ON P-SASAKIAN MANIFOLDS
SATYABROTA KUNDU
Communicated by the former editorial board
The object of the present paper is to study P-Sasakian manifolds admitting φ-pseudo-quasi-conformal structure. It has been shown that aφ-pseudo-quasi- conformally symmetric P-Sasakian manifold isφ-symmetric. Some observations for a 3-dimensional locally φ-pseudo-quasi-conformally symmetric P-Sasakian manifold are also given.
AMS 2010 Subject Classication: 53C15, 53C25, 53C21.
Key words: P-Sasakian manifolds,φ-symmetric manifolds, pseudo-quasi-conformal curvature tensor.
1. INTRODUCTION
In [8], Yano and Sawaki introduced the notion of quasi-confomal curvature tensor on an n(≥3)-dimensional Riemannian manifold. Recently, the authors of [5] introduced the notion of pseudo-quasi-conformal curvature tensorCe on a Riemannian manifold of dimensionn(≥3)which includes the projective, quasi- conformal, Weyl conformal and concircular curvature tensor as special cases.
This tensor is dened by
C(X, Ye )Z = (p+d)R(X, Y)Z+
q− d n−1
h
S(Y, Z)X−S(X, Z)Y i
+qh
g(Y, Z)QX −g(X, Z)QYi
− r
n(n−1){p+ 2(n−1)q}h
g(Y, Z)X−g(X, Z)Yi (1.1)
whereX, Y, Z ∈χ(M), andp, q, dare real constants such thatp2+q2+d2>0. In particular, if
(1) p = q = 0, d = 1;
(2) p6=0, q6=0, d=0;
(3) p = 1, q = −n−21 , d=0;
(4) p = 1, q = d = 0;
MATH. REPORTS 15(65), 3 (2013), 221232
then Ce reduces to the projective curvature tensor, quasi-conformal curvature tensor, conformal curvature tensor and concircular curvature tensor, respec- tively.
From (1.1), we obtain
(∇WC)(X, Ye )Z = (p+d)(∇WR)(X, Y)Z +
q− d n−1
h
(∇WS)(Y, Z)X−(∇WS)(X, Z)Y i
+qh
g(Y, Z)(∇WQ)X−g(X, Z)(∇WQ)Yi
− dr(W)
n(n−1)[p+ 2(n−1)q]h
g(Y, Z)X−g(X, Z)Yi (1.2) .
In [3], authors introduced the notion of φ-quasi-conformal symmetric structure on a contact metric manifold. In the present paper we introduce the notion ofφ-pseudo-quasi-conformal structure on a paracontact metric manifold, as follows:
Denition 1.1. A paracontact metric manifold is said to be locally φ- pseudo-quasi-conformally symmetric if the pseudo-quasi-conformal curvature tensor Ce satises the condition
(1.3) φ2
(∇XC)(Y, Z)We
= 0, for all X, Y, Z, W ∈χ(M) which are orthogonal to ξ.
Denition 1.2. A paracontact metric manifold is said to be φ-pseudo- quasi-conformally symmetric if the pseudo-quasi-conformal curvature tensorCe satises the condition
(1.4) φ2
(∇XC)(Y, Z)We
= 0, ∀X, Y, Z, W ∈χ(M).
It is shown that if a P-Sasakian manifold is φ-pseudo-quasi-conformally symmetric, then the manifold is Einstein providedn
p+ (n−2)q+n−1d o 6= 0. It is also shown that an Einstein P-Sasakian manifold admitting a φ-pseudo- quasi-conformally symmetric structure isφ-symmetric. We study 3-dimensional φ-pseudo-quasi-conformally symmetric P-Sasakian manifolds in the next sec- tion. We prove that a 3-dimensional P-Sasakian manifold is φ-pseudo-quasi- conformally symmetric if and only if the scalar curvaturer is constant provided
2p+ 2q+32d
6= 0. Finally, we give an example of a 3-dimensional φ-pseudo- quasi-conformally symmetric P-Sasakian manifold.
