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arXiv:1612.07063v2 [math.DG] 2 Aug 2017
A NEW CLASS OF METRIC f-MANIFOLDS
PABLO ALEGRE, LUIS M. FERN ´
ANDEZ, AND ALICIA PRIETO-MART´
IN
Abstract. We introduce a new general class of metric f-man-
ifolds which we call (almost) trans-S-manifolds and includes S-
manifolds, C-manifolds, s-th Sasakian manifolds and generalized
Kenmotsu manifolds studied previously. We prove their main prop-
erties and we present many examples which justify their study.
1. Introduction
In complex geometry, the relationships between the different classes
of manifolds can be summarize in the well known diagram by Blair [3]:
Complex metric
//Hermitian dΩ=0 //Kaehler
Almost
Complex
[J,J]=0
OO
metric
//
Almost
Hermitian
[J,J]=0
OO
dΩ=0 //
∇J=0
88
r
r
r
r
r
r
r
r
r
r
r
Almost
Kaehler
[J,J]=0
OO
And the same for contact geometry:
Normal Almost
Contact
metric
//
Normal Almost
Contact Metric
Φ=dη
//Sasakian
Almost
Contact
normal
OO
metric
//
Almost
Contact Metric
normal
OO
Φ=dη
//
(1)
66
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
Contact
Metric
normal
OO
2010 Mathematics Subject Classification. 53C15,53C25,53C99.
Key words and phrases. Almost trans-S-manifold, trans-S-manifold, generalized
D-conformal deformation, warped product.
The first and the second authors are partially supported by the project
MTM2014-52197-P (MINECO, Spain).
1
2 PABLO ALEGRE, LUIS M. FERN´
ANDEZ, AND ALICIA PRIETO-MART´
IN
In the above diagram the almost contact structure (φ, η, ξ) is said to
be normal if [φ, φ] + 2dη ⊗ξ= 0 and condition (1) is
(∇Xφ)Y=g(X, Y )ξ−η(Y)X,
for any tangent vector fields Xand Y.
Moreover, an almost contact metric manifold (M, φ, ξ , η, g) is said to
have an (α, β) trans-Sasakian structure if (see [11] for more details)
(1.1) (∇Xφ)Y=α{g(X, Y )ξ−η(Y)X}+β{g(φX, Y )ξ−η(Y)φX},
where α, β are differentiable functions (called characteristic functions)
on M. Particular cases of trans-Sasakian manifolds are Sasakian (α=
1, β = 0), cosymplectic (α=β= 0) or Kenmotsu (α= 0, β = 1)
manifolds. In fact, we can extend the above diagram to
Normal Almost
Contact Metric
(1.3)
//T rans −Sasakian
Almost
Contact Metric
normal
OO
(1.2)
//
(1.1)
77
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
♥
Almost
T rans −Sasakian
normal
OO
where
(1.2) dΦ = Φ ∧(φ∗(δΦ) −(δη)η), dη =1
2n{δΦ(ξ)Φ −2η∧φ∗(δΦ)}.
and:
(1.3) dΦ = −1
nδη(Φ ∧η), dη =1
2nδΦ(ξ)Φ, φ∗(δΦ) = 0.
More in general, K. Yano [15] introduced the notion of f-structure
on a (2n+s)-dimensional manifold as a tensor field fof type (1,1) and
rank 2nsatisfying f3+f= 0. Almost complex (s= 0) and almost
contact (s= 1) structures are well-known examples of f-structures. In
this context, D.E. Blair [2] defined K-manifolds (and particular cases of
S-manifolds and C-manifolds). Then, K-manifolds are the analogue of
Kaehlerian manifolds in the almost complex geometry and S-manifolds
(resp., C-manifolds) of Sasakian manifolds (resp., cosymplectic mani-
folds) in the almost contact geometry. Consequently, one can obtain
a similar diagram for metric f-manifolds, that is, manifolds endowed
with an f-structure and a compatible metric.
The purpose of the present paper is to introduce a new class of metric
f-manifolds which generalizes that one of trans-Sasakian manifolds. In
this context, we notice that there has been a previous generalization of
A NEW CLASS OF METRIC f-MANIFOLDS 3
(α, 0)-trans-Sasakian manifolds for metric f-manifolds. It was due to I.
