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Abstract

Generalized Sasakian-space-forms are introduced and studied. Many examples of these manifolds are presented, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations. New results on generalized complex-space-forms are also obtained.
Proceedings of The Ninth International
Workshop on Diff. Geom. 9(2005) 31-39
On generalized Sasakian-space-forms
Alfonso Carriazo
From a joint work with David E. Blair and Pablo Alegre
Department of Geometry and Topology, Faculty of Mathematics, University of
Sevilla, Apdo. Correos 1160, 41080 Sevilla, Spain
e-mail :carriazo@us.es
(2000 Mathematics Subject Classification : 53C25, 53D15.)
Abstract. We study contact metric and trans-Sasakian generalized Sasakian-space-forms.
We also give some interesting examples of generalized Sasakian-space-forms by using
warped products and conformal changes of metric.
1 Introduction
The study of the curvature tensor Rof a Riemannian manifold and the elements
which allow us to determine it has always been a main topic in differential geometry.
In this sense, Sasakian-space-forms play an important role in contact Riemannian
geometry.
A Sasakian manifold (M, φ, ξ , η, g) is said to be a Sasakian-space-form if all the
φ-sectional curvatures K(XφX) are equal to a constant c, where K(XφX)
denotes the sectional curvature of the section spanned by the unit vector field X,
orthogonal to ξ, and φX. In such a case, the Riemann curvature tensor of Mis
given by
R(X, Y )Z=c+ 3
4{g(Y, Z )Xg(X, Z)Y}
+c1
4{g(X, φZ )φY g(Y, φZ)φX + 2g(X, φY )φZ}
+c1
4{η(X)η(Z)Yη(Y)η(Z)X
+g(X, Z )η(Y)ξg(Y, Z)η(X)ξ}.
(1.1)
These spaces can be modeled, depending on c > 3, c=3 or c < 3.
As a natural generalization of these manifolds, P. Alegre, D. E. Blair and the
author introduced in [1] the notion of generalized Sasakian-space-forms. They were
defined as almost contact metric manifolds with Riemann curvature tensors satis-
fying an equation similar to (1.1), in which the constant quantities, (c+ 3)/4 and
31
32 Alfonso Carriazo
(c1)/4 are replaced by differentiable functions, i.e., such that
R(X, Y )Z=f1{g(Y, Z )Xg(X, Z)Y}
+f2{g(X, φZ )φY g(Y, φZ)φX + 2g(X, φY )φZ}
+f3{η(X)η(Z)Yη(Y)η(Z)X
+g(X, Z )η(Y)ξg(Y, Z)η(X)ξ}.
(1.2)
Let us notice that a generalized Sasakian-space-form is not, in general, a
Sasakian manifold, just like a generalized complex-space-form is not always a
Kaehler manifold (see [8, 9]).
In this lecture, we will first study the structure of a generalized Sasakian-space-
form. In particular, we will focus on both contact metric and trans-Sasakian ones.
On the other hand, we will show some procedures to obtain interesting examples
(i.e., examples with non-constant functions f1, f2, f3) by using warped products
and conformal changes of metric. For more details and the proofs of the theorems
presented in the following sections, we refer to [1, 2].
Acknowledgements The author is partially supported by the MEC-FEDER
grant MTM2004-04934-C04-04, the PAI group FQM-327 and the BK21 Proj. of
KRF. He wants to express his deepest gratitude to Prof. Young Jin Suh and Prof.
Young Ho Kim for the invitation to participate in this Workshop and the warm
hospitality received during his visit to Kyungpook National University.
2 Preliminaries
In this section, we recall some definitions and basic formulas which we will use
later. For more background on almost contact metric manifolds, we recommend the
reference [3].
An odd-dimensional Riemannian manifold (M, g) is said to be an almost contact
metric manifold if there exist on Ma (1,1) tensor field φ, a vector field ξ(called the
structure vector field) and a 1-form ηsuch that η(ξ) = 1, φ2(X) = X+η(X)ξand
g(φX, φY ) = g(X, Y )η(X)η(Y), for any vector fields X, Y on M. In particular,
in an almost contact metric manifold we also have φξ = 0 and ηφ= 0.
Such a manifold is said to be a contact metric manifold if dη= Φ, were
Φ(X, Y ) = g(X, φY ) is called the fundamental 2-form of M. If, in addition,
ξis a Killing vector field, then Mis said to be a K-contact manifold. It is
well-known that a contact metric manifold is a K-contact manifold if and only
if Xξ=φX, for any vector field Xon M. On the other hand, the almost con-
tact metric structure of Mis said to be normal if [φ, φ](X, Y ) = 2(X, Y )ξ, for
any X, Y , where [φ, φ] denotes the Nijenhuis torsion of φ, given by [φ, φ](X, Y ) =
φ2[X, Y ] + [φX, φY ]φ[φX, Y ]φ[X, φY ]. A normal contact metric manifold is
called a Sasakian manifold. It can be proved that an almost contact metric manifold
On generalized Sasakian-space-forms 33
is Sasakian if and only if
(Xφ)Y=g(X, Y )ξη(Y)X,
for any X, Y .
