INTERNATIONAL JOURNAL OF MATHEMATICS AND ANALYSIS
January-June 2011, Volume 3, No. 1, pp. 1-12
ON GENERALIZED SASAKIAN-SPACE-FORMS
SATISFYING CERTAIN CONDITIONS
D. Narain, S. Yadav and P. K. Dwivedi
ABSTRACT: The object of the present paper is to study generalized Sasakianspace-forms with generalized recurrent and Ricci semi symmetric generalized
Sasakian space-forms. Such space-forms with D-Conformal curvature tensor
are also considered.
Mathematics Subject classification (2000): 53C25, 53D25, 53D15.
Keywords: Generalized Sasakian-space-forms, Generalized recurrent,
Generalized concircular recurrent, Ricci symmetric, and D-conformal
curvature tensor.
1. INTRODUCTION
The nature of a Riemannian manifold mostly depends on the curvature tensor R
of the manifold. It is well known that the sectional curvature of a manifold
determine curvature tensor completely A Riemannian manifold with constant
sectional curvature c is known as real-space forms and its curvature tensor is
given by
R (X, Y ) Z = c {g (T, Z) X – g (X, Z) Y}.
A Sasakian manifold with constant sectional curvature is a Sasakian-spaceform and it has a specific form of its curvature. Similar notion also hold for
Kenmotsu and cosymplectic space forms. In order to generalized space-forms in
a common frame. P. Alegre, D. E. Blair and A. Carriazo introduced the notion of
generalized Sasakian-space-forms in 2004[1]. In this connection it should
mentioned that in 1989, Z. Olszak [9] studied generalized complex-space-forms
and proved that its existence. A generalized Sasakian-space-form is defined in [1]:
Given an almost contact metric manifold M (φ, ξ, η, g), we say that M is a
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D. Narain, S. Yadav and P. K. Dwivedi
generalized Sasakian-space-form if there exist three functions f1, f2, f3 on M such
that the curvature tensor R is given by
R (X, Y ) Z = f1{g (Y, Z ) X – g (X, Z ) Y }
f2{g (X, φZ ) φY – g (Y, φZ ) φX + 2g (X, φY ) φZ }
f3{η (X ) η (Z ) Y – η (Y ) η (Z ) X + g (X, Z ) η (Y ) ξ – g (Y, Z ) Y }
for any vector fields X, Y, Z on M.
In such case we denote the manifold as M (f1, f2, f3). In [1] the authors cited
the several examples of such manifolds. If f1 = c 4+ 1 , f 2 = c 4− 1 and f 3 = c 4− 1 then
a generalized Sasakian-space-form with Sasakian structure becomes Sasakianspace-forms. Generalized Sasakian-space-forms and Sasakian-space-forms have
been studied by many authors ([1] [2] [3] [10]). Symmetry of the manifold is the
most important properties among its all its geometrical properties. thise properties
have been studied by many authors ([8] [9]). As a weaker notion of locally
symmetric manifolds. Symmetry of the manifold basically depends on curvature
tensor and the Ricci tensor of the manifold. The main purpose of this paper is to
investigate the class of almost contact metric manifolds which are called
generalized Sasakian space-forms.
2. PRELIMINARIES
A (2n + 1)-dimensional Riemannian manifold (M, g) is called an almost contact
manifold if the following results hold [2]:
(a) φ2(X ) = – X + η (X ) ξ,
(b) φξ = 0
(2.1)
(a) η (ξ) = 1,
(b) g (X, ξ) = η (X ), (c) η (φX) = 0
(2.2)
g (φX, φY ) = g (X, Y ) – η (X ) η (Y )
(a) g (φX, Y ) = – g (X, φY ) (b) g (φX, X ) = 0
(∇X η) (Y ) = g (∇X ξ, Y )
An almost contact metric manifold is called contact metric manifold if
dη (X, Y ) = Φ (X, Y ) = g (X, φY )
(2.3)
(2.4)
(2.5)
On Generalized Sasakian-Space-Forms Satisfying Certain Conditions
3
where Φ is called the fundamental two-form of the manifold. If ξ is a killing
vector field the manifold is called a k-contact manifold. It is well known that a