Hasegawa, Y. Okuyama and T. Abe who introduced the so-called ho-
mothetic s-contact Riemannian manifolds in [8] as metric f-manifolds
such that 2cig(fX, Y ) = dηi(X, Y ) for certain nonzero constants ci,
i= 1,...,s (actually, they use pinstead of s). In particular, if the
structure vector fields ξiare Killing vector fields and the f-structure
is also normal, the manifold is called a homothetic s-th Sasakian man-
ifold. They proved that a homothetic s-contact Riemannian manifold
is a homothetic s-th Sasakian manifold if and only if
(∇Xf)Y=−
s
X
i=1
ci{g(f X, fY )ξi+ηi(Y)f2X},
and
∇Xξi=cifX,
for any tangent vector fields Xand Yand any i= 1,...,s.
More recently, M. Falcitelli and A.M. Pastore have introduced f-
structures of Kenmotsu type as those normal f-manifolds with dF =
2η1∧Fand dηi= 0 for i= 1,...,s [5]. In this context, L. Bhatt and
K.K. Dube [1] and A. Turgut Vanli and R. Sari [14] have studied a
more general type of Kenmotsu f-manifolds for which all the structure
1-forms ηiare closed and:
dF = 2
s
X
i=1
ηi∧F.
These examples justify the idea of introducing the mentioned new
more general class of metric f-manifolds, including the above ones,
which we shall call trans-S-manifolds because trans-Sasakian manifolds
become to be a particular case of them.
The paper is organized as follows: after a preliminaries section con-
cerning metric f-manifolds, in Section 3 we define almost trans-S-
manifolds and trans-S-manifolds in terms of the derivative of the f-
structure and some characteristic functions and study their main prop-
erties. Specially, we prove a characterization theorem which gives a
necessary and sufficient condition for an almost trans-S-manifold to
be a trans-S-manifold, that is, for the normality of the structure, con-
cerning the derivative of the structure vector fields in any direction.
Moreover, we observe that S-manifolds, C-manifolds and Kenmotsu
f-manifolds actually are trans-S-manifolds. On the other hand, we
get some desirable conditions to be satisfied for trans-S-manifolds in
order to generalize those ones of trans-Sasakian manifolds. By using
them, we characterize what trans-S-manifolds are K-manifolds and we
4 PABLO ALEGRE, LUIS M. FERN´
ANDEZ, AND ALICIA PRIETO-MART´
IN
justify that these two classes of metric f-manifolds are not related by
inclusion.
Finally, in the last section, we present many non-trivial examples,
that is, with non-constant characteristic functions, of (almost) trans-S-
manifolds. To this end, we use generalized D-conformal deformations
and warped products as tools.
Acknowledgement: The first and the second authors are partially
supported by the project MTM2014-52197-P (MINECO, Spain).
2. Metric f-manifolds
A (2n+s)-dimensional Riemannian manifold (M, g) endowed with an
f-structure f(that is, a tensor field of type (1,1) and rank 2nsatisfying
f3+f= 0 [15]) is said to be a metric f-manifold if, moreover, there
exist sglobal vector fields ξ1,...,ξson M(called structure vector fields)
such that, if η1,...,ηsare the dual 1-forms of ξ1,...,ξs, then
fξα= 0; ηα◦f= 0; f2=−I+
s
X
α=1
ηα⊗ξα;
(2.1) g(X, Y ) = g(f X, fY ) +
s
X
i=1
ηi(X)ηi(Y),
for any X, Y ∈ X (M) and i= 1,...,s. The distribution on Mspanned
by the structure vector fields is denoted by Mand its complementary
orthogonal distribution is denoted by L. Consequently, T M =L ⊕ M.
Moreover, if X∈ L, then ηα(X) = 0, for any α= 1,...,s and if
X∈ M, then f X = 0.
For a metric f-manifold Mwe can construct very useful local or-
thonormal basis of tangent vector fields. To this end, let Ube a
coordinate neighborhood on Mand X1any unit vector field on U,
orthogonal to the structure vector fields. Then, f X1is another unit
vector field orthogonal to X1and to the structure vector fields too.
Now, if it is possible, we choose a unit vector field X2orthogonal to
the structure vector fields, to X1and to fX1. Then, f X2is also a unit
vector field orthogonal to the structure vector fields, to X1, to f X1
and to X2. Proceeding in this way, we obtain a local orthonormal basis
{Xi, f Xi, ξj},i= 1,...,n and j= 1,...,s, called an f-basis.
