabia.
b
Department of Pure Mathematics, University of Calcutta, 35, B.C. Road, Kol- 700019, West Bengal, India.
c
Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, P.R. China.
Abstract. The object of the present paper is to characterize Ricci semisymmetric almost Kenmotsu man-
ifolds with its characteristic vector field ξ belonging to the (k, µ)
0
-nullity distribution and (k, µ)-nullity
distribution respectively. Finally, an illustrative example is given.
1. Introduction
Among Riemannian manifolds, the most interesting and most important for applications are the sym-
metric ones. From the local point of view it was introduced independently by Shirokov [15] as a Riemannian
manifold with covariant constant curvature tensor R, that is, with ∇R = 0, where ∇ is the Levi-Civita con-
nection. An extensive theory of symmetric Riemannian manifolds was worked out by Cartan in 1927. As a
generalization of symmetric manifolds Cartan in 1946 introduced the notion of semisymmetric manifolds.
A Riemannian manifold is called semisymmetric if the curvature tensor R satisfies R(X, Y) · R = 0, where
R(X, Y) is considered as a field of linear operators, acting on R. Semisymmetric manifolds were classified
by Szab
´
o, locally in [16]. The classification results of Szab
´
o were presented in the book [4].
A Riemannian manifold is said to be Ricci semisymmetric if R(X, Y) ·S = 0 where S denotes the Ricci tensor
of type (0, 2). A general classification of these manifolds has been worked out recently by V.A. Mirzoyan
[11]. Recently, De and Velimirovi
´
c [5] studied spacetimes with semisymmetric energy momentum tensor.
On the other hand, an odd dimensional manifold M
2n+1
(n ≥ 1) is said to admit an almost contact structure,
sometimes called a (φ, ξ, η)-structure, if it admits a tensor field φ of type (1, 1), a vector field ξ and a 1-form
η satisfying [1, 2]
φ
2
= −I + η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0. (1)
The first and one of the remaining three relations in (1) imply the other two relations in (1). An almost
contact structure is said to be normal if the induced almost complex structure J on M
2n+1
× R defined by
2010 Mathematics Subject Classification. Primary 53C25; Secondary 53C35
Keywords. Almost Kenmotsu manifolds, nullity distribution, Ricci semisymmetric, Einstein manifold.
Received: 29 February 2016; Accepted: 15 June 2016
Communicated by Ljubica Velimirovi
´
c
This work is supported by King Saud University, Deanship of Scientific Research, College of Science Research Centre. The third
author is thankful to National Natural Science Foundation of Chaina No. 11371194 for financial support.
Email addresses: shariefd@ksu.edu.sa (Sharief Deshmukh), uc de@yahoo.com (Uday Chand De), pbzhao@njust.edu.cn (Peibiao
Zhao)
S. Deshmukh et al. / Filomat 32:1 (2018), 179–186 180
J(X, f
d
dt
) = (φX − f ξ, η(X)
d
dt
) is integrable, where X is tangent to M, t is the coordinate of R and f is a smooth
function on M
2n+1
× R. Let 1 be a compatible Riemannian metric with (φ, ξ, η), that is,
1(φX, φY) = 1(X, Y) −η(X)η(Y) (2)
or equivalently,
1(X, φY) = −1(φX, Y) and 1(X, ξ) = η(X)
for all vector fields X, Y ∈ χ(M) = the set of all differentiable vector fields on M.
A Kenmotsu manifold [10] can be defined as a normal almost contact metric manifold such that dη = 0
and dΦ = 2η ∧ Φ where Φ = 1(X, φY). It is well known that Kenmotsu manifolds can be characterize by
(∇
X
φ)Y = 1(φX, Y) −η(Y)φX, for any vector fields X, Y, Z.
Recently in ([7],[8],[12],[13]), almost contact metric manifolds such that η is closed and dΦ = 2η ∧ Φ
are studied and they are called almost Kenmotsu manifolds. Obviously, a normal almost Kenmotsu
manifold is a Kenmotsu manifold. In [6] G. Dileo and A.M. Pastore studied locally symmetric almost
Kenmotsu manifolds. Moreover almost Kenmotsu manifolds satisfying some nullity conditions were also
investigated by G. Dileo and A.M. Pastore [7]. Also for more results on (k, µ)
0
-nullity distribution and (k, µ)-
nullity distribution on almost Kenmotsu manifolds, we refer to A.M. Pastore and V. Saltarelli ([12],[13]).