contact metric manifold is k-contact if and only if ∇X ξ = – φX, for any vector
field X on (M, g).
An almost contact metric manifold is Sasakian if and only if
(∇X φ) (Y ) = g (X, Y ) ξ – η (Y ) X,
for any vector fields X, Y.
In 1967, D. E. Blair introduced the notion of quasi-Sasakian manifold to
unify Sasakian and cosymplectic manifolds [4]. Again in 1987, Z. Olszak
introduced and characterized three-dimensional quasi-Sasakian manifolds [8].
An almost contact metric manifold of dimension three is quasi-Sasakian if and
only if
∇X ξ = – βφ X,
for all X ∈ TM and a function β such that ξβ = 0 (2.6)
As the consequence of (3.6), we get
(∇X η) (Y ) = g (∇X ξ, Y ) = – βg (φX, Y )
(2.7)
(∇X η) (ξ) = – βg (φX, ξ) = 0
(2.8)
Clearly such a quasi-Sasakian manifold is cosymplectic if and only if β = 0.
It is known that [5] for a three-dimensional quasi-Sasakian manifold the
Riemannian curvature tensor satisfies
R(X, Y)ξ = β2{η(Y)X – η(X)Y} + dβ(Y)φX – dβ(X )φY
(2.9)
for a (2n + 1) –dimensional generalized Sasakian-space-form, we have
R(X, Y)Z = f1{g(Y, Z)X – g(X, Z)Y }
f2{g(X, φZ)φY – g(Y, φZ)φX + 2g(X, φY)φZ }
f3{η(X)η(Z)Y – η(Y)η(Z)X + g(X, Z)η(Y)ξ – g(Y, Z)Y } (2.10)
R(X, Y)ξ = (f1 – f2) {η(Y)X – η(X)Y}
(2.11)
R(ξ, Y)Y = (f1 – f3) {g(X,Y)ξ – η(Y)X}
(2.12)
g(R(ξ, X)Y, ξ) = (f1 – f3)g(φX, φY)
(2.13)
R(ξ, X)ξ = (f1 – f3)φ2X
(2.14)
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D. Narain, S. Yadav and P. K. Dwivedi
S(X, Y) = (2nf1 + 3f2 – f3)g(X, Y) – (3f2 + (2n – 1) f3)η(X)η(Y) (2.15)
Qξ = 2n(f1 – f3)ξ
(2.16)
S(φX, φY) = S(X, Y) + 2n(f3 – f1)η(X)η(Y)
r = 2n(2n – 1) f1 + 6nf2 – 4nf3
(2.17)
(2.18)
Here S is the Ricci tensor and r is the scalar curvature tensor of the spaceforms. A generalized Sasakian space-form of dimension greater than three is said
to be conformally flat if and only if Weyl-conformal curvature tensor vanishes. It
is known that [10] a (2n + 1)-dimensional (n > 1) generalized Sasakian-spaceform is conformally flat if and only if f2 = 0.
3. GENERALIZED RECURRENT SASAKIAN-SPACE-FORMS
Definition: A Riemannian manifold (M, g) is called generalized recurrent
Riemannian manifold [11] if its curvature tensor R satisfies the following conditions
(∇X R) (Y, Z)W = ψ(X)R(X, Z)W + β(X) [g(Z, W)X – g(Y, W)Z]
(3.1)
where ψ and β are two 1-form, β is non-zero and defined by
ψ(X) = g(X, A),
β(X) = g(X, B)
(3.2)
where A, B are the vector field associated with 1-form and ∇ is the Riemannian
connection of g.
Theorem 3.1: If (M, g) is a generalized recurrent Sasakian-space-form. Then
β – ( f2 – f1) ψ is everywhere zero.
Proof: Assume that (M, g) be a generalized recurrent Sasakian-space-forms.
Then the curvature tensor R of (M, g) satisfies (3.1) for any X, Y, Z, W ∈ χ(M).
Taking Y = W = ξ in (3.1), we have
(∇X R) (ξ, Z)ξ = ψ(X)R(ξ, Z)ξ + β(X) [η(Z)ξ – Z].
(3.3)
On the other hand, it is well-know that
(∇X R) (ξ, Z)ξ = ∇X R(ξ, Z)ξ – R(∇X ξ, Z)ξ – R(ξ, ∇X Z)ξ – R(ξ, Z)∇X ξ.
Using (2.6) (2.11) (2.14) in above, we get (∇X R) (ξ, Z)ξ = 0. Then from (3.3),
we get
β(X) + (f1 – f2)ψ(X) [Z – η(Z)ξ] = 0.
(3.4)
On Generalized Sasakian-Space-Forms Satisfying Certain Conditions
5
Which implies that either η(Z)ξ = Z or β(X) = (f2 – f1)ψ(X).
Since η(Z)ξ = Zdoes not hold for generalized Sasakian-space-forms. Therefore,
we get β(X) = (f2 – f1)ψ(X).
Hence we get the result
Definition: A Riemannian manifold (M, g) is called generalized Ricci recurrent
[11] if its Ricci tensor S satisfies the condition
(∇ X S ) (Y , Z ) = ψ ( X ) S (Y , Z ) + (n − 1) β ( X ) g (Y , Z )
(3.5)
where ψ and β are defined as in (3.2).
Theorem 3.2: In a generalized Ricci-recurrent Sasakian-space-form the 1-form
ψ(X) and β(X) cannot be closed simultaneously.
