Ricci tensors on three-dimensional almost coKähler manifolds

TLDR: In this paper, the Ricci curvature of the Reeb vector field is invariant to the Riemannian curvature tensor in a 3D almost co-Kahler manifold.

Abstract: Let M3 be a three-dimensional almost coKahler manifold such that the Ricci curvature of the Reeb vector field is invariant along the Reeb vector field. In this paper, we obtain some classification results of M3 for which the Ricci tensor is η-parallel or the Riemannian curvature tensor is harmonic.

Full text

Y. WANG
KODAI MATH. J.
39 (2016), 469–483
RICCI TENSORS ON THREE-DIMENSIONAL ALMOST
COKA
¨
HLER MANIFOLDS
Yaning Wang
Abstract
Let M
3
be a three-dimensional almost coKa
¨
hler manifold such that the Ricci
curvature of the Reeb vector field is invariant along the Reeb vector field. In this
paper, we obtain some classification results of M
3
for which the Ricci tensor is
h-parallel or the Riemannian curvature tensor is harmonic.
1. Introduction
In the last several decades, the study of almost contact geometry has been an
interesting research field both from pure mathematical and physical viewpoints.
One important class of di¤erentiable manifolds in the framework of almost
contact geometry is know n as the coKa
¨
hler manifolds, which were first intro-
duced by Blair [1] and studied by Blair [2], Goldberg and Yano [7] and Olszak
et al. [5, 11]. We point out here that the coKa
¨
hler manifolds in this paper are
just the cosymplectic manifolds shown in the above early literatures. The new
terminology was recently adopted by many authors mainly due to Li [8], in which
the author gave a topology constru ction of coKa
¨
hler manifolds via Ka
¨
hler
mapping tori. According to Li’s work, we see that the coK a
¨
hler manifolds are
really odd dimensional analogues of Ka
¨
hler manifolds. We also refer the readers
to a recent survey by Cappelletti-Montano et al. [3] and many references therein
regarding geometric and topological results on coKa
¨
hler manifolds.
As a generalization of coKa
¨
hler manifolds and an analogy of almost Ka
¨
hler
manifolds, the almost coKa
¨
hler manifolds were widely studied by many authors
recently. In particular, D. Perrone in [12] obtained a complete classification of
homogeneous almost coKa
¨
hler manifolds of dimension three and also gave a
local characterization of such manifolds under a condition of local symmetry.
Recently, the present author in [15] proved that on a three-dimensional almost
coKa
¨
hler manifold the conditions of local symmetry and f-symmetry are
469
2010 Mathematics Subject Classification. Primary 53D15; Secondary 53C25.
Key words and phrases. 3-dimensional almost coKa
¨
hler manifolds, h-parallel Ricci tensor,
harmonic curvature tensor, Lie group.
Received May 20, 2015; revised October 27, 2015.

equivalent. Also, D. Perrone in [13] characterized the minimality of the Reeb
vector field of three-dimensional almost coKa
¨
hler manifolds. In addition, a new
local classification of three-dimensional almost coKa
¨
hler manifolds under the
condition ‘‘‘
x
h ¼ 2f fh and kgradðlÞk is a non-zero constant, where f is a
smooth function and l denotes a positive eigen value function of h :¼
1
2
L
x
f’’ was
also provided by Erken and Murathan [6].
In this paper, we aim to study a three-dimensiona l almost coKa
¨
hler manifold
M
3
such that the Ricci curvature of the Reeb vector field is invariant along the
Reeb vector field (this is equivalent to ‘
x
h ¼ 2f fh, where f denotes a smooth
function). Some examples of such manifolds were also provided in Section 3.
If, in addition, the Ricci tensor of M
3
is of Codazzi-type (this is equivalent to
that the Riemannian curvature tensor of M
3
is harmonic), we prove that M
3
is
locally isometric to the product R  N
2
ðcÞ, where N
2
ðcÞ denotes a Ka
¨
hler surface
of constant curvature c ðc ¼ 0 means that M
3
is locally the flat Euclidean space
R
3
Þ. We also prove that if the Ricci tensor of M
3
is h-parallel, then either M
3
is locally the product R  N
2
ðcÞ or M
3
is locally isometric to a Lie group
equipped with a left invariant almost coKa
¨
hler structure. Some applications and
corollaries of our main results are also provided.
2. Preliminaries
On a ð2n þ 1Þ-dimensional smooth manifold M
2nþ1
if there exist a ð1; 1Þ-type
tensor field f, a global vector field x and a 1-form h such that
f
2
¼id þ h n x; hðxÞ¼1;ð2:1Þ
where id denotes the identity endomorphism, then we say that M
2nþ1
admits
an almost contact structure which is denoted by the triplet ðf; x; hÞ and x is called
the characteristic or the Reeb vector field. It follows from relation (2.1) that
fðxÞ¼0, h  f ¼ 0 and rankðfÞ¼2n. We denote by ðM
2nþ1
; f; x; hÞ a smooth
manifold M
2nþ1
endowed with an almost contact structure, which is called
an almost contact manifold. We define an almost com plex structure J on the
product manifold M
2nþ1
 R by
JX; f
d
dt

