DEMONSTRATIO MATHEMATICA
Vol. XLVII
No 3
2014
A. Sarkar, Ali Akbar
GENERALIZED SASAKIAN-SPACE-FORMS WITH
PROJECTIVE CURVATURE TENSOR
Abstract. The object of the present paper is to study φ-projectively flat generalized
Sasakian-space-forms, projectively locally symmetric generalized Sasakian-space-forms
and projectively locally φ-symmetric generalized Sasakian-space-forms. All the obtained
results are in the form of necessary and sufficient conditions. Interesting relations between
projective curvature tensor and conformal curvature tensor of a generalized Sasakian-spaceform of dimension greater than three have been established. Some of these properties are
also analyzed in the light of quarter-symmetric metric connection, in addition with the
Levi-Civita connection. Obtained results are supported by illustrative examples.
1. Introduction
Recently, P. Alegre, D. Blair and A. Carriazo [1] introduced and studied
generalized Sasakian-space-forms. These space-forms are defined as follows:
Given an almost contact metric manifold M pφ, ξ, η, gq, we say that M is
generalized Sasakian-space-form if there exist three functions f1 , f2 , f3 on
M such that the curvature tensor R is given by
RpX, Y qZ “ f1 tgpY, ZqX ´ gpX, ZqY u
` f2 tgpX, φZqφY ´ gpY, φZqφX ` 2gpX, φY qφZu
` f3 tηpXqηpZqY ´ ηpY qηpZqX
` gpX, ZqηpY qξ ´ gpY, ZqηpXqξu,
for any vector fields X, Y, Z on M. In such a case, we denote the manifold
as M pf1 , f2 , f3 q. These kind of manifolds appear as a generalization of the
well known Sasakian-space-forms, which can be obtained as a particular case
c´1
of generalized Sasakian-space-forms by taking f1 “ c`3
4 , f2 “ f3 “ 4 .
But, it is to be noted that generalized Sasakian-space-forms are not merely
2010 Mathematics Subject Classification: 53C25, 53D15.
Key words and phrases: generalized Sasakian-space-forms, φ-projectively flat, projectively locally symmetric, projectively locally φ-semisymmetric, conformally flat.
DOI: 10.2478/dema-2014-0058
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A. Sarkar, A. Akbar
generalization of Sasakian-space-forms. It also contains a large class of almost
contact manifolds. For example, it is known that [2] any three-dimensional
pα, βq-trans Sasakian manifold with α, β depending on ξ is a generaliged
Sasakian-space-form. However, we can find generalized Sasakian-space-forms
with non-constant functions and arbitrary dimensions. In [1], the authors
cited several examples of generalized Sasakian-space-forms in terms of warped
product spaces. In [8], U. K. Kim studied conformally flat generalized
Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms.
In our recent paper [6], we have studied generalized Sasakian-space-forms with
vanishing quasi-conformal curvature tensor and some symmetry properties
have also been considered. In Riemannian geometry, one of the basic interests
is curvature property and to what extent this determines the manifold itself.
Two important curvature properties are flatness and symmetry. In [9], the
authors studied φ-projectively flat LP-Sasakian manifolds. In the paper [4], we
have studied projectively flat generalized-Sasakian-space-forms. In [5], we also
have studied locally φ-symmetric generalized Sasakian-space-forms. In this
connection it should be mentioned that in [10], T. Takahashi introduced the
notion of locally φ-symmetric manifolds in the context of Sasakian geometry.