Proof: Assume that (M, g) be a generalized Ricci-recurrent Sasakian-spaceforms. Then the Ricci tensor S of (M, g) satisfies the condition (3.5) for any
X, T, Z ∈ χ(M). Taking Y = Z = ξ in (3.5), we obtain
(∇ X S ) (ξ, ξ) = ψ ( X ) S (ξ, ξ) + ( n − 1) β ( X ) g (ξ, ξ) .
(3.6)
On the other hand, it is well-known that
(∇X S ) (ξ, ξ) = ∇X S (ξ, ξ) − S (∇X ξ, ξ) − S (ξ, ∇X ξ) .
Using (2.6)(2.17),we get
(∇X S ) (ξ, ξ) = df ( X ),
where
f = 2n ( f1 − f3 ) .
Using this result in (3.6), we have
df ( X ) = f ψ ( X ) + ( n − 1) β ( X ) .
(3.7)
Also
d 2 f ( X , W ) = 2n ( f1 − f 3 ) [ df (W ) ψ ( X ) + (∇W ψ ) ( X ) f ]
+ ( n − 1) [ df (W ) β ( X ) + (∇W β) ( X ) f ].
(3.8)
Interchanging X, W in (3.8), we get
d 2 f (W , X ) = 2n ( f1 − f 3 ) [ df ( X ) ψ (W ) + (∇X ψ ) (W ) f ]
+ ( n − 1) [ df ( X ) β (W ) + (∇X β) (WRD ) f ].
(3.9)
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D. Narain, S. Yadav and P. K. Dwivedi
from (3.8) and (3.9),we get
2n ( f1 − f 3 ) [(∇W ψ ) ( X ) f − (∇X ψ ) (W ) f ]
+ ( n − 1) [(∇W β) ( X ) f − (∇X β) (W ) f ] = 0.
(3.10)
We get the result as we required.
Corollary 3.1: If a (2n + 1)-dimensional generalized Sasakian-space-form
has generalized Ricci recurrent. Then 2n(f1 – f3) can never be a non-zero constant
Theorem 3.3: If a (2n + 1)-dimensional generalized Sasakian-space-form
has generalized Ricci recurrent then 2nfψ + (n – 1) β = 0 provided f is constant.
We know that if a (2n + 1)-dimensional generalized Sasakian-space-forms
admits contact structure then f1 – f3 is constant [13]. Hence f3 is constant if and
only if f1 is constant
Theorm 3.4: [13] A (2n + 1)-dimensional conformally flat generalized
Sasakian-space-form is locally φ-symmetric if and only if f1 is constant.
From theorem 3.3 and theorem 3.4, we have the result
Theorem 3.5: A (2n + 1)-dimensional conformally flat contact metric
generalized Sasakian space-form has generalized Ricci recurrent if and only if it
is locally φ-symmetric.
Corollary 3.2: In a generalized Ricci recurrent Sasakian space form the
non-zero 1-form is closed if f1 – f3.
Theorem 3.6: If generalized Sasakian space form (M, f1, f2, f3) is generalized
recurrent then the scalar curvature r of the Sasskian space form satisfies the
condition rη (A) = 4n ( f – f ) η (A) – n (n – 1) η (B).
Proof: Suppose that generalized Sasakian-space-form is a generalized
recurrent Then by using the second Bainchi’s identity, we get
ψ (X ) R (Y, Z ) W + β (X ) [g (Z, W ) Y – g (Y, W ) Z ]
+ ψ (Y ) R (Z, X ) W + β (Y ) [g (X, W ) Z – g (Z, W ) X ]
+ ψ (Z ) R (X, Y ) W + β (X ) [g (Y, W ) X – g (X, W ) Y ] = 0
(3.11)
On Generalized Sasakian-Space-Forms Satisfying Certain Conditions
7
Contracting (3.11), with respect to Y, we have
ψ (Y ) S (Z, W ) + nβ (X ) g (Z, W ) + R (Z, X, W, A) ξ (Z ) g (X, W )
– β (X ) g (Z, W ) – ψ (Z ) S (Z, W ) + nβ (Z ) g (X, W ) = 0. (3.12)
Again contracting (3.12), over Z and W,we get
1
S ( X , A) = {r ψ ( X ) + (n − 1) β ( X )} .
2
(3.13)
Putting X = ξ in (3.13) and using (3.2), we get
rη (A) = 4n( f – f ) η (A) – n (n – 1) η (B).
Hence we get the result.
Definition: A Riemannian manifold (M, g) is called generalized concircular
~
recurrent if its concircular curvature tensor C satisfies the condition
(∇ X C )(Y , Z )W = ψ ( X )C (Y , Z )W + β( X )[ g ( Z , W )Y − g (Y , W ) Z ] (3.14)
~
where the concircular curvature tensor C defined as
C ( X , Y ) Z = R ( X , Y ) Z −
r
[ g (Y , Z ) X − g ( X , Z ) Y ]
n (n − 1)
(3.15)
where ψ and β are defined in (3.2).and r is the scalar curvature tensor of the
manifold (M, g).
Theorem 3.7: Let (M, g) be a generalized concircular recurrent generalized
Sasakian-space-form then the following relation hold