In [7], J. A. Oubi˜na introduced the notion of a trans-Sasakian manifold. An
almost contact metric manifold Mis a trans-Sasakian manifold if there exist two
functions αand βon Msuch that
(Xφ)Y=α(g(X, Y )ξη(Y)X) + β(g(φX, Y )ξη(Y)φX),
for any X, Y on M. If β= 0, Mis said to be an α-Sasakian manifold. Sasakian
manifolds appear as examples of α-Sasakian manifolds, with α= 1. If α= 0, M
is said to be a β-Kenmotsu manifold. Kenmotsu manifolds are particular examples
with β= 1. If both αand βvanish, then Mis a cosymplectic manifold. Actually,
in [6], Marrero showed that a trans-Sasakian manifold of dimension greater than or
equal to 5 is either α-Sasakian, β-Kenmotsu or cosymplectic.
In the following two sections we will show some results giving us more informa-
tion about the relationship between the structure on a generalized Sasakian-space-
form and the functions f1, f2, f3. In this sense, we have the following theorem from
[5], which we adapt to our notation:
Theorem 2.1. ([5]) Let (M, φ, ξ, η, g )be a connected generalized Sasakian-space-
form with f2=f3not identically zero. If dim(M)5and g(X, Xξ) = 0 for any
vector field Xorthogonal to ξ, then f1and f2are constant functions and f1f20.
Moreover, if f1f2= 0, then (M, φ, ξ, η, g )is a cosymplectic-space-form and if
f1f2=α2>0then (M, φ, ξ , η, g)or (M, φ, ξ , η, g)is an α-Sasakian manifold
with constant φ-sectional curvature cand a generalized Sasakian-space-form with
f1= (c+ 3α2)/4and f2=f3= (cα2)/4.
3 Contact metric generalized Sasakian-space-forms
By using (1.2) and the well-known fact which establishes that in a K-contact
manifold, the sectional curvature of any plane section containing ξis equal to 1, we
can first prove:
Proposition 3.1. ([1]) Let M(f1, f2, f3)be a generalized Sasakian-space-form. If
Mis a K-contact manifold, then f3=f11.
Moreover, it is well-known that any Sasakian manifold is a K-contact manifold.
For a generalized Sasakian-space-form, the converse is also true:
34 Alfonso Carriazo
Theorem 3.1. ([1]) Every generalized Sasakian-space-form with a K-contact struc-
ture is a Sasakian manifold.
For a contact metric generalized Sasakian-space-form, we can obtain the follow-
ing theorems:
Theorem 3.2. ([1]) Let M(f1, f2, f3)be a generalized Sasakian-space-form. If M
is a contact metric manifold with f3=f11, then it is a Sasakian manifold.
Theorem 3.3. ([1]) Let M(f1, f2, f3)be a generalized Sasakian-space-form. If M
is a contact metric manifold, then f1f3is constant on M.
Let us notice that the assumption f3=f11 in Theorem 3.2 is coherent with
Theorem 3.3. In that case, the constant is just 1.
On the other hand, by working with some curvature identities, we can also
prove:
Theorem 3.4. ([1]) Let M(f1, f2, f3)be a generalized Sasakian-space-form. If M
is a Sasakian manifold, then f2=f3=f11.
Thus, by using the obstruction established in Theorem 2.1, we deduce that, in
dimensions greater than or equal to 5, any contact metric connected generalized
Sasakian-space-form, such that f3=f11, must be a Sasakian-space-form. In
particular, this fact is true for either K-contact or Sasakian generalized Sasakian-
space-forms.
But, is it still possible to find interesting examples of contact metric generalized
Sasakian-space-forms non-satisfying the above conditions? To answer this question,
we have to point out that a contact metric generalized Sasakian-space-form is a
particular case of (κ, µ)-space, with κ=f1f3and µ= 0. Let us recall that a
contact metric manifold is said to be a (κ, µ)-space ([4]) if
R(X, Y )ξ=κ{η(Y)Xη(X)Y}+µ{η(Y)hX η(X)hY },
for any vector fields X, Y on M, where hX = 1/2(Lξφ)X,Lbeing the usual Lie
derivative.
Then, by virtue of Theorem 1 of [4] we can obtain the following results:
Lemma 3.1. ([2]) If M(f1, f2, f3)is a non-Sasakian contact metric generalized
Sasakian-space-form, with dimension greater than or equal to 5, then f2=κ.