Let Fbe the 2-form on Mdefined by F(X, Y ) = g(X, f Y ), for any
X, Y ∈ X (M). Since fis of rank 2n, then
η1∧ · · · ∧ ηs∧Fn6= 0
A NEW CLASS OF METRIC f-MANIFOLDS 5
and, in particular, Mis orientable. A metric f-manifold is said to be
ametric f-contact manifold if F= dηi, for any i= 1,...,s.
The f-structure fis said to be normal if
[f, f ] + 2
s
X
i=1
ξi⊗dηi= 0,
where [f, f ] denotes the Nijenhuis tensor of f. If fis normal, then [7]
(2.2) [ξi, ξj] = 0,
for any i, j = 1,...,s.
A metric f-manifold is said to be a K-manifold [2] if it is normal
and dF= 0. In a K-manifold M, the structure vector fields are Killing
vector fields [2]. A K-manifold is called an S-manifold if F= dηi, for
any iand a C-manifold if dηi= 0, for any i. Note that, for s= 0, a
K-manifold is a Kaehlerian manifold and, for s= 1, a K-manifold is
a quasi-Sasakian manifold, an S-manifold is a Sasakian manifold and
aC-manifold is a cosymplectic manifold. When s≥2, non-trivial
examples can be found in [2, 8]. Moreover, a K-manifold Mis an
S-manifold if and only if
(2.3) ∇Xξi=−fX, X ∈ X (M), i = 1,...,s,
and it is a C-manifold if and only if
(2.4) ∇Xξi= 0, X ∈ X (M), i = 1,...,s.
It is easy to show that in an S-manifold,
(2.5) (∇Xf)Y=
s
X
i=1 g(f X, fY )ξi+ηi(Y)f2X,
for any X, Y ∈ X (M) and in a C-manifold,
(2.6) ∇f= 0.
3. Definition of trans-S-manifolds and main properties
The original idea to define (α, β) trans-Sasakian manifolds is to gen-
eralize cosymplectic, Kenmotsu and Sasakian manifolds.
6 PABLO ALEGRE, LUIS M. FERN ´
ANDEZ, AND ALICIA PRIETO-MART´
IN
Kenmotsu:
dη = 0, normal
Cosymplectic:
dΦ = 0, dη = 0,
normal
Sasakian:
Φ = dη, normal
Quasi-Sasakian:
dΦ = 0,
normal
Trans-Sasakian:
dΦ = 2β(Φ ∧η),
dη =αΦ,
φ∗(δΦ) = 0,
normal
In the same way, our idea is to define trans-S-manifolds generalizing
C-manifolds, f-Kenmotsu and S-manifolds.
As we said in the Introduction, an almost contact manifold is trans-
Sasakian if and only if it (1.1) holds. This property aims us to introduce
trans-S-manifolds.
Definition 3.1. A(2n+s)-dimensional metric f-manifold Mis said
to be a almost trans-S-manifold if it satisfies
(∇Xf)Y=
s
X
i=1 αi{g(fX, f Y )ξi+ηi(Y)f2X}
+βi{g(f X, Y )ξi−ηi(Y)fX}],
(3.1)
for certain smooth functions (called the characteristic functions)
αi, βi,i= 1....s, on Mand any X, Y ∈ X (M). If, moreover, Mis
normal, then it is said to be a trans-S-manifold.
So, if s= 1, a trans-S-manifold is actually a trans-Sasakian manifold.
Furthermore, in this case, condition (3.1) implies normality. However,
for s≥2, this does not hold. In fact, it is straightforward to prove
that, for any X, Y ∈ X (M),
[f, f ](X, Y )+2
s
X
i=1
dηi(X, Y )ξi
=
s
X
i,j=1
[ηj(∇Xξi)ηj(Y)−ηj(∇Yξi)ηj(X)] ξi,
(3.2)
which is not zero in general. But, in a trans-S-manifold, (3.2) implies
that, for any X∈ X (M) and any i= 1,...,s:
s
X
j=1
ηj(∇Xξi)ηj(Y)−
s
X
j=1
ηj(∇Yξi)ηj(X) = 0.
A NEW CLASS OF METRIC f-MANIFOLDS 7
If we put Y=ξk, from (2.2) we get that
(3.3) ηk(∇Xξi) = 0,
for any i, k = 1,...,s. Using this fact, from (3.1), we deduce that
(3.4) ∇Xξi=−αifX −βif2X,
for any X∈ X (M) and any i= 1,...,s.