r 
X [r ]
+ β ( X ) = 0 .
 ( f1 − f3 ) − n(n − 1)  ψ ( x) +
n (n − 1)



for every vector field X on M, where is derivative of r with respect to X.
Proof: Suppose (M, g) be a generalized concircular recurrent generalized
~
Sasakian-space-forms. Then concircular curvature tensor C of M satisfies the
condition (3.15), for all X, Y, Z, W ∈ χ(M). Taking Y = W = ξ in (3.14), we get
(∇ X C ) (ξ, Z ) ξ = ψ ( X ) C (ξ, Z ) ξ + β ( X ) [η ( Z ) ξ − Z ) Z ] .
(3.16)
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D. Narain, S. Yadav and P. K. Dwivedi
On the other hand, it is well-known that
(∇X C )(ξ, Z )ξ=∇X C (ξ, Z )ξ− C (∇X ξ, Z )ξ− C (ξ, ∇X Z )ξ− C (ξ, Z )∇X ξ . (3.17)
Using (2.9)(3.16) in (3.17),we get
X [r ]
(∇X C ) (ξ, Z ) ξ =
( Z − ( Z ) ξ) .
n (n − 1)
(3.18)
By virtue of equation (3.16)and(3.18),we get


r 
X [r ]
+ β ( X )  {η ( Z ) ξ − Z } = 0 .
 ( f1 − f3 ) − n(n − 1)  ψ ( x) +
n (n − 1)