¼ fX  f x; hðX Þ
d
dt

;ð2:2Þ
where X denotes the vector field tangent to M
2nþ1
, t is the coordinate of R and
f is a smooth function defined on the product M
2nþ1
 R.
An almost contact structure is said to be normal if the above almost complex
structure J is integrable, i.e., J is a complex structure. According to Blair [2],
the normality of an almost contact structure is expressed by ½f; f¼2 dh n x,
where ½f; f denotes the Nijenhuis tensor of f defined by
½f; fðX ; Y Þ¼f
2
½X; Y þ½fX ; fY f½fX ; Y f½X ; fY 
for any vector fields X , Y on M
2nþ1
.
470 yaning wang

If on an almost contact manifold there exists a Riemannian metric g satisfying
gðfX ; fY Þ¼gðX; YÞhðXÞhðYÞð2:3Þ
for any vector fields X and Y ,theng is said to be compatible with the given
almost contact structure. In general, an almost contact manifold equipped with
a compatible Riemannian metric is said to be an almost contact metric manifold
and is denoted by ðM
2nþ1
; f; x; h; gÞ. The fundamental 2-form F on an almost
contact metric manifold M
2nþ1
is defined by FðX; YÞ¼gðX; fY Þ for any vector
fields X and Y .
In this paper, by an almost coKa
¨
hler manifold we mean an almost contact
metric manifold such that both the 1-form h and 2-form F are closed (see [3]).
In particular, an almost coKa
¨
hler manifold is said to be a coKa
¨
hler manifold
(see [8]) if the associated almost contact structure on it is normal, which is also
equivalent to ‘f ¼ 0, or equivalently, ‘F ¼ 0. Notice that (almost) coKa
¨
hler
manifolds are just the (almost) cosymplectic manifolds studied in [1, 2, 5, 7, 11].
The simplest example of (almost) coKa
¨
hler manifolds is the Riemannian product
of a real line or a circle and a (almost) Ka
¨
hler manifold. However, there do
exist some examples of (almost) coKa
¨
hler manifolds which are not globally the
product of a (alm ost) Ka
¨
hler manifold and a real line or a circle (see, for
example, Dacko [11, Section 3]).
On an almost coKa
¨
hler manifold ðM
2nþ1
; f; x; h; gÞ, we shall set h ¼
1
2
L
x
f
and h
0
¼ h  f (notice that both h and h
0
are symmetric operators wi th respect
to the metric g). Then the following formulas can be found in Olszak [11] and
Perrone [12]:
hx ¼ 0; hf þ fh ¼ 0; trðhÞ¼trðh
0
Þ¼0;ð2:4Þ
‘
x
f ¼ 0; ‘x ¼ h
0
; div x ¼ 0;ð2:5Þ
‘
x
h ¼h
2
f  fl;ð2:6Þ
flf  l ¼ 2h
2
;ð2:7Þ
where l :¼ Rð; xÞ x is the Jacobi operator along the Reeb vector field and the
Riemannian curvature tensor R is defined by
RðX ; Y ÞZ ¼ ‘
X
‘
Y
Z  ‘
Y
‘
X
Z  ‘
½X ; Y 
Z;
and tr and div denote the trace and divergence operators, respectively. The well-
known Ricci tensor S is defined by
SðX; Y Þ¼gðQX ; YÞ¼trfZ ! RðZ; XÞY g;
where Q denotes the associated Ricci operator with respective to the metric g.
3. Three-dimensional almost coKa
¨
hler manifolds
In the following, we denote by ðM
3
; f; x; h; gÞ an almost coKa
¨
hler manifold
of dimension three. According to the second term of relation (2.5) we obtain
that ðL
x
gÞðX ; Y Þ¼2gðh
0
X; Y Þ, then we have
471ricci tensors on three-dimensional almost coka
¨
hler manifolds