Now, we can prove:
Theorem 3.1. A almost trans-S-manifold Mis a trans-S-manifold if
and only if (3.4) holds for any X∈ X (M)and any i= 1,...,s.
Proof. From (3.1) we have that, for any X∈ X (M) and any i=
1,...,s:
∇Xξi=−αifX −βif2X+
s
X
j=1
ηj(∇Xξi)ξj.
Comparing this equality and (3.4) we have that ηj(∇Xξi) = 0, for
any i, j = 1, . . . , s. So, from (3.2), the metric f-manifold Mis normal
and, consequently, a trans-S-manifold. The converse is obvious.
Observe that (3.1) can be re-written as
(∇XF)(Y, Z ) =
s
X
i=1
[αi{g(f X, fZ)ηi(Y)−g(f X, fY )ηi(Z)}
+βi{g(X, f Y )ηi(Z)−g(X, f Z)ηi(Y)}],
for any X, Y, Z ∈ X (M). Then, if X∈ L is a unit vector field, we
have:
(∇XF)(X, ξi) = −αi,(∇XF)(ξi, f X) = βi, i = 1,...,s.
Moreover, from (3.4), we deduce
(∇Xηi)Y=αig(X, f Y ) + βig(f X, fY ),
for any X, Y ∈ X (M) and any i= 1,...,s. Again, if X∈ L is a unit
vector field, we get:
(∇Xηi)fX =−αi,(∇Xηi)X=βi, i = 1,...,s.
For trans-S-manifolds, we can prove the following theorem.
Theorem 3.2. Let Mbe a trans-S-manifold. Then, (δ F )ξi= 2nαi
and δηi=−2nβi, for any i= 1,...,s.
8 PABLO ALEGRE, LUIS M. FERN ´
ANDEZ, AND ALICIA PRIETO-MART´
IN
Proof. Taking a f-basis {X1,...,Xn, fX1,...,fXn, ξ1,...,ξs}, since
(δF )X=−
n
X
k=1
{(∇XkF)(Xk, X) + (∇fXkF)(f Xk, X )}
−
s
X
j=1
(∇ξjF)(ξj, X)
=
n
X
k=1
{g(Xk,(∇Xkφ)X) + g(fXk,(∇fXkφ)X)},
for any X∈ X (M), by using (3.1) it is straightforward to obtain
(3.5) (δF )X= 2n
s
X
j=1
αjηj(X)
and, putting X=ξi, it follows that (δF )ξi= 2nαi.
Moreover,
δηi=−
n
X
k=1
{(∇Xkηi)Xk+ (∇fXkηi)fXk} −
s
X
j=1
(∇ξjηi)ξj,
for any i= 1, . . . , s. But, from (3.3) we get that (∇ξjηi)ξj= 0, for any
j= 1,...,s. Consequently, by using (3.4)
δηi=−
n
X
k=1
{g(Xk,∇Xkξi) + g(fXk,∇fXkξi)}
=−
n
X
k=1
βi{g(Xk, Xk) + g(fXk, fXk)}=−2nβi.
which concludes the proof.
The above theorem generalizes the result given by D.E. Blair and
J.A. Oubi˜na in [4] for trans-Sasakian manifolds. Moreover, trans-S-
manifolds verify certain desirable conditions.
Proposition 3.1. Let Mbe a trans-S-manifold. The following equa-
tions are verified:
(i) dF = 2F∧
s
X
i=1
βiηi;
(ii) dηi=αiF,i= 1,...,s;
(iii) f∗(δF ) = 0.
A NEW CLASS OF METRIC f-MANIFOLDS 9
Proof. From (3.1), a direct computation gives, for any X, Y, Z ∈ X (M):
dF (X, Y, Z) = −g((∇Xf)Y, Z) + g((∇Yf)X, Z)−g((∇Zf)X, Y )
=2
s
X
i=1
{−βiηi(Z)g(fX, Y ) + βiηi(Y)g(f X, Z)
−βiηi(X)g(f Y, Z)}
=2(F∧
s
X
i=1
βiηi)(X, Y, Z).
Next, from (3.4) it is obtained the second statement. Finally, from
(3.5) we get (iii).
From (ii) of the above proposition we observe that if one of the
functions αiis a non-zero constant function, then the 2-form Fis closed
and the trans-S-manifold Mis a K-manifold. Moreover we can prove:
Theorem 3.3. A trans-S-manifold Mis a K-manifold if and only if
β1=···=βs= 0.