Since the equality η (Z) ξ = Z does not hold in generalized Sasakian-Space
form, therefore, we get


r 
X [r ]
+ β ( X ) = 0 .
 ( f1 − f3 ) − n(n − 1)  ψ ( x) +
n (n − 1)



(3.19)
We get the result as we required.
4. RICCI SEMI-SYMMETRIC GENERALIZED
SASAKIAN-SPACE-FORMS
Theorem 4.1: A generalized Sasakian-space-forms is Ricci-symmetric then it is
an η-Einstein manifold.
Proof: We suppose that a-dimensional generalized Sasakian-space-form which
satisfies the following condition
(R (X, Y )  S) (Z, U ) = 0.
(4.1)
Since
(R(X, Y)  S) (Z, U ) = (R(X, Y)S(Z, U) – S(R(X, Y) (Z, U) – S(Z, R(X, Y)U). (4.2)
From (4.1) and (4.2), we get
R(X, Y)S(Z, U ) = S(R(X, Y) (Z, U ) – S(Z, R(X, Y)U),
for any X, Y, Z, U ∈ χ(M).
(4.3)
9
On Generalized Sasakian-Space-Forms Satisfying Certain Conditions
Taking X = Z = ξ in (4.3) and using (2.12, 13, 14, 15), we get
S (Y , U ) = − ( f1 − f 3 ) [(2nf1 + 3 f 2 − f 3 ) − 2n ( f1 − f 3 )] g (Y , U )
+ ( f1 − f 3 ) [(2nf1 + 3 f 2 − f 3 ) + 2n ( f1 − f 3 )] η (Y ) η (U ). (4.4)
We get the result as we required.
Corollary 4.1: The necessary condition for generalized Sasakian-space-form
is Ricci semi symmetric is that f1 = f2.
Corollary 4.2: The necessary condition for Ricci semi symmetric generalized
sasakian space form is η-Einstein manifold is that f1 ≠ f2.
5. D-CONFORMAL CURVATURE TENSOR ON GENERALIZED
SASAKIAN-SPACE-FORMS
In Riemannian manifold the D-conformal curvature tensor is defined as
 S ( X , Z )Y − S (Y , Z ) X + g ( X , Z )QY 
1 
B( X , Y )Z = R( X , Y )Z +
− S ( X , Z )η(Y )ξ + S (Y , Z )η( X )ξ 

(n − 1)
 + η( X )η( Z )QY + η(Y )η( Z )QX 
(k − 2)
−
[ g ( X , Z )Y − g (Y , Z ) Z ]
(n − 3)
+
where k =
k  g ( X , Z )η(Y )ξ − g (Y , Z )η( X )ξ + η( X )η( Z )Y  . (5.1)
− η(Y )η( Z ) X 
(n − 3) 
r + 2( n − 1)
( n − 2)
and r is the scalar curvature of the manifold M.
Definition: The rotation (curl) of the D-conformal curvature tensor B on
Riemannian manifold is given by
Rot B =(∇V B)( X ,Y )Z + (∇X B)(V ,Y )Z + (∇Y B)(V ,Y )Z −(∇Z B)( X ,Y ,V ) . (5.2)
By virtue of the Banchi’s identity, we have
(∇V B )( X , Y ) Z + (∇X B )(V , Y ) Z + (∇Y B )(V , Y ) Z = 0 .
(5.3)
From (5.2) and (5.3), we obtain
Curl B = − (∇ Z B )( X , Y )V .
(5.4)
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D. Narain, S. Yadav and P. K. Dwivedi
If the conformal curvature tensor is irrotational then curl B = 0. Then from
(5.4), we have
(∇ Z B )( X , Y )V = 0
which implies that
∇Z {B ( X , Y )V } = B (∇Z X , Y )V + B ( X , ∇Z Y )V + B ( X , Y )∇Z V
Taking V = ξ, we get
∇Z {B ( X , Y )ξ} = B (∇Z X , Y )ξ + B ( X , ∇Z Y )ξ + B ( X , Y )∇Z ξ .
(5.5)
Since for V = ξ from (5.1), we get
 S ( X , ξ)Y − S (Y , ξ) X + g ( X , ξ)QX 
1 
B ( X , Y )ξ = R ( X , Y )ξ +
− S ( X , ξ)η(Y )ξ + S (Y , ξ)η( X )ξ 