Lemma 3.2. ([2]) If M(f1, f2, f3)is a non-Sasakian contact metric generalized
Sasakian-space-form, with dimension greater than or equal to 5, then f2= 0.
On generalized Sasakian-space-forms 35
Lemma 3.3. ([2]) If M(f1, f2, f3)is a non-Sasakian contact metric generalized
Sasakian-space-form, with dimension greater than or equal to 5, then f1=f3= 0.
It follows from 1.2 and the above lemmas that a non-Sasakian contact metric
generalized Sasakian-space-form, with dimension greater than or equal to 5 should
be a flat manifold. But, as a contact metric manifold with such dimension cannot
be flat, we have:
Theorem 3.5. ([2]) Any contact metric generalized Sasakian-space-form M(f1, f2, f3),
with dimension greater than or equal to 5, is a Sasakian manifold. Therefore, f1,
f2and f3must be constant functions.
And, what can we say about 3-dimensional contact metric generalized Sasakian-
space-forms? First, let us mention that the writing of the curvature tensor of a
3-dimensional generalized Sasakian-space-form is not unique. Actually, if M3is an
almost contact metric manifold such that its curvature tensor can be written as
R(X, Y )Z=f1R1(X, Y )Z+f2R2(X, Y )Z+f3R3(X, Y )Z
and
R(X, Y )Z=f
1R1(X, Y )Z+f
2R2(X, Y )Z+f
3R3(X, Y )Z,
where Ri(X, Y )Zdenotes the corresponding term in (1.2), then the functions fi
and f
iare related as follows,
f
1=f1+f, f
2=f2f/3, f
3=f3+f,
where fis a function on M. Conversely, if M3(f1, f2, f3) is a generalized Sasakian-
space-form and we define the functions f
ias above, for any function fon M, then
it is also a generalized Sasakian-space-form M3(f
1, f
2, f
3). Therefore, in order to
consider an unique writing of the curvature tensor of a three-dimensional generalized
Sasakian-space-form, we will chose that satisfying f2= 0.
Now, we can show the following results from [2]:
Proposition 3.2. ([2]) If M3(f1, f2, f3)is a 3-dimensional, non-Sasakian, contact
metric generalized Sasakian-space-form, then 2f1+ 3f2f3= 0.
Theorem 3.6. ([2]) If M3(f1, f2, f3)is a 3-dimensional, non-Sasakian, contact
metric generalized Sasakian-space-form, then we can write
R(X, Y )Z=κR1(X, Y )Z2κR3(X, Y )Z,
for any X, Y, Z vector fields on M, where κ < 1is a constant.
36 Alfonso Carriazo
4 Examples
Even if from the results in the previous section we could think that it must
be difficult to find examples of generalized Sasakian-space-forms with non-constant
functions f1, f2, f3, in this section we will show some construction procedures to
obtain them.
Given an almost Hermitian manifold (N, J, G), we consider the warped product
M=R×fN, where f > 0 is a function on R. This manifold can be endowed with
an almost contact metric structure (φ, ξ, η, gf). In fact,
gf=π(gR)+(fπ)2σ(G),
is the warped product metric, where πand σare the projections from R×Non R
and N, respectively; φ(X) = (JσX), for any vector field Xon M, and ξ=∂/∂t,
where tdenotes the coordinate of R. Then, we have:
Theorem 4.1. ([1]) Let N(F1, F2)be a generalized complex-space-form. Then, the
warped product M=R×fN, endowed with the almost contact metric structure
(φ, ξ, η, gf), is a generalized Sasakian-space-form M(f1, f2, f3)with functions:
f1=(F1π)f02
f2, f2=F2π
f2, f3=(F1π)f02
f2+f00
f.
In particular, if N(c) is a complex-space-form, we obtain the generalized
Sasakian-space-form
Mµc4f02
4f2,c
4f2,c4f02
4f2+f00
f.
Hence, for example, the warped products R×fCn,R×fCPn(4) and R×fCHn(4)
are generalized Sasakian-space-forms.
Moreover, we can obtain more examples by using conformal and related changes
of metric. We just summarize now the most important results concerning these
procedures. For more results and all the details, we refer once again to [1].
Given an almost contact metric manifold (M, φ, ξ, η, g), if we now consider a
conformal change of metric
g=ρ2g,
where ρis a positive function on M, then it is easy to prove that (M, φ, ξ, η, g)
is also an almost contact metric manifold, where we put
φ=φ, ξ=1
ρξ, η=ρη.