Proof. Firstly, if all the functions βiare equal to zero, from (i) of
Proposition 3.1 we get dF = 0 and Mis a K-manifold.
Conversely, it is known (see [6]) that, for K-manifolds, the following
formula holds, for any X, Y, Z ∈ X (M):
g((∇Xf)Y, Z ) =
s
X
i=1
{dηi(f Y, X)ηi(Z)−dηi(fZ, X)ηi(Y)}.
Consequently, from (ii) of Proposition 3.1 and (3.1) we conclude that
βi= 0, for any i= 1,...,s.
From Theorem 3.2 we deduce:
Corollary 3.1. A trans-S-manifold Mis a K-manifold if and only if
δηi= 0, for any i= 1,...,s.
Furthermore, taking into account (2.3) and (2.4), we have:
Corollary 3.2. Any trans-S-manifold is an S-manifold if and only if
αi= 1,βi= 0 and it is a C-manifold if and only if αi=βi= 0, in
both cases for any i= 1,...,s.
In next section, we shall present some examples of trans-S-manifolds
which are not K-manifolds due to not all their characteristic functions
βiare zero. Now, the natural question is if any K-manifold is a trans-
S-manifold. In general, the answer in negative and to this end, we can
consider the following example.
10 PABLO ALEGRE, LUIS M. FERN ´
ANDEZ, AND ALICIA PRIETO-MART´
IN
Let (N, J, G) be a Kaehler manifold, (M, f, ξ1,...,ξs, η1,...,ηs, g)
be an S-manifold and f
M=N×M.
If e
X=U+X, e
Y=V+Y∈ X (f
M), where U, V ∈ X (N) and
X, Y ∈ X (M), respectively, we can define a metric f-structure on f
M
by the following structure elements:
e
f(U+X) = JU +fX, e
ξi= 0 + ξi,eηi(U+X)ηi(X), i = 1,...,s,
eg(U+X, V +Y) = G(U, V ) + g(X, Y ).
It is straightforward to check that f
Mwith this structure is a K-
manifold. However, it is not a trans-S-manifold. In fact, since N
is a Kaehler manifold and so, Jis parallel, if ∇and e
∇denote the
Riemannian connections of Mand f
M, respectively, then
(e
∇e
Xe
f)e
Y= 0 + (∇Xf)Y
and, consequently, (3.1) does not hold for f
M.
However, we can observe that, from (2.5) and (2.6), the particular
cases of S-manifolds and C-manifolds are trans-S-manifolds.
On the other hand, it is known [2] that, in a K-manifold, all the
structure vector fields are Killing vector fields. For trans-S-manifolds
we can prove:
Proposition 3.2. Let Mbe a trans-S-manifold. Then, the structure
vector field ξiis a Killing vector field if and only if the corresponding
characteristic function βi= 0.
Proof. A direct computation by using (3.4) gives
(Lξig)(X, Y ) = 2βig(f X, fY ),
for any X, Y ∈ X (M). This completes the proof.
4. Examples of trans-S-manifolds
As we have mentioned above, it is obvious that, from (2.5) and (2.6),
S-manifolds and C-manifolds are trans-S-manifolds. Moreover, the
homothetic s-th Sasakian manifolds of [8] are also trans-S-manifolds
with the function αiconstant and βi= 0, for any i.
From Propositions 2.2 and 2.5 of [5], we see that f-manifolds of
Kenmotsu type, introduced by M. Falcitelli and A.M. Pastore, actually
are trans-S-manifolds with functions α1=··· =αs=β2=···=βs=
0 and β1= 1.
Also, from Theorem 2.4 in [14], we see that generalized Kenmotsu
manifolds studied by L. Bhatt and K.K. Dube [1] and A. Turgut Vanli
A NEW CLASS OF METRIC f-MANIFOLDS 11
and R. Sari [14] are trans-S-manifolds with functions α1=···=αs= 0
and β1=···=βs= 1.
Then, we are going to look for examples with different non-constant
functions αiand βi. We shall obtain these examples by using D-
conformal deformations and warped products.
Firstly, given a metric f-manifold (M, f, ξ1,...,ξs, η1,...,ηs, g), let
us consider the generalized D-conformal deformation given by
(4.1) e
f=f, e
ξi=1
aξi,eηi=aηi,eg=bg + (a2−b)
s
X
i=1
ηi⊗ηi,
for any i= 1, . . . , s, where a, b are two positive differentiable functions
on M. Then, it is easy to see that (M, e
f, e
ξ1,...,e
ξs,eη1,...,eηs,eg) is also
a metric f-manifold. Let us notice that we can obtain conformal, D-
homothetic (see [13]) or D-conformal (in the sense of S. Suguri and S.