2. PRELIMINARIES
An n-dimensional dierentiable manifoldM is called an almost paracon- tact manifold (see [1, 2, 4]) if it admits an almost paracontact structure (φ, ξ, η) consisting of a (1,1) tensor eldφ, a vector eldξ and its dual 1-form η satis- fying
φ2 =I−η⊗ξ, η(ξ) = 1, φξ = 0, η◦φ= 0.
(2.1)
Let gbe a Riemannian metric compatible with (φ, ξ, η), i.e., (2.2) g(X, Y) =g(φX, φY) +η(X)η(Y),
or equivalently,
(2.3) g(X, φY) =g(φX, Y), g(X, ξ) =η(X),
for allX,Y ∈χ(M)whereχ(M)is the collection of all smooth vector elds on M. Then M becomes an almost paracontact Riemannian manifold equipped with an almost paracontact Riemannian structure (φ, ξ, η, g).
An almost paracontact Riemannian manifold(Mn, g)is called a P-Sasakian manifold if it satises (see [1]),
(2.4) (∇Xφ)(Y) =−g(X, Y)ξ−η(Y)X+ 2η(X)η(Y)ξ, ∀X, Y ∈χ(M) where∇is the Levi-Civita connection of the Riemannian manifold.
From the above equations it follows that,
(2.5) ∇Xξ =φX,
(2.6) (∇Xη)(Y) =g(φX, Y) = (∇Yη)(X), ∀X, Y ∈χ(M).
In an n-dimensional P-Sasakian manifoldM, the curvature tensorR, the Ricci tensor S and the Ricci operatorQ satisfy (see [2, 4]):
R(X, Y)ξ=η(X)Y −η(Y)X, (2.7)
R(ξ, X)Y =η(Y)X−g(X, Y)ξ, (2.8)
R(ξ, X)ξ=X−η(X)ξ, (2.9)
S(X, ξ) =−(n−1)η(X), (2.10)
Qξ=−(n−1)ξ ∀X, Y ∈χ(M).
(2.11)
Further, in ann-dimensional P-Sasakian manifold, the following relations hold:
η R(X, Y)Z
=g(X, Z)η(Y)−g(Y, Z)η(X), (2.12)
η R(X, Y)ξ (2.13) = 0,
η R(ξ, X)Y
=η(X)η(Y)−g(X, Y), (2.14)
S(φX, φY) =S(X, Y) + (n−1)η(X)η(Y) ∀X, Y, Z ∈χ(M). (2.15)
A Riemannian manifold is said to be an Einstein manifold if there exists a real constant λ such that ∀ X, Y ∈ χ(M) the Ricci tensor S is of the form S(X, Y) =λg(X, Y).
3.φ-PSEUDO-QUASI-CONFORMALLY SYMMETRIC P-SASAKIAN MANIFOLDS
Let M be a φ-pseudo-quasi-conformally symmetric P-Sasakian manifold.
Then equation (1.4) holds onM and, from (2.1), we obtain (3.1) (∇WC)(X, Ye )Z−η (∇WC)(X, Ye )Z
ξ = 0.
Using (1.1) in (3.1), we get, 0 = (p+d)(∇WR)(X, Y)Z
+
q− d n−1
h
(∇WS)(Y, Z)X−(∇WS)(X, Z)Y i
+q h
g(Y, Z)(∇WQ)X−g(X, Z)(∇WQ)Y i
− dr(W)
n(n−1)[p+ 2(n−1)q]
h
g(Y, Z)X−g(X, Z)Y i
−(p+d)η (∇WR)(X, Y)Z ξ
−
q− d n−1
h
(∇WS)(Y, Z)η(X)−(∇WS)(X, Z)η(Y)i ξ
−q h
g(Y, Z)η (∇WQ)X
−g(X, Z)η (∇WQ)Yi ξ
+ dr(W)
n(n−1)[p+ 2(n−1)q]
h
g(Y, Z)η(X)−g(X, Z)η(Y) i
ξ.