(n − 1)

+ η( X )QY + η(Y )QX 
(k − 2)
−
[η( X )Y − ξ(Y ) X ]
(n − 3)
+
k
[ξ( X )η(Y )ξ − ξ(Y )η( X )ξ + η( X )Y − η(Y ) X ]. (5.6)
(n − 3)
Using (2.11) (2.15) in (5.6), we obtain
(k − 2)
 k

−
( f1 − f3 )  {η( X )Y − η(Y ) X }
B ( X , Y )ξ = 
 n − 3 (k − 3)

1
.
+
[η( X )QY − η(Y )QX ]
n−3
(5.7)
Theorem 5.1: The D-conformal curvature tensor B on generalized Sasakianspace-forms satisfies the relation (5.7).
Theorem 5.2: If D-conformal curvature tensor B on generalized Sasakianspace-forms is irrotational then the manifold is an η-Einstein manifold.
On Generalized Sasakian-Space-Forms Satisfying Certain Conditions
11
Proof: Also from (5.1) and (5.7), we have
R ( X , Y ) Z = k / {η( X )Y − η(Y ) X } +
1
[ g ( X , Z )QX − g (Y , Z )QX ]
n−3
 S ( X , Z )Y − S (Y , Z ) X + g ( X , Z )QY 

− g (Y , Z )QX − S ( X , Z )η(Y )ξ 
1 
+
+ S (Y , Z )η( X )ξ + η( X )η( z )QY 
(n − 1) 


+ η(Y )η( Z )QX 

(k − 2)
+
[ g ( X , Z )Y − g (Y , Z ) X ]
(n − 3)
k  g ( X , Z )η(Y )ξ − g (Y , Z )η( X )ξ 
(5.8)
.
(n − 3)  + η( X )η( Z )Y − η(Y )η( Z ) X 
Let {e1, e2, ..., e2n + 1} be the orthonormal basis of the tangent space at any
point. First we apply the inner product with any vector field W both side. Then
summing over i, 1 ≤ i ≤ 2n + 1 put X = W = ei equation (8.8), yields
+
2 ( nf1 − nf3 + 1) 

S (Y , Z ) =  2n ( f1 − f3 ) − k / +
 g (Y , Z )
(n − 3)


2 ( nf1 + nf3 + 1) 

+ k / −
 η (Y ) η ( Z )
(n − 3)


where k / =
k
n−3
−
( k − 2)
( k − 3)
− ( f1 − f3 ) . Which is an η-Einstein manifold.
We get the result as we required.
Claim: A generalized sasakian space-forms with semi symmetric and
φ-Ricci-symmetric is an Einstein manifold.
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D. Narain
Department of Mathematics &Statistics,
D.D.U. Gorakhpur University, Gorakhpur.
E-mail: [email protected]
S. Yadav
Department of Mathematics,
Alwar Institute of Engineering & Technology, Alwar.
E-mail:[email protected]
P. K. Dwivedi
Department of Mathematics,
Institute of Engineering & Technology, Alwar.
E-mail: [email protected]