Nakayama [12]) deformations, by putting a2=b,a=b=constant
or a=bin (4.1), respectively. In [9] Z. Olszack considered aand b
constants, a6= 0, b > 0 but not necesarily equal and he also called the
resulting transformation a D-homothetic deformation.
Moreover, let us suppose that Mis a trans-S-manifold and that
a, b depend only on the directions of the structure vector fields ξi,i=
1,...,s. Therefore, we can calculate e
∇from ∇and egby using Koszul’s
formula and (3.4). It follows that the Riemannian connection e
∇of eg
is given by
e
∇XY=∇XY+
s
X
i=1
2(a2−b)βi−ξib
2a2g(f X, fY )ξi
−1
2b{(Xb)f2Y+ (Y b)f2X}
+1
2a2
s
X
i=1 (Xa2)ηi(Y) + (Y a2)ηi(X)
−(ξia2)
s
X
j=1
ηj(X)ηj(Y)}ξi
−a2−b
b
s
X
i=1
αi{ηi(Y)fX +ηi(X)fY },
(4.2)
for any vector fields X, Y ∈ X (M).
Theorem 4.1. Let (M, f, ξ1,...,ξs, η1,...,ηs, g)be a trans-S-manifold
and consider a generalized D-conformal deformation on M, with a, b
positive functions depending only on the directions of the structure
12 PABLO ALEGRE, LUIS M. FERN ´
ANDEZ, AND ALICIA PRIETO-MART´
IN
vector fields. Then (M, e
f, e
ξ1,...,e
ξs,eη1,...,eηs,eg)is also a trans-S-
manifold with functions:
eαi=αia
b,e
βi=ξib
2ab +βi
a, i = 1,...,s.
Proof. By using (4.2) and taking into account that bonly depends on
the directions of the structure vector fields, we have
(e
∇Xe
f)Y= (∇Xf)Y−
s
X
i=1
2(a2−b)βi−ξib
2a2g(f X, Y )ξi
−1
2b
s
X
i=1
(ξib)ηi(Y)fX +a2−b
b
s
X
i=1
αiηi(Y)f2X,
for any X, Y ∈ X (M). Now, since Mis trans-S-manifold, from (3.1)
and (4.1) we obtain
(e
∇Xe
f)Y=
s
X
i=1
{αia
b(eg(e
fX, e
fY )e
ξi+eηi(Y)X)
+ξib
2ab +βi
a(eg(e
f X, Y )e
ξi−eηi(Y)e
fX)},
and this completes the proof.
Note that if Mis a Sasakian manifold, that is, if s= 1, α= 1 and
β= 0, this method does not produce an (α, β ) trans-Sasakian manifold
but a (α, 0) one because, by Darboux’s theorem, if a, b only depend of
the direction of ξ, they should be constants.
Corollary 4.1. Let (M, f , ξ1,...,ξs, η1,...,ηs, g)be an S-manifold and
consider a generalized D-conformal deformation on M, with a, b posi-
tive functions depending only on the directions of the structure vector
fields. Then (M, e
f, e
ξ1,...,e
ξs,eη1,...,eηs,eg)is a trans-S-manifold with
functions:
eαi=a
b,e
βi=ξib
2ab, i = 1,...,s.
Corollary 4.2. Let (M, f, ξ1,...,ξs, η1,...,ηs, g)be an C-manifold
and consider a generalized D-conformal deformation on M, with a, b
positive functions depending only on the directions of the structure vec-
tor fields. Then (M, e
f, e
ξ1,...,e
ξs,eη1,...,eηs,eg)is a trans-S-manifold
with functions:
eαi= 0,e
βi=ξib
2ab, i = 1,...,s.
A NEW CLASS OF METRIC f-MANIFOLDS 13
Corollary 4.3. Let (M, f, ξ1,...,ξs, η1,...,ηs, g)be a generalized Ken-
motsu manifold and consider a generalized D-conformal deformation
on M, with a, b positive functions depending only on the directions
of the structure vector fields. Then (M, e
f, e
ξ1,...,e
ξs,eη1,...,eηs,eg)is a
trans-S-manifold with functions:
eαi= 0,e
βi=ξib
2ab +1
a, i = 1,...,s.