Taking inner product with U, we have, 0 = (p+d)(∇WR)(X, Y, Z, U)
+
q− d n−1
h
(∇WS)(Y, Z)g(X, U)−(∇WS)(X, Z)g(Y, U) i
+q h
g(Y, Z)g
(∇WQ)X, U
−g(X, Z)g
(∇WQ)Y, U i
− dr(W)
n(n−1)[p+ 2(n−1)q]
h
g(Y, Z)g(X, U)−g(X, Z)g(Y, U) i
−(p+d)η (∇WR)(X, Y)Z η(U)
−
q− d n−1
h
(∇WS)(Y, Z)η(X)−(∇WS)(X, Z)η(Y)i η(U)
−q h
g(Y, Z)η (∇WQ)X
−g(X, Z)η (∇WQ)Yi η(U) + dr(W)
n(n−1)[p+ 2(n−1)q]
h
g(Y, Z)η(X)−g(X, Z)η(Y) i
η(U).
Putting X=U =ei, where {ei},i= 1,2, . . . , n, is an orthonormal basis of the tangent space at each point of the manifold, and taking summation over i, the above equation reduces to
0 =
p+ (n−2)q+ d n−1
(∇WS)(Y, Z) +n
qg (∇WQ)ei, ei
− dr(W)
n(n−1) p+ 2(n−1)q
(n−2)
−qη (∇WQ)(ei) η(ei)
o
g(Y, Z)
−qg (∇WQ)Y, Z
−(p+d)η (∇WR)(ei, Y)Z η(ei) +
q− d n−1
(∇WS)(ξ, Z)η(Y) +qη (∇WQ)(Y) η(Z)
− dr(W)
n(n−1) p+ 2(n−1)q
η(Y)η(Z).
Putting Z =ξ, and using (2.1), we obtain the from above equation, 0 =
p+ (n−2)q+ d n−1
(∇WS)(Y, ξ) +n
qg (∇WQ)ei, ei
− dr(W)
n(n−1)(p+ 2(n−1)q)(n−2)
−qη
(∇WQ)(ei) η(ei)o
η(Y)
−(p+d)η((∇WR)(ei, Y)ξ)η(ei) +
q− d n−1
(∇WS)(ξ, ξ)η(Y)
− dr(W)
n(n−1) p+ 2(n−1)q η(Y).
Now,
g (∇WQ)ei, ei
= (∇WS)(ei, ei),
=dr(W).
Hence, using the above equation, we have 0 =
p+ (n−2)q+ d n−1
(∇WS)(Y, ξ)
+n
q− dr(W)
n(n−1)(p+ 2(n−1)q)(n−2)−qη (∇WQ)(ei) η(ei)o
η(Y)
−(p+d)η (∇WR)(ei, Y)ξ
η(ei) +
q− d n−1
(∇WS)(ξ, ξ)η(Y)
− dr(W)
n(n−1) p+ 2(n−1)q η(Y).
(3.2)
Also, by (2.1), (2.5), (2.10), (2.11), we get η (∇WQ)(ei)
η(ei) =g (∇WQ)ξ, ξ , (3.3) = 0.
Now, equation g (∇WR)(ei, Y)ξ, ξ
=g ∇WR(ei, Y)ξ, ξ
−g R(∇Wei, Y)ξ, ξ
−g R(ei,∇WY)ξ, ξ
−g R(ei, Y)∇Wξ, ξ , leads to
g (∇WR)(ei, Y)ξ, ξ
=g ∇WR(ei, Y)ξ, ξ
−g R(ei,∇WY)ξ, ξ (3.4)
−g R(ei, Y)∇Wξ, ξ (3.5) ,
since from (2.7), we have g R(∇Wei, Y)ξ, ξ
=g η(∇Wei)Y −η(Y)∇Wei, ξ ,
= 0.
Also, using (2.7), we nd g R(ei,∇WY)ξ, ξ
=g η(ei)∇WY −η(∇WY)ei, ξ ,
=η(∇WY)η(ei)−η(ei)η(∇WY),
= 0.