Lemma 3.1 ([7, Proposition 3]). Any 3-dimensional almost coKa
¨
hler mani fold
is coKa
¨
hler if and only if x is a Killing vector field, or equivalently, h ¼ 0.
Following Perrone [13], let U
1
be the open subset of M
3
on which h 0 0 and
U
2
the open subset defined by U
2
¼fp A M
3
: h ¼ 0 in a neighborhood of pg.
Therefore, U
1
[ U
2
is an open dense subset of M
3
. For any point p A U
1
[ U
2
,
we find a local orthonormal basis fx; e
1
; e
2
¼ fe
1
g of three distinct unit eigen-
vector fields of h in certain neighborhood of p.OnU
1
we assume that he
1
¼ le
1
and hence he
2
¼le
2
, where l is a positive function. Notice that l is con-
tinuous on M
3
and smooth on U
1
[ U
2
.
Lemma 3.2 ([13, Lemma 2.1]). On U
1
we have
‘
x
e
1
¼ fe
2
; ‘
x
e
2
¼fe
1
; ‘
e
1
x ¼le
2
; ‘
e
2
x ¼le
1
;
‘
e
1
e
1
¼
1
2l
ðe
2
ðlÞþsðe
1
ÞÞe
2
; ‘
e
2
e
2
¼
1
2l
ðe
1
ðlÞþsðe
2
ÞÞe
1
;
‘
e
2
e
1
¼ lx 
1
2l
ðe
1
ðlÞþsðe
2
ÞÞe
2
; ‘
e
1
e
2
¼ lx 
1
2l
ðe
2
ðlÞþsðe
1
ÞÞe
1
;
‘
x
h ¼
1
l
xðlÞ id þ 2f f

h;
where f is a smooth function and s is the 1-form defined by sðÞ ¼ Sð; xÞ.
Using the above Lemma 3.2, one obtains that the Ricci operator Q is
expressed as follows (see [13, Proposition 4.1]):
Q ¼ a id þ bh n x þ f‘
x
h  sðf
2
Þn x þ sðe
1
Þh n e
1
þ sðe
2
Þh n e
2
;ð3:1Þ
where a ¼
1
2
ðr þ trðh
2
ÞÞ, b ¼
1
2
ðr þ 3trðh
2
ÞÞ and r denotes the scalar curvature.
Moreover, using Lemma 3.2 we have the following Poisson brackets
½x; e
1
¼ðf þ lÞe
2
; ½e
2
; x¼ðf  lÞe
1
;
½e
1
; e
2
¼
1
2l
ðe
1
ðlÞþsðe
2
ÞÞe
2

1
2l
ðe
2
ðlÞþsðe
1
ÞÞe
1
:
ð3:2Þ
Therefore, from the well-known Jaco bi identity
½½x; e
1
; e
2
þ½½e
1
; e
2
; xþ½½e
2
; x; e
1
¼0;
we get from relation (3.2) that
e
2
ðf þ lÞx
e
1
ðlÞþsðe
2
Þ
2l

þ
e
2
ðlÞþsðe
1
Þ
2l
ðf þ lÞ¼0;
e
1
ðf  lÞþx
e
2
ðlÞþsðe
1
Þ
2l

þ
e
1
ðlÞþsðe
2
Þ
2l
ðf  lÞ¼0:
ð3:3Þ
472 yaning wang

By Lemma 3.2 and relation (3.1) we obtain
Sðx; xÞ¼trðh
2
Þ¼2l
2
:ð3:4Þ
Comparing this with the last term of Lemma 3.2 then we get
Proposition 3.1. On a three-dimensional almost coKa
¨
hler manifold, the Ricci
curvature of the Reeb vector field is invariant along the Reeb vector field if and
only if
‘
x
h ¼ 2f fh;ð3:5Þ
where f is a smooth function.
Next, we present several examples of three-dimensional almost coKa
¨
hler
manifolds satisfying condition (3.5). Firstly, from ðL
x
gÞðX ; Y Þ¼2gðh
0
X ; Y Þ we
see that relation (3.5) holds trivially on any almost coKa
¨
hler manifold with x a
Killing vector field.
Example 3.1. Let M
3
be an almost coKa
¨
hler manifold of dimension 3 such
that the Reeb vector field x belongs to the ðk; m; nÞ-nullity distribution (see [5]),
i.e.,
RðX; Y Þx ¼ kðhðY ÞX  hðX ÞY ÞþmðhðY ÞhX  hðX ÞhY Þ
þ nðhðY Þh
0
X h ðXÞ h
0
YÞ
for any vector fields X , Y , Z, where we have assumed that k is a non-zero
constant and m, n are smooth functions. From (2.4) and the above relation we
know that Sðx; xÞ¼2k is a non-zero constant. Then, applying Lemma 3.2 we
see that relation (3.5) holds on M
3
.
Example 3.2 ([6]). Let us recall the following example constructed in [6].
On a three-dimensional manifold M
3
¼fðx; y; zÞ A R
3
jz > 0g we denote by
x ¼
q
qx
; e ¼ z
2
q
qx
þ 2xz 
z þ y
2z

q
qy
þ
q
qz
; fe ¼
q
qy
:
Consider a Riemannian metric g and a ð1; 1Þ-type tensor field f defined by
g ¼
10 
a
1
a
3
01 
a
2
a
3

a
1
a
3

a
2
a
3
1 þ a
2
1
þ a
2
2
a
3
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
and f ¼
0 a
1
a
1
a
2
a
3
0 a
2
1 þ a
2
2
a
3
0 a
3
a
2
0
B
B
B
B
B
@
1
C
C
C
C
C
A
473ricci tensors on three-dimensional almost coka
¨
hler manifolds