Next, we are going to construct more examples of trans-S-manifolds
by using warped products. For later use, we need the following lemma
from [10] to compute the Riemannian connection of a warped product:
Lemma 4.1. Let us consider M=B×hFand denote by ∇,∇Band
∇Fthe Riemannian connections on M,Band F. If X, Y are tangent
vector fields on Band V, W are tangent vector fields on F, then:
(i) ∇XYis the lift of ∇B
XY.
(ii) ∇XV=∇VX= (Xh/h)V.
(iii) The component of ∇VWnormal to the fibers is:
−(gh(V, W )/h)grad h.
(iv) The component of ∇VWtangent to the fibers is the lift of ∇F
VW.
In this context, given an almost Hermitian manifold (N, J, G), the
warped product f
M=Rs×hNcan be endowed with a metric f-
structure ( e
f, e
ξ1,...,e
ξs,eη1,...,eηs, gh), with the warped metric
gh=−π∗(gRs) + (h◦π)2σ∗(G),
where h > 0 is a differentiable function on Rsand πand σare the
projections from Rs×Non Rsand N, respectively. In fact, e
f(e
X) =
(Jσ∗e
X)∗, for any vector field e
X∈ X (f
M) and e
ξi=∂/∂ti,i= 1,...,s,
where tidenotes the coordinates of Rs. Note that this metric is the
one used to construct the Robertson-Walker spaces (see [10]).
Now, we study the structure of this warped product.
Theorem 4.2. Let Nbe an almost Hermitian manifold. Then, the
warped product (f
M=Rs×hN, e
f, e
ξ1,...,e
ξs,eη1,...,eηs, gh)is a trans-S-
manifold with functions eα1=··· =eαs= 0 and e
βi=hi)/h,i= 1,...,s,
if and only if Nis a Kaehlerian manifold, where hi)are denoting the
components of the gradient of the function h, for i= 1,...,s.
Proof. Consider e
X=U+Xand e
Y=V+Y, where U, V and X, Y
are tangent vector fields on Rsand N, respectively. Then, taking into
14 PABLO ALEGRE, LUIS M. FERN ´
ANDEZ, AND ALICIA PRIETO-MART´
IN
account Lemma 4.1, if e
∇and ∇Ndenote the Riemannian connections
of f
Mand N, respectively, we have:
(e
∇e
Xe
f)e
Y=e
∇UJX +∇XJY
−e
f(e
∇UV+e
∇XV+e
∇UY+e
∇XY)
=U(h)
hJY −gh(X, JY )
hgrad(h) + ∇N
XJY
−f(∇UV+V(h)
hX+U(h)
hY−gh(X, Y )
hgrad(h) + ∇XY)
=−gh(X, J Y )
hgrad(h)−V(h)
hJX + (∇N
XJ)Y
=gh(JX, Y )
h
s
X
i=1
hi)e
ξi−
s
X
i=1 eηi(V)hi)
hJX + (∇N
XJ)Y
=gh(e
fe
X, e
Y)
h
s
X
i=1
hi)e
ξi−
s
X
i=1 eηi(V)hi)
he
fe
X+ (∇N
XJ)Y.
Therefore, (3.1) holds if and only if (∇N
UJ)V= 0, that is, if and only
if Nis a Kaehlerian manifold. Moreover, for any i= 1,...,s,
e
∇e
Xe
ξi=∇Ue
ξi+∇Xe
ξi=hi)
hX=hi)
h(e
X−
s
X
i=1 eη(e
X)e
ξi) = −hi)
he
f2e
X
and then, Theorem 3.1 gives the result.
Corollary 4.4. The warped product Rs×hN, being Na Kaehlerian
manifold and ha constant function, is a C-manifold. In particular, if
h= 1, the Riemannian product Rs×Nis a C-manifold.
Combining these examples with a generalized D-conformal deforma-
tion, a great variety of non-trivial trans-S-manifolds can be presented.
Moreover, if we do the warped product of Rswith a (2n+s1)-
dimensional (almost) trans-S-manifold (M, f, ξ1,...,ξs1, η1,...,ηs1, g),
we obtain a new metric f-manifold
(f
M=Rs×hM, e
f, e
ξ1,...,e
ξs+s1,eη1,...,eηs+s1, gh),
where e
f(e
X) = (fσ∗e
X)∗and:
e
ξi=
∂
∂ti
if 1 ≤i≤s,
1
hξi−sif s+ 1 ≤i≤s+s1.