Hence, (3.4) reduces to, g (∇WR)(ei, Y)ξ, ξ
=g ∇WR(ei, Y)ξ, ξ
−g R(ei, Y)∇Wξ, ξ (3.6) .
From the denition of the Levi-Civita connection of g, we have
∇Wg
R(ei, Y)ξ, ξ
= 0, and then, using (2.7),
g ∇WR(ei, Y)ξ, ξ
−g R(ei, Y)∇Wξ, ξ
= 0.
Hence, from (3.6), it follows,
g (∇WR)(ei, Y)ξ, ξ
=η (∇WR)(ei, Y)ξ (3.7) = 0.
By making use of (3.3) and (3.7), we obtain from (3.2),
p+ (n−2)q+ d n−1
(∇WS)(Y, ξ) (3.8)
= 1
n p+ 2(n−1)q +q
dr(W)η(Y).
PuttingY =ξin (3.8), we obtaindr(W) = 0, provided p+(3n−2)q 6= 0, which implies r is constant. Thus, we have the following,
Theorem 3.1. If a P-Sasakian manifold admitsφ-pseudo-quasi-conformal structure then the scalar curvature of the manifold is constant provided {p+ (3n−2)q} 6= 0.
Hence, from (3.8) it follows (3.9) (∇WS)(Y, ξ) = 0, provided
p+ (n−2)q+ d n−1
6= 0.
Using (2.5), (2.6) and (2.10) we obtain from (3.9), (3.10) S(Y, φW) + (n−1)g(Y, φW) = 0.
Replacing Y by φY, and using (2.2) and (2.15) we have, (3.11) S(Y, W) =−(n−1)g(Y, W).
Hence, we have the following
Theorem 3.2. If a P-Sasakian manifold is φ-pseudo-quasi-conformally symmetric, then the manifold is an Einstein manifold provided n
p+ (n−2)q+n−1d o 6= 0.
Moreover, if p, q 6= 0 and d = 0, then pseudo-quasi-conformal curva- ture tensor reduces to quasi-conformal curvature tensor, hence, from Theo- rem (3.2) we can state as follows
Corollary 3.3. Aφ-quasi-conformal P-Sasakian manifold is an Einstein manifold.
Conversely, let us assume the P-Sasakian manifold to be Einstein one.
Then
S(X, Y) =λg(X, Y), whereλis constant and X, Y ∈χ(M). Then,
QX =λX.
Thus, from (1.1) we have, C(X, Ye )Z = (p+d)R(X, Y)Z
+
λ
q− d n−1
+qλ− r
n(n−1) p+ 2(n−1)q
×h
g(Y, Z)X−g(X, Z)Y i
, or,
(∇WC)(X, Ye )Z = (p+d)(∇WR)(X, Y)Z
−
p+ 2(n−1)q n(n−1)
dr(W) g(Y, Z)X−g(X, Z)Y .
Applying φ2 to both sides of the above equation, we have φ2
(∇WC)(X, Ye )Z
= (p+d)φ2
(∇WR)(X, Y)Z
−
p+ 2(n−1)q n(n−1)
dr(W) g(Y, Z)φ2X−g(X, Z)φ2Y .
Since the manifold is Einstein one, therefore, the scalar curvature r is constant and hence, from above we can state as follows
Theorem 3.4. An Einstein P-Sasakian manifold admittingφ-pseudo-quasi- conformally symmetric structure is also φ-symmetric (see [7]) .
4. 3-DIMENSIONALφ-PSEUDO-QUASI-CONFORMALLY SYMMETRIC P-SASAKIAN MANIFOLDS
For a 3-dimensional P-Sasakian manifold, we have,
R(X, Y)Z =g(Y, Z)QX −g(X, Z)QY +S(Y, Z)X−S(X, Z)Y +r
2 h
g(X, Z)Y −g(Y, Z)Xi (4.1) .
Replacing Z by ξ in (4.1) and using (2.7), (2.10) and (2.11), r
2+ 1 h
η(Y)X−η(X)Yi
=η(Y)QX −η(X)QY.