In recent papers ([17],[18],[19],[20]) Y. Wang and X.M. Liu study almost Kenmotsu manifolds with nullity
distributions. In [18] Y. Wang and X.M. Liu study ξ-Riemannian semisymmetric almost Kenmotsu manifolds
satisfying (k, µ)
0
-nullity distribution and (k, µ)-nullity distribution. Since semisymmetry (R · R = 0) implies
Ricci semisymmetry (R · S = 0), but the converse is not true, in general, in the present paper we generalize
the results of [18] and [6].
The paper is organized as follows:
In section 2, some basic results of almost Kenmotsu manifolds are given. Section 3 deals with Ricci
semisymmetic almost Kenmotsu manifolds with ξ belonging to (k, µ)
0
-nullity distribution. In the next
section we consider Ricci semisymmetric almost Kenmotsu manifolds with ξ belonging to the (k, µ)-nullity
distribution. As a consequence of these results we obtain several corollaries. Finally, an illustrative example
is given.
2. Almost Kenmotsu Manifolds
Let M
2n+1
be an almost Kenmotsu manifold with structure (φ, ξ, η, 1). Let h =
1
2
£
ξ
φ on an almost
Kenmotsu manifold, where £ is the Lie differentiation. We denote by l = R(·, ξ)ξ. The two (1, 1)- type tensor
l and h are symmetric and satisfy [12]
hξ = 0, lξ = 0, tr(h) = 0, tr(hφ) = 0, hφ + φh = 0. (3)
Also we have the following results
∇
X
ξ = −φ
2
X −φhX, (4)
φlφ − l = 2(h
2
− φ
2
), (5)
R(X, Y)ξ = η(X)(Y + h
0
Y) −η(Y)(X + h
0
X) + (∇
X
h
0
)Y −(∇
Y
h
0
)X, (6)
where h
0
= h ◦ φ.
Finally, we recall the definition of the nullity distribution. D.E. Blair, T. Koufogiorgos and B.J. Papantoniou
[3] introduced (k, µ)-nullity distribution on a contact metric manifold (M
2n+1
, φ, ξ, η, 1), which is defined for
any p ∈ M
2n+1
as follows:
N
p
(k, µ) = {Z ∈ T
p
M : R(X, Y)Z = k[1(Y, Z)X −1(X, Z)Y]
+µ[1(Y, Z)hX − 1(X, Z)hY]}, (7)
S. Deshmukh et al. / Filomat 32:1 (2018), 179–186 181
where h =
1
2
£
ξ
φ and (h, k) ∈ R
2
.
In [7], G. Dileo and A.M. Pastore introduced the notion of (k, µ)
0
-nullity distribution on an almost Kenmotsu
manifold (M
2n+1
, φ, ξ, η, 1), which is defined for any p ∈ M
2n+1
as follows:
N
p
(k, µ)
0
= {Z ∈ T
p
M : R(X, Y)Z = k[1(Y, Z)X −1(X, Z)Y]
+µ[1(Y, Z)h
0
X −1(X, Z)h
0
Y]}, (8)
where h
0
= h ◦ φ and (k, µ) ∈ R
2
.
3. ξ Belongs to the (k, µ)
0
-Nullity Distribution
In this section we consider an almost Kenmotsu manifold with ξ belonging to the (k, µ)
0
-nullity distri-
bution and we recall some results stated in [7]. From (8) we have
R(X, Y)ξ = k[η(Y)X −η(X)Y] + µ[η(Y)h
0
X −η(X)h
0
Y], (9)
where (k, µ) ∈ R
2
. We denote by D the contact distribution defined by D = ker(η) = Im(φ). Replacing Y by ξ
in (9) gives lX = k(X −η(X)ξ) + µh
0
X. Using (1) and (3) in the above equation we obtain
φlφX = −k(X −η(X)ξ + µh
0
X).