On generalized Sasakian-space-forms 37
Theorem 4.2. ([1]) Given a complex-space-form N(c)and two positive functions
f=f(t),ρ=ρ(t), the conformal change of metric with function ρendows the
warped product M=R×fN(c)with the structure of a generalized Sasakian-space-
form M(f
1, f
2, f
3)with functions
f
1=1
ρ2Ãc
4f2µf0
f+ρ0
ρ2!, f
2=1
ρ2
c
4f2,
f
3=1
ρ2Ãc
4f2µf0
f+ρ0
ρ2
+ρ0f0
ρf +f00
f+ (log(ρ))00!.
On the other hand, if we consider on (M, φ, ξ, η, g), a D–homothetic deformation
defined by
φ=φ, ξ=1
aξ, η=aη, g=ag +a(a1)ηη,
where ais a positive constant, we can prove:
Theorem 4.3. ([1]) Given a complex-space-form N(c), a positive constant aand a
function f=f(t)>0, the D–homothetic deformation with constant aendows the
warped product M=R×fN(c)with the structure of a generalized Sasakian-space-
form M(f
1, f
2, f
3)with functions
f
1=ac 4f02
4a2f2, f
2=c
4af2, f
3=ac 4f02
4a2f2+f00
a2f.
Finally, if we consider on M=R×fN(c) the D-conformal deformation given
by
φ=φ, ξ=1
δ2ξ, η=δ2η, g=δ2g+δ2(δ21)ηη,
δ=δ(t)>0, then we have:
Theorem 4.4. ([1]) Given a complex-space-form N(c)and two functions f=
f(t)>0,δ=δ(t)>0, the D–conformal deformation with function δendows
the warped product M=R×fN(c)with the structure of a generalized Sasakian-
space-form M(f
1, f
2, f
3)with functions
f
1=1
δ2Ãc
4f21
δ2µf0
f+δ0
δ2!, f
2=c
4δ2f2,
f
3=1
δ2Ãc
4f21
δ2õf0
f+δ0
δ2
µf00
f+δ00
δ+ 2 µδ0
δ2!!.
38 Alfonso Carriazo
5 Trans-Sasakian generalized Sasakian-space-forms
First of all, let us point out why trans-Sasakian generalized Sasakian-space-
forms are interesting:
Proposition 5.1. ([1]) Let Nbe an almost Hermitian manifold. Then, R×fNis
a(0, β)trans-Sasakian manifold, with β=f0/f, if and only if Nis a Kaehlerian
manifold.
Therefore, the useful warped products R×fN(c) of the previous section are
examples of trans-Sasakian generalized Sasakian-space-forms.
Actually, as we already said, the important examples of trans-Sasakian mani-
folds are either α-Sasakian or β-Kenmotsu manifolds.
By working with the second Bianchi identity and after some very long calcula-
tions, we can show the following results concerning α-Sasakian generalized Sasakian-
space-forms:
Proposition 5.2. ([2]) Let M(f1, f2, f3)be an α-Sasakian generalized Sasakian-
space-form, with dimension greater than or equal to 5. Then αdepends only on the
direction of ξand the functions f1,f3and αsatisfy the equation f1f3=α2.
Theorem 5.1. ([2]) Let M(f1, f2, f3)be a connected α-Sasakian generalized
Sasakian-space-form, with dimension greater than or equal to 5. Then, f1and
f2are constant functions and, if either α= 0 or α6= 0 at every point of M, then
f3is also a constant function.
With respect to β-Kenmotsu generalized Sasakian-space-forms, we have:
Proposition 5.3. ([2]) Let M(f1, f2, f3)be a β-Kenmotsu generalized Sasakian-
space-form. Then, βdepends only on the direction of ξand the functions f1,f3
and βsatisfy the equation f1f3+ξ(β) + β2= 0.
Theorem 5.2. ([1]) Let M(f1, f2, f3)be a β-Kenmotsu generalized Sasakian-space-
form, with dimension greater than or equal to 5. Then, X(fi) = 0 for any X
orthogonal to ξ,i= 1,2,3, and the following equations hold:
ξ(f1)+2βf3= 0, ξ(f2)+2βf2= 0.
Actually, by integrating with respect to tin the above equations, we deduce
that, locally,
f1=˜
F12Zβf3dt, f2=˜
F2e2Rβdt ,
where ˜
F1/∂t =˜
F2/∂t = 0.
On generalized Sasakian-space-forms 39
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In this paper, we introduce two new classes of almost contact structures, called trans-Sasakian and almost trans-Sasakian structures, which are obtained from certain classes of almost Hermitian manifolds closely related to locally conformal Kahler or almost Kahler manifolds, respectively. In particular, although transSasakian structures are normal almost contact metric structures containing both cosymplectic and Sasakian structures, they are different from quasi-Sasakian structures, as it is shown constructing explicit examples, and in fact no inclusion relation between these classes exists.