Putting Y =ξ above, we have on using (2.1) (4.2) S(X, Y) =
r 2 + 1
g(X, Y)−r 2 + 3
η(X)η(Y).
Using (4.2) in (4.1), we get R(X, Y)Z =
r 2+ 2
h
g(Y, Z)X−g(X, Z)Y i
−r 3 + 2
h
g(Y, Z)η(X)ξ−g(X, Z)η(Y)ξ +η(Y)η(Z)X−η(X)η(Z)Y
i (4.3) .
Using (4.2) and (4.3), we obtain from (1.1), C(X, Ye )Z =r
6 + 1
2p+ 2q+3 2d
g(Y, Z)X−g(X, Z)Y + 2p+ 3q+ 2dr
6 + 1 h
g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ i
+
2p+ 3q+d 2
r 6 + 1
h
η(X)η(Z)Y −η(Y)η(Z)X i (4.4) .
Covariantly dierentiating both sides of (4.4) with respect toW, we have (∇WC)(X, Ye )Z= 1
6dr(W)
2p+ 2q+3 2d
g(Y, Z)X−g(X, Z)Y +1
6dr(W) 2p+ 3q+ 2dh
g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξi +1
6dr(W)
2p+ 3q+d 2
h
η(X)η(Z)Y −η(Y)η(Z)Xi + 2p+ 3q+ 2dr
6+ 1 n
g(X, Z)(∇Wη)(Y)ξ
+g(X, Z)η(Y)∇Wξ−g(Y, Z)(∇Wη)(X)ξ
−g(Y, Z)η(X)∇Wξ o
−
2p+ 3q+d 2
r
6 + 1 n
(∇Wη)(X)η(Z)Y
−(∇Wη)(Y)η(Z)X+ (∇Wη)(Z)η(X)Y
−(∇Wη)(Z)η(Y)X o (4.5) .
Assuming X, Y and Z orthogonal to ξ, we have from (4.5), (∇WC)(X, Ye )Z = 1
6dr(W)
2p+ 2q+3 2d
g(Y, Z)X−g(X, Z)Y
+ 2p+ 3q+ 2dr 6 + 1
n
g(X, Z)(∇Wη)(Y)ξ
−g(Y, Z)(∇Wη)(X)ξ o (4.6) .
Taking φ2 on both sides of (4.6), φ2
(∇WC)(X, Y˜ )Z
= 1 6dr(W)
2p+ 2q+3 2d
g(Y, Z)φ2X
−g(X, Z)φ2Y (4.7) .
If possible, let us assume φ2 (∇WC)(X, Ye )Z
= 0, then dr(W) = 0 provided 2p+ 2q+32d
6= 0. Hence, dr(W) = 0 impliesr is constant.
For the converse part, if the scalar curvature r is constant then from (4.7) we can say that the P-Sasakian manifold is φ-pseudo-quasi-conformally symmetric. Thus, we have the following
Theorem 4.1. A 3-dimensional P-Sasakian manifold is locally φ-pseudo- quasi-conformally symmetric if and only if the scalar curvature r is constant provided 2p+ 2q+32d
6= 0.
5. EXAMPLE OF Aφ-PSEUDO-QUASI-CONFORMALLY SYMMETRIC P-SASAKIAN STRUCTURE
Let us consider the 3-dimensional Riemannian manifold M =R3 with a rectangular cartesian coordinate system (xi).
Let us choose the vector elds
e1, e2, e3 as e1 = ∂
∂x1
, e2 =e−x1 ∂
∂x2
, e3 =e−x1 ∂
∂x3
(5.1) .
Thus,
e1, e2, e3 forms a basis of χ(M) =χ(R3). Let gbe the Riemannian metric dened by
g(e1, e1) =g(e2, e2) =g(e3, e3) = 1, g(e1, e2) =g(e1, e3) =g(e2, e3) = 0.