Substituting the above equation in (5) gives
h
0
2
= (k + 1)φ
2
(⇔ h
2
= (k + 1)φ
2
). (10)
Now let X ∈ D be the eigen vector of h
0
corresponding to the eigen value λ. Then from (10) it follows that
λ
2
= −(k + 1). Hence k ≤ −1 and λ = ±
√
−k −1. We denote the eigenspaces associated with h
0
by [λ]
0
and
[−λ]
0
corresponding to the eigen value λ , 0 and −λ of h
0
respectively. Before proving our main theorem,
we state the following result due to G. Dileo and A.M. Pastore [6, Prop. 4.3].
Lemma 3.1. Let (M
2n+1
, φ, ξ, η, 1) be an almost Kenmotsu manifold such that ξ belongs to the (k, µ)
0
-nullity distri-
bution and h
0
, 0. Then k < −1, µ = −2 and Spec (h
0
) = {0, λ, −λ}, with 0 as simple eigen value and λ =
√
−k −1.
The distribution [ξ] ⊕ [λ]
0
and [−λ]
0
are integrable with totally umbilical leaves, respectively. Furthermore, the
sectional curvature are given as following:
(a) K(X, ξ) = k − 2λ if X ∈ [λ] and
K(X, ξ) = k + 2λ if X ∈ [−λ]
0
,
(b) K(X, Y) = k − 2λ if X, Y ∈ [λ]
0
;
K(X, Y) = k + 2λ if X, Y ∈ [−λ]
0
and
K(X, Y) = −(k + 2) if X ∈ [λ]
0
, Y ∈ [−λ]
0
,
(c) M
2n+1
has constant negative scalar curvature r = 2n(k − 2n).
Theorem 3.2. Let (M
2n+1
, φ, ξ, η, 1) be an almost Kenmotsu manifold with ξ belonging to the (k, µ)
0
-nullity dis-
tribution and h
0
, 0. If M
2n+1
is Ricci semisymmetric, then either 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, or
the manifold is an Einstein manifold.
Proof. From (9) we get on contraction by using (3),
S(X, ξ) = 2nkη(X). (11)
We suppose that the manifold is Ricci semisymmetric. Then (R(X, Y) · S)(U, V) = 0 for all vector fields
X, Y, U, V, which implies
S(R(X, Y)U, V) + S(U, R(X, Y)V) = 0. (12)
S. Deshmukh et al. / Filomat 32:1 (2018), 179–186 182
Also from (9) it follows that
R(X, ξ)Y = k[η(Y)X −1(X, Y)ξ] + µ[η(Y)h
0
X −1(h
0
X, Y)ξ]. (13)
Substituting Y = ξ in (12) and using (13) we obtain
kη(U)S(X, V) −k1(X, U)S(ξ, V) + µη(U)S(h
0
X, V)
−µ1(h
0
X, U)S(ξ, V) + kη(V)S(U, X)
−k1(X, V)S(U, ξ) + µη(V)S(U, h
0
X) −µ1(h
0
X, V)S(U, ξ) = 0. (14)
Again putting U = ξ in (14) and using (11) we get
kS(X, V) + µS(h
0
X, V) − 2nk
2
1(X, V) − 2nkµ1(h
0
X, V) = 0. (15)
Replacing X by h
0
X in (15) and using the fact h
0
2
= (k + 1)φ
2
yields
kS(h
0
X, V) − µ(k + 1)S(X, V) − 2nk
2
1(h
0
X, V) + 2nkµ(k + 1)1(X, V) = 0. (16)
Subtracting k multiple of (15) and µ multiple of (16) we have
(k
2
+ µ
2
(k + 1))[S(X, V) − 2nk1(X, V)] = 0. (17)
Since µ = −2, the above equation reduces to
(k + 2)
2
[S(X, V) − 2nk1(X, V)] = 0. (18)
Now we consider the following two cases:
case 1: k , −2. It follows from (18) that
S(X, V) = 2nk1(X, V), (19)
which implies that the manifold is an Einstein manifold.