In this paper, we like to study φ-projectively flat generalized Sasakian-spaceforms, projectively locally symmetric generalized Sasakian-space-forms and
projectively locally φ-symmetric generalized Sasakian-space-forms, because
after conformal and quasi-conformal curvature tensor, projective curvature
tensor is an important one from the geometric point of view. The projective
curvature tensor is a measure of the failure of a Riemannian manifold to be of
constant curvature [11]. The present paper is organized as follows: In Section
2, we review some preliminary results. In Section 3, we study φ-projectively
flat generalized Sasakian-space-forms and obtain that a generalized Sasakianspace-form of dimension greater than three is φ-projectively flat if and only
if it is projectively flat. Section 4 deals with projectively locally symmetric
generalized Sasakian-space-forms. In this section, we show that a generalized
Sasakian-space-form of dimension greater than three is projectively locally
symmetric if and only if it is conformally flat. Section 5 is devoted to study
projectively locally φ-symmetric generalized Sasakian-space-forms. Here we
find that a projectively locally φ-symmetric generalized Sasakian-space-form
of dimension greater than three is also conformally flat and hence projectively
locally symmetric. We show that the converse is also true. As a corollary, we
prove that for a generalized Sasakian-space-form of dimension greater than
three the condition that φ-projectively flat, projectively flat, projectively
locally symmetric, projectively locally φ-symmetric and conformally flat
are equivalent. We also investigate these properties of the space-forms
with respect to quarter-symmetric metric connection as well as Levi-Civita
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connection. The last section contains illustrative examples to ensure the
validity of the obtained results.
2. Preliminaries
In an almost contact metric manifold, we have [3]
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
φ2 X “ ´X ` ηpXqξ, φξ “ 0,
ηpξq “ 1, gpX, ξq “ ηpXq, ηpφXq “ 0,
gpφX, φY q “ gpX, Y q ´ ηpXqηpY q,
gpφX, Y q “ ´gpX, φY q, gpφX, Xq “ 0,
p∇X ηqpY q “ gp∇X ξ, Y q,
where φ is a p1, 1q tensor, ξ is a vector field, η is an 1-form and g is a
Riemannian metric. The metric g induces an inner product on the tangent
space of the manifold. Again we know that [1] in a generalized Sasakianspace-form
(2.6)
RpX, Y qZ “ f1 tgpY, ZqX ´ gpX, ZqY u
` f2 tgpX, φZqφY ´ gpY, φZqφX ` 2gpX, φY qφZu
` f3 tηpXqηpZqY ´ ηpY qηpZqX
` gpX, ZqηpY qξ ´ gpY, ZqηpXqξu,
for any vector fields X, Y, Z on M, where R denotes the curvature tensor of M
and f1 , f2 , f3 are smooth functions on the manifold. The Ricci tensor S and
the scalar curvature r of the manifold of dimension p2n ` 1q are respectively
given by
(2.7) SpX, Y q “ p2nf1 ` 3f2 ´ f3 qgpX, Y q ´ p3f2 ` p2n ´ 1qf3 qηpXqηpY q,
(2.8)
r “ 2np2n ` 1qf1 ` 6nf2 ´ 4nf3 .
For a p2n ` 1q-dimensional pn ą 1q almost contact metric manifold, the Weyl
projective curvature tensor P is given by
(2.9)
P pX, Y qZ “ RpX, Y qZ ´
1
rSpY, ZqX ´ SpX, ZqY s.
2n
3. φ-projectively flat generalized Sasakian-space-forms
Definition 3.1. A p2n ` 1q-dimensional pn ą 1q generalized Sasakianspace-form is called φ-projectively flat if it satisfies
φ2 P pφX, φY qφZ “ 0,
for any vector fields X, Y, Z on the manifold [9].
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From the definition, it follows that every projectively flat generalized
Sasakian-space-form is φ-projectively flat but the converse may not be true.
Interestingly, in this section we prove that for a generalized Sasakian-spaceform of dimension greater than three, the converse also holds. Here lies the
importance of the study of φ-projectively flat generalized Sasakian-space
forms.
Let us consider a φ-projectively flat generalized Sasakian-space-form.
Then by definition
φ2 P pφX, φY qφZ “ 0.
In view of p2.9q, the above equation yields
φ2 rRpφX, φY qφZ ´
1
pSpφY, φZqφX ´ SpφX, φZqφY qs “ 0.