(5.2)
Let ξ = e1 be the vector eld associated with the 1-form η. The (1,1)- tensor eldφbe dened by,
φ(e1) = 0, φ(e2) =e2, φ(e3) =e3. (5.3)
Since
e1, e2, e3 is a basis, therefore, any vector eldsXandY inM can be expressed as
X=ae1 +be2+ce3 and
Y =a1e1+b1e2 +c1e3 wherea, b, c, a1, b1, c1 are functions.
Now, using the linearity of φand g, and takingξ =e1 we have, η(ξ) = 1, φ2X=X−η(X)ξ, g(φX, φY) =g(X, Y)−η(X)η(Y), for any vector elds X and Y in M. Thus, (φ, ξ, η, g) denes an almost para- contact metric structure onM.
Let∇be the Levi-Civita connection with respect to the Riemannian met- ric g. Then we have,
[e1, e2] =−e2, [e1, e3] =−e3, [e2, e3] = 0.
By using Koszul's formulae (see [6]), we have
∇e
1e3 = 0, ∇e
1e2 = 0, ∇e
1e1 = 0,
∇e
2e3 = 0, ∇e
2e1 =e2, ∇e
2e2 =−e1,
∇e
3e1 =e3, ∇e
3e2 = 0, ∇e
3e3 =−e1. Also, the Riemannian curvature tensor R is given by,
R(X, Y)Z =∇X∇YZ− ∇Y∇XZ− ∇[X,Y]Z.
Then,
R(e1, e2)e2 =−e1, R(e1, e3)e3 =−e1, R(e2, e1)e1 =−e2, R(e2, e3)e3 =−e2, R(e3, e1)e1 =−e3, R(e3, e2)e2 =−e3, R(e1, e2)e3 = 0, R(e2, e3)e1 = 0, R(e3, e1)e2 = 0.
Then, the Ricci tensor S is given by
S(e1, e1) =−2, S(e2, e2) =−2, S(e3, e3) =−2, S(e1, e2) = 0, S(e1, e3) = 0, S(e2, e3) = 0.
Thus, the scalar curvature r = S(e1, e1) +S(e2, e2) +S(e3, e3) = −6 is constant. The conditions (2.4) and (2.5) for any vector elds X and Y inM holds. It can be shown that all the properties of P-Sasakian manifold hold for any vector eldsX, Y inM. Since the given 3-dimensional P-Sasakian manifold is of constant scalar curvature r = 6, therefore, by virtue of Theorem(4.1), it implies that it is locally φ-pseudo-quasi-conformally symmetric in nature.
Acknowledgments. The author was nancially supported by University Grants Com- mission (India). The author is also thankful to the honorable referee for his valuable suggestions for the improvement of the paper.
REFERENCES
[1] T. Adati and T. Miyazawa, Some properties of P-Sasakian manifold. TRU Math 13 (1977), 2532.
[2] T. Adati and T. Miyazawa, On P-Sasakian manifold satisfying certain conditions. Tensor (N.S) 33 (1979), 173178.
[3] U.C. De, C. Ozgur and A.K. Mondal, Onφ-quasi-conformally symmetric Sasakian man- ifolds. Indag. Math. (N.S.) 20 (2009), 191200.
[4] I. Sato, On a structure similar to almost contact structure. Tensor (N.S.) 30 (1976), 219224.
[5] A.A. Shaikh and S.K. Jana, A pseudo-quasi-conformal curvature tensor on a Riemannian manifold. South East Asian J. Math. Math. Sci. 4 (2005), 1, 1520.
[6] J.A. Schouten, Ricci Calculus. Springer-Verlag, Berlin, 2nd Ed.(1954), p. 332.
[7] T. Takahashi, Sasakian φ-symmetric spaces. Tohoku Math. J. (2) 29 (1977), 1, 91113.
[8] K. Yano and S. Sawaki, Riemannian manifolds admitting a conformal transformation group. J. Dierential Geom. 2 (1968), 161184.
Received 29 July 2011 Loreto College,
Department of Mathematics, Kolkata-700071, India