A NEW CLASS OF METRIC f-MANIFOLDS 15
These manifolds, under certain hypothesis about the function h, ver-
ify (3.1) but not (3.4), so from Theorem 3.1 they are not normal. Con-
sequently, they are examples of almost trans-S-manifolds not trans-S-
manifolds.
Theorem 4.3. Let Mbe a (2n+s1)-dimensional (almost) trans-S-
manifold with functions (αi, βi),i= 1,...,s1. Then, the warped prod-
uct f
M=Rs×hM, with the metric f-structure defined above, is a
(2n+s+s1)-dimensional almost trans-S-manifold with fuctions
eαi=
0 for i= 1,...,s,
αi−s
hfor i=s+ 1,...,s+s1.
and:
e
βi=
hi)
hfor i= 1,...,s,
βi−s
hfor i=s+ 1 ...,s+s1.
Proof. Consider e
X=U+Xand e
Y=V+Y, where U, V and X, Y
are tangent vector fields on Rsand M, respectively. Then, taking
into account Lemma 4.1, if ∇is the Riemannian connection of M, we
deduce:
(∇e
Xe
f)e
Y=−gh(X, f Y )
hgrad(h)−V(h)
hfX + (∇Xf)Y
=gh(f X, Y )
h
s
X
i=1
hi)∂
∂ti
−
s
X
i=1
V(ti)hi)
hfX
+
s+s1
X
i=s+1 αi−sg(f X, fY )ξi−s+ηi−s(Y)f2X
+βi−s{g(f X, Y )ξi−s−ηi−s(Y)fX}]
=
s
X
i=1
hi)
h{gh(e
fe
X, e
Y)e
ξi−eηi(e
Y)e
fe
X}
+
s+s1
X
i=s+1 hαi−s
h{gh(e
fe
X, e
fe
Y)e
ξi−s+eηi−s(e
Y)e
f2e
X}
+βi−s
h{gh(e
fe
X, e
Y)e
ξi−s−eηi−s(e
Y)e
fe
X}.
16 PABLO ALEGRE, LUIS M. FERN ´
ANDEZ, AND ALICIA PRIETO-MART´
IN
Joining the addends appropriately, it takes the form of (3.1) with
the desired functions. Therefore, f
Mis a almost trans-S-manifold.
Observe that, in the above conditions, (3.4) is not verified in general.
In fact, consider e
ξiwith 1 ≤i≤s. Then, for any e
X∈ X (f
M),
e
∇e
Xe
ξi=hi)
hU=hi)
h(e
X−
s
X
j=1 eηj(e
X)ξj)
and so, if his not a constant function, from Theorem 3.1, we get that
f
Mis not a trans-S-manifold.
Corollary 4.5. The warped product Rs×hM, being Ma trans-S-
manifold, is a trans-S-manifold if and only if his constant. In par-
ticular, the Riemannian product Rs×Mis a trans-S-manifold with
functions
(0,s)
. . ., 0, α1,...,αs1,0,s)
. . ., 0, β1,...,βs1),
where (αi, βi),i= 1,...,s1, denote the characteristic functions of M.
Corollary 4.6. Let Mbe a Sasakian manifold. Then, the warped
product R×hMis a almost trans-S-manifold with functions:
α1= 0, α2=1
h, β1=h′
hand β2= 0.
Corollary 4.7. Let Mbe a three dimensional trans-Sasakian, with
non-constant characteristic functions αand β. Then, the warped prod-
uct R×hMis a four dimensional almost trans-S-manifold not trans-
S-manifold with functions:
α1= 0, α2=α
h, β1=h′
hand β2=β
h.
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Departamento de Econom´
ıa, M´
etodos Cuantitativos e Historia de
la Econom´
ıa, Universidad Pablo de Olavide, Ctra. de Utrera, km. 1,
41013-Sevilla, Spain
E-mail address:psalerue@upo.es
Departamento de Geometr´
ıa y Topolog´
ıa, Facultad de Matem´
aticas,
Universidad de Sevilla, C./ Tarfia, s.n., 41012-Sevilla, Spain.
E-mail address:lmfer@us.es
Departamento de Geometr´
ıa y Topolog´
ıa, Facultad de Matem´
aticas,
Universidad de Sevilla, C./ Tarfia, s.n., 41012-Sevilla, Spain.
E-mail address:aliciaprieto@us.es