2n
Using p2.2q, p2.6q and p2.7q, we obtain from above
(3.1)
φ2 rf1 pgpφY, φZqφX ´ gpφX, φZqφY q
` f2 pgpφX, φ2 Zqφ2 Y ´ gpφY, φ2 Zqφ2 X ` 2gpφX, φ2 Y qφ2 Zqs
1
“
p2nf1 ` 3f2 ´ f3 qφ2 pgpφY, φZqφX ´ gpφX, φZqφY q.
2n
Applying p2.3q, we get from the above equation
(3.2)
φ2 rf1 pgpY, ZqφX ´ ηpY qηpZqφX ´ gpX, ZqφY ` ηpXqηpZqφY q
` f2 pgpX, φZqφ2 Y ´ gpY, φZqφ2 X ` 2gpX, φY qφ2 Zqs
1
p2nf1 ` 3f2 ´ f3 qφ2 pgpY, ZqφX ´ ηpY qηpZqφX
“
2n
´ gpX, ZqφY ` ηpXqηpZqφY q.
By virtue of p2.1q and p2.2q, the above equation yields
(3.3)
f1 pgpY, ZqφX ´ ηpY qηpZqφX ´ gpX, ZqφY ` ηpXqηpZqφY q
` f2 pgpX, φZqφ2 Y ´ gpY, φZqφ2 X ` 2gpX, φY qφ2 Zq
1
“
p2nf1 ` 3f2 ´ f3 qpgpY, ZqφX ´ ηpY qηpZqφX
2n
´ gpX, ZqφY ` ηpXqηpZqφY q.
In the above equation, taking the inner product g in both sides with respect
to W, we get
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(3.4)
729
f1 pgpY, ZqgpφX, W q ´ ηpY qηpZqgpφX, W q ´ gpX, ZqgpφY, W q
` ηpXqηpZqgpφY, W qq ` f2 pgpX, φZqgpφ2 Y, W q
´ gpY, φZqgpφ2 X, W q ` 2gpX, φY qgpφ2 Z, W qq
“
1
p2nf1 ` 3f2 ´ f3 qpgpY, ZqgpφX, W q ´ ηpY qηpZqgpφX, W q
2n
´ gpX, ZqgpφY, W q ` ηpXqηpZqgpφY, W qq.
Putting Y “ Z “ ei , where tei u is an orthonormal basis of the tangent space
at each point of the manifold and taking summation over i, i “ 1, 2, . . . , 2n`1,
we get
3f2 ´ f3
3f2 gpX, φW q “
p2n ´ 1qgpX, φW q.
2n
The above equation is true for any vector fields X and W. Let W ­“ X. Then,
it follows from the above equation that
3f2 “
3f2 ´ f3
p2n ´ 1q.
2n
The above result implies
3f2
.
1 ´ 2n
From [4], it is known that a generalized Sasakian-space-form of dimension
3f2
greater than three is projectively flat if and only if f3 “ 1´2n
. Hence, we see
that a φ-projectively flat generalized Sasakian-space-form is projectively flat.
Conversely, if the manifold is projectively flat, then P pX, Y qZ “ 0. From
which it trivially follows that φ2 P pφX, φY qφZ “ 0. Therefore, the manifold
is φ-projectively flat. Now we are in a position to state the following:
f3 “
Theorem 3.1. A p2n ` 1q-dimensional pn ą 1q generalized Sasakian-spaceform is φ-projectively flat if and only if it is projectively flat.
It is also known that [4] a generalized Sasakian-space-form of dimension
greater than three is projectively flat if and only if it is Ricci semisymmetric.
Hence we can state the following:
Corollary 3.1. A p2n ` 1q-dimensional pn ą 1q generalized Sasakianspace-form is φ-projectively flat if and only if it is Ricci semisymmetric.
We also know that [4] every flat generalized Sasakian-space-form is projectively flat but the converse is true when f1 “ f3 . Therefore, we conclude
the following:
Corollary 3.2. Every flat generalized Sasakian-space-form is φ-projectively flat but the converse is true when f1 “ f3 .
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In general, generalized Sasakian-space-forms are almost contact metric
manifolds. If it admits a contact structure, then we know that [5] if a
p2n ` 1q-dimensional contact metric generalized Sasakian-space-form admits
3f2
.
an infinitesimal non-isometric conformal transformation then f3 “ 1´2n
Hence, we can state the following:
Corollary 3.3. If a p2n ` 1q-dimensional pn ą 1q contact metric generalized Sasakian-space-form admits an infinitesimal non-isometric conformal
transformation, then it is φ-projectively flat and hence projectively flat.
In the following, we prove that the relation
f3 “
(3.5)
3f2
1 ´ 2n
implies f2 “ f3 “ 0.
In view of p2.7q and p2.6q, we can write the equation p2.9q as
(3.6)
P̃ pX, Y, Z, W q “ f2 tgpX, φZqgpφY, W q ´ gpY, φZqgpφX, W q
` 2gpX, φY qgpφZ, W qu ` f3 tηpXqηpZqgpY, W q
´ ηpY qηpZqgpX, W q ` gpX, ZqηpY qηpW q
´ gpY, ZqηpXqηpW q ` gpY, ZqgpX, W q ´ gpX, ZqgpY, W q,
where P̃ pX, Y, Z, W q “ gpP pX, Y qZ, W q.
Replacing X by φX and Y by φY, we get from p3.6q
(3.7)
P̃ pφX, φY, Z, W q “ f2 tgpφX, φZqgpφ2 Y, W q ´ gpφW, φZqgpφ2 X, W q
` 2gpφX, φ2 Y qgpφZ, φW qu
` f3 tgpφY, ZqgpφX, W q ´ gpφX, ZqgpφY, W qu.
Putting Y “ W “ ei , where tei u, i “ 1, 2, . . . , 2n ` 1, is an orthonormal basis
of the tangent space at each point of the manifold, and taking summation
over i, we get
(3.8)
2n`1
ÿ
P̃ pφX, φei , Z.ei q “ f2 t´gpφX, φZqgpφei , φei q ` gpφ2 Z, φ2 Xq
i“1
` 2gpφ2 X, φ2 Zqu ´ f3 gpφZ, φXq.
In the above equation again putting X “ Y “ ei , and taking summation
over i, we get by virtue of p3.5q, f2 “ 0. Which in view of p3.5q yields f3 “ 0.
Conversely, f2 “ f3 “ 0 trivially implies f3 “
3f2
1´2n ,
for n ą 1.
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In view of the above discussion, we can state the following:
Theorem 3.2. A p2n ` 1q-dimensional pn ą 1q generalized Sasakian-spaceform is φ-projectively flat or projectively flat if and only if f2 “ f3 “ 0.
It is known that [8] a generalized Sasakian-space-form of dimension
greater than three is conformally flat if and only if f2 “ 0. Hence, we have
the following:
Corollary 3.4. A p2n ` 1q-dimensional pn ą 1q generalized Sasakianspace-form is φ-projectively flat or projectively flat if and only if it is conformally flat.
Remark 3.1. It is well known that a Riemannian manifold of dimension
greater than three is projectively flat if and only if it is of constant curvature.
On the other hand, a manifold of constant curvature is conformally flat but
the converse does not hold always. The converse is true when the manifold is
an Einstein manifold. So, for arbitrary Riemannian manifold, the property
projectively flat and conformally flat are not equivalent. But, interestingly,
they are equivalent for a generalized Sasakian-space-form of dimension greater
than three.
4. Projectively locally symmetric generalized Sasakian-spaceforms
Definition 4.1. A p2n ` 1q-dimensional pn ą 1q generalized Sasakianspace-form will be called projectively locally symmetric if it satisfies
p∇W P qpX, Y qZ “ 0, for all X, Y, Z orthogonal to ξ.
From p2.6q, p2.8q and p2.10q, we have
(4.1) P pX, Y qZ “ f1 tgpY, ZqX ´ gpX, ZqY u
` f2 tgpX, φZqφY ´ gpY, φZqφX ` 2gpX, φY qφZu
` f3 tηpXqηpZqY ´ ηpY qηpZqX
` gpX, ZqηpY qξ ´ gpY, ZqηpXqξu
1
´ rSpY, ZqX ´ SpX, ZqY s.
2n
From p4.1q, we get by covariant differentiation
(4.2)
p∇W P qpX, Y qZ “ df1 pW qtgpY, ZqX ´ gpX, ZqY u
` df2 pW qtgpX, φZqφY ´ gpY, φZqφX ` 2gpX, φY qφZu
` f2 tgpX, φZqp∇W φqY ` gpX, p∇W φqZqφY
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´ gpY, φZqp∇W φqX ´ gpY, p∇W φqZqφX
` 2gpX, φY qp∇W φqZ ` 2gpX, p∇W φqY qφZu
` df3 pW qtηpXqηpZqY ´ ηpY qηpZqX
` gpX, ZqηpY qξ ´ gpY, ZqηpXqξu
` f3 tp∇W ηqpXqηpZqY ` ηpXqp∇W ηqpZqY
´ p∇W ηqpY qηpZqX ´ ηpY qp∇W ηqpZqX
` gpX, Zqp∇W ηqpY qξ ` gpX, ZqηpY qp∇W ξq
´ gpY, Zqp∇W ηqpXqξ ´ gpY, ZqηpXqp∇W ξqu
1
rp∇W SqpY, ZqX ´ p∇W SqpX, ZqY s,
´
2n
where ∇ denotes the Levi-Civita connection on the manifold. Differentiating
p2.7q covariantly with respect to W, we obtain
(4.3)
p∇W SqpX, Y q “ dp2nf1 ` 3f2 ´ f3 qpW qgpX, Y q
´ p3f2 ` p2n ´ 1qf3 qpp∇W ηqpXqηpY q
` ηpXqp∇W ηqpY qq ´ dp3f2 ` p2n ´ 1qf3 qpW qηpXqηpW q.
In view of p4.2q and p4.3q, it follows that
(4.4)
p∇W P qpX, Y qZ “ df1 pW qtgpY, ZqX ´ gpX, ZqY u
` df2 pW qtgpX, φZqφY ´ gpY, φZqφX ` 2gpX, φY qφZu
` f2 tgpX, φZqp∇W φqY ` gpX, p∇W φqZqφY
´ gpY, φZqp∇W φqX ´ gpY, p∇W φqZqφX
` 2gpX, φY qp∇W φqZ ` 2gpX, p∇W φqY qφZu
` df3 pW qtηpXqηpZqY ´ ηpY qηpZqX
` gpX, ZqηpY qξ ´ gpY, ZqηpXqξu
` f3 tp∇W ηqpXqηpZqY ` ηpXqp∇W ηqpZqY
´ p∇W ηqpY qηpZqX ´ ηpY qp∇W ηqpZqX
` gpX, Zqp∇W ηqpY qξ ` gpX, ZqηpY qp∇W ξq
´ gpY, Zqp∇W ηqpXqξ ´ gpY, ZqηpXqp∇W ξqu
1
´
rdp2nf1 ` 3f2 ´ f3 qpW qgpY, ZqX
2n
´ p3f2 ` p2n ´ 1qf3 qpp∇W ηqpY qηpZq ` ηpY qp∇W ηqpZqqX
´ dp3f2 ` p2n ´ 1qf3 qηpY qηpZqX
´ dp2nf1 ` 3f2 ´ f3 qpW qgpX, ZqY
` p3f2 ` p2n ´ 1qf3 qpp∇W ηqpXqηpZq ` ηpXqp∇W ηqpZqq
` dp3f2 ` p2n ´ 1qf3 qpW qηpXqηpZqY s.
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Taking X, Y, Z orthogonal to ξ, we get from p4.4q
(4.5)
p∇W P qpX, Y qZ “ df1 pW qtgpY, ZqX ´ gpX, ZqY u
` df2 pW qtgpX, φZqφY ´ gpY, φZqφX ` 2gpX, φY qφZu
` f2 tgpX, φZqp∇W φqY ` gpX, p∇W φqZqφY
´ gpY, φZqp∇W φqX ´ gpY, p∇W φqZqφX
` 2gpX, φY qp∇W φqZ ` 2gpX, p∇W φqY qφZu
1
dp2nf1 ` 3f2 ´ f3 qpW qpgpY, ZqX ´ gpX, ZqY q.
´
2n
If the manifold is projectively locally symmetric then from the above
equation, we get
(4.6)
1
dp2nf1 ` 3f2 ´ f3 qpW qpgpY, ZqX ´ gpX, ZqY q
2n
“ df1 pW qtgpY, ZqX ´ gpX, ZqY u
` df2 pW qtgpX, φZqφY ´ gpY, φZqφX ` 2gpX, φY qφZu
` f2 tgpX, φZqp∇W φqY ` gpX, p∇W φqZqφY
´ gpY, φZqp∇W φqX ´ gpY, p∇W φqZqφX
` 2gpX, φY qp∇W φqZ ` 2gpX, p∇W φqY qφZu.
Taking the inner product g in both sides of the above equation with a vector
field V, we have
1
dp3f2 ´ f3 qpW qpgpY, ZqgpX, V q ´ gpX, ZqgpY, V qq
2n
` df2 pW qtgpX, φZqgpφY, V q ´ gpY, φZqgpφX, V q ` 2gpX, φY qgpφZ, V qu
` f2 tgpX, φZqgpp∇W φqY, V q ` gpX, p∇W φqZqgpφY, V q
´ gpY, φZqgpp∇W φqX, V q ´ gpY, p∇W φqZqgpφX, V q
` 2gpX, φY qgpp∇W φqZ, V q ` 2gpX, p∇W φqY qgpφZ, V qu.
(4.7)
Putting Z “ V “ ei , where tei u is an orthonormal basis of the tangent space
at each point of the manifold, and taking summation over i, i “ 1, 2, . . . , 2n`1,
we get
ÿ
(4.8)
f2 t´gpφX, p∇W φqY q ` gpX, p∇W φqei qgpφY, ei q
i
` gpφY, p∇W φqXq ´
ÿ
gpY, p∇W φqei qgpφX, ei q
i
`2
ÿ
gpX, φY qgpp∇W φqei , ei qu “ 0.
i
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We know that for the Levi-Civita connection ∇, p∇W gqpX, Y q “ 0, which
gives
∇W gpX, Y q ´ gp∇W X, Y q ´ gpX, ∇W Y q “ 0.
In the above equation putting X “ ei and Y “ φei , we have
´gp∇W ei , φei q ´ gpei , ∇W φei q “ 0.
The above equation can be written as
gpei , φ∇W ei q ´ gpei , ∇W φei q “ 0.
Thus we have
gpei , p∇W φqei q “ 0.
(4.9)
By virtue of p4.9q, p4.8q takes the form
ÿ
(4.10)
f2 t´gpφX, p∇W φqY q ` gpX, p∇W φqei qgpφY, ei q
i
` gpφY, p∇W φqXq ´
ÿ
gpY, p∇W φqei qgpφX, ei q “ 0.
i
The above equation is true for any vector fields X, Y on the manifold.
For X ­“ Y, p4.10q yields
f2 “ 0.
It is known that [8] a generalized Sasakian-space-form of dimension greater
than three is conformally flat if and only if f2 “ 0. Hence, the manifold under
consideration is conformally flat.
Conversely, suppose that the manifold is conformally flat. Hence, f2 “ 0.
In addition, if we consider X, Y, Z orthogonal to ξ, then p2.6q yields
RpX, Y qZ “ f1 pgpY, ZqX ´ gpX, ZqY q.
The above equation gives
(4.11)
r “ 2np2n ` 1qf1 .
In view of p2.8q and p4.11q, we have f3 “ 0. Hence, from p4.5q, we get
p∇W P qpX, Y qZ “ 0.
Therefore, the manifold is projectively locally symmetric. The above discussion helps us to state the following:
Theorem 4.1. A p2n ` 1q-dimensional pn ą 1q generalized Sasakianspace-form is projectively locally symmetric if and only if it is conformally
flat.
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5. Projectively locally φ-symmetric generalized Sasakian-spaceform
Defintion 5.1.
A generalized Sasakian-space-form of dimension
greater than three is called projectively locally φ-symmetric if it satisfies
φ2 p∇W P qpX, Y qZ “ 0, for all vector fields X, Y, Z orthogonal to ξ.
In this connection, it should be mentioned that the notion of locally
φ-symmetric manifolds was introduced by T. Takahashi in the context of
Sasakian geometry [10]. Let us consider a projectively locally φ-symmetric
generalized Sasakian-space-form of dimension greater than three. Then from
the definition and p2.1q, we get
(5.1)
´ p∇W P qpX, Y qZ ` ηpp∇W P qpX, Y qZqξ “ 0.
Taking the inner product g in both sides of the above equation with respect
to W , we get
(5.2)
´ gpp∇W P qpX, Y qZ, W q ` ηpp∇W P qpX, Y qZqηpW q “ 0.
If we take W orthogonal to ξ, then the above equation yields
(5.3)
gpp∇W P qpX, Y qZ, W q “ 0.
The above equation is true for all W orthogonal to ξ. If we choose W ­“ 0
and not orthogonal to p∇W P qpX, Y qZ, then it follows that
p∇W P qpX, Y qZ “ 0.
Hence, the manifold is projectively locally symmetric and hence by Theorem
4.1, it is conformally flat.
Conversely, let the manifold is conformally flat and hence f2 “ 0. Again,
for X, Y, Z orthogonal to ξ, f2 “ 0 implies f3 “ 0, as before. From p4.5q,
using p2.1q we have
φ2 p∇W P qpX, Y qZ “ df1 pW qtgpX, ZqY ´ gpY, ZqXu
` df2 tgpY, φZqφX ´ 2gpX, φY qφZ ´ gpX, φZqφY u
` f2 tgpX, φZqφ2 pp∇W φqY q ´ gpX, p∇W φqφZqφY
´ gpY, φZqφ2 pp∇W φqXq ` gpY, p∇W φqZqφX
` 2gpX, φY qφ2 pp∇W φqZq ´ 2gpX, p∇W φqY qφZu
1
dp2nf1 ` 3f2 ´ f3 qpW qpgpX, ZqY ´ gpY, ZqXq.
´
2n
Hence, for f2 “ f3 “ 0, the above equation yields
φ2 p∇W P qpX, Y qZ “ 0,
where X, Y, Z are orthogonal to ξ. Therefore, the manifold is projectively
locally φ-symmetric.
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736
A. Sarkar, A. Akbar
Now we are in a position to state the following:
Theorem 5.1. A p2n ` 1q-dimensional pn ą 1q generalized Sasakianspace-form is projectively locally φ-symmetric if and only if it is conformally
flat.
Combining the results of Section 3, Section 4 and Section 5, we find the
following:
Corollary 5.1. For a p2n ` 1q-dimensional pn ą 1q generalized Sasakianspace-form, the following conditions are equivalent:
(i)
(ii)
(iii)
(iv)
(v)
the
the
the
the
the
manifold
manifold
manifold
manifold
manifold
is
is
is
is
is
projectively flat,
φ-projectively flat,
conformally flat,
projectively locally symmetric,
projectively locally φ-symmetric.
Remark. The notion of quarter-symmetric metric connection was introduced by S. Golab [7]. The torsion tensor of the quarter-symmetric metric
connection is given by
T pX, Y q “ ηpY qX ´ ηpXqY.
If X, Y are orthogonal to ξ, then the torsion tensor vanishes and the quartersymmetric metric connection reduces to Levi-Civita connection. Therefore,
all the results of Section 4 and Section 5 are of the same form with respect
to quarter-symmetric metric connection and Levi-Civita connection.
6. Examples
Example 6.1. Let us now give an example of a generalized Sasakianspace-form which is projectively flat, φ-projectively flat, conformally flat,
projectively locally symmetric and projectively locally φ-symmetric.
In [1], it is shown that R ˆf Cm is a generalized Sasakian-space-form with
f1 “ ´
pf 1 q2
,
f2
f2 “ 0,
f3 “ ´
pf 1 q2 f 2
` ,
f2
f
where f “ f ptq, t P R and f 1 denotes derivative of f with respect to t. If
we choose m “ 4, and f ptq “ et , then M is a 5-dimensional conformally flat
generalized Sasakian-space-form because f2 “ 0. We also see that f3 “ 0.
Therefore, by the results obtained in the present paper, M is projectively
flat, φ-projectively flat, conformally flat, projectively locally symmetric and
projectively locally φ-symmetric.
Example 6.2. For a Sasakian-space-form of dimension greater than three
and of constant φ-sectional curvature 1, f1 “ 1, f2 “ f3 “ 0. So, by the
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Generalized Sasakian-space-forms with projective curvature tensor
737
results obtained in the present paper, M is projectively flat, φ-projectively
flat, conformally flat, projectively locally symmetric and projectively locally
φ-symmetric.
References
[1] P. Alegre, D. Blair, A. Carriazo, Generalized Sasakian-space-forms, Israel J. Math. 14
(2004), 157–183.
[2] P. Alegre, A. Cariazo, Structures on generalized Sasakian-space-forms, Differential
Geom. Appl. 26 (2008), 656–666.
[3] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture notes in Mathematics, 509, Springer-Verlag, Berlin, 1976.
[4] U. C. De, A. Sarkar, On the projective curvature tensor of generalized Sasakian
space-forms, Quaestiones Math. 33 (2010), 245–252.
[5] U. C. De, A. Sarkar, Some results on generalized Sasakian-space-forms, Thai. J. Math.
8 (2010), 1–10.
[6] U. C. De, A. Sarkar, Some curvature properties of generalized Sasakian space forms,
Lobachevskii J. Math. 33 (2012), 22–27.
[7] S. Golab, On semi-symmetric and quarter-symmetric linear connections, Tensor (N.S.)
29 (1975), 249–254.
[8] U. K. Kim, Conformally flat generalized Sasakian-space-forms and locally symmetric
generalized Sasakian-space-forms, Note Mat. 26 (2006), 55–67.
[9] C. Özgür, On φ-conformally flat LP-Sasakian manifolds, Rad. Mat. 12 (2003), 99–106.
[10] T. Takahashi, Sasakian φ-symmetric spaces, Tohoku Math J. 29 (1977), 91–113.
[11] K. Yano, S. Bochner, Curvature and Betti Numbers, Ann. of Math. Stud. 32, Princeton
Univ. Press, 1953.
A. Sarkar, A. Akbar
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF KALYANI
KALYANI 741235
NADIA, WEST BENGAL, INDIA
E-mail: [email protected]
Received November 7, 2012.
Communicated by W. Domitrz.
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