case 2: k = −2. It follows from h
0
2
X = (k + 1)φ
2
X for any X ∈ χ(M) that the nonzero eigenvalue of h
0
is
either 1 or −1, that is, λ = ±1. Without loss of any generality we now choose λ = 1, noticing µ = −2 and then
it follows from Lemma 3.1 that K(X, ξ) = −4 and for any X ∈ [λ]
0
and K(X, ξ) = 0 for any X ∈ [−λ]
0
. Also from
Lemma 3.1 we see that K(X, Y) = −4 for any X, Y ∈ [λ]
0
; K(X, Y) = 0 for any X, Y ∈ [−λ]
0
and K(X, Y) = 0 for
any X ∈ [λ]
0
, Y ∈ [−λ]
0
. As is shown in [7] that the distribution [ξ] ⊕ [λ]
0
is integrable with totally geodesic
leaves and the distribution [−λ]
0
is integrable with totally umbilical leaves by H = −(1 − λ)ξ, where H is
the mean curvature vector field for the leaves of [−λ]
0
immersed in M
2n+1
. Noticing that λ = 1, then we
known that two orthogonal distributions [ξ]⊕[λ]
0
and [−λ]
0
are both integrable with totally geodesic leaves
immersed in M
2n+1
. Then we conclude that M
2n+1
is locally isometric to H
n+1
(−4) × R
n
.
This completes the proof.
In [18] the authors studied semisymmetric almost Kenmotsu manifolds with ξ belonging to the (k, µ)
0
-nullity
distribution and in this case they obtain k = −2. Also R · R = 0 implies R · S = 0. Therefore from Theorem
3.2 we obtain the result of [18] by Y. Wang and X. Liu.
Corollary 3.3. Let (M
2n+1
, φ, ξ, η, 1) be an almost Kenmotsu manifold with ξ belonging to the (k, µ)
0
-nullity dis-
tribution and h
0
, 0. If M
2n+1
is semisymmetric, 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.
Again Ricci symmetry (∇S = 0) implies R · S = 0, therefore we can state the following:
S. Deshmukh et al. / Filomat 32:1 (2018), 179–186 183
Corollary 3.4. A Ricci symmetric almost Kenmotsu manifold with ξ belonging to the (k, µ)
0
-nullity distribution
with h
0
, 0 is locally isometric to the Riemannian product of an (n + 1)-dimensional manifold of constant sectional
curvature −4 and a flat n-dimensional manifold, or an Einstein manifold.
The above corollary generalizes the results of [6].
A Riemannain manifold is said to be Ricci-recurrent [14] if the Ricci tensor S is non-zero and satisfies the
condition
(∇
X
S)(Y, Z) = A(X)S(Y, Z),
where A is a non-zero 1-form.
In [9] J.B. Jun, U.C. De and Gautam Pathak prove that a Ricci-recurrent Riemannian manifold is Ricci
semisymmetric. Hence we conclude the following:
Corollary 3.5. A Ricci-recurrent almost Kenmotsu manifold M
2n+1
with ξ belonging to the (k, µ)
0
-nullity distribution
and h
0
, 0 is either locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional
curvature −4 and a flat n-dimensional manifold, or an Einstein manifold.
4. ξ Belongs to the (k, µ)-Nullity Distribution
This section is devoted to study Ricci semisymmetric almost Kenmotsu manifolds with ξ belonging to
the (k, µ)-nullity distribution.
From (7) we have
R(X, Y)ξ = k[η(Y)X −η(X)Y] + µ[η(Y)hX − η(X)hY], (20)
where (k, µ) ∈ R
2
. Now we state the following:
Lemma 4.1. ( [7], Theorem 4.1) Let M
2n+1
be an almost Kenmotsu manifold of dimension 2n + 1. Suppose that the
characteristic vector field ξ belongs to the (k, µ)-nullity distribution. Then k = −1, h = 0 and M
2n+1
is locally a
warped product of an open interval and an almost K¨ahler manifold.
From (20) we get by using Lemma 4.1,
S(X, ξ) = −2nη(X). (21)
By hypothesis the manifold under consideration is Ricci semisymmetric, therefore
(R(X, Y) · S)(U, V) = 0.
Replacing Y by ξ in the above equation gives
S(R(X, ξ)U, V) + S(U, R(X, ξ)V) = 0. (22)
From (20) it follows that
R(X, ξ)U = k[η(U, X) − 1(X, U)ξ] + µ[η(U)hX −1(hX, U)ξ].
By Lemma 4.1 we get
R(X, ξ)U = 1(X, U)ξ − η(U)X. (23)
Now using (23) and (21) in (22) yields
S(X, V) = −2n1(X, V),
which implies that the manifold is an Einstein manifold.
Conversely, if the manifold is an Einstein manifold, then obviously the manifold is Ricci semisymmetric.
Hence we can state the following: