The object of the present paper is to study ξ-conformally at and ϕ-conformally
at generalized Sasakian-space-forms.peerReviewe

Pradip Majhi, U. C. De

Extracta Mathematicae

English

Institutional Repository University of Extremadura

E extracta mathematicae Vol. 27, Num. 2, 301 { 308 (2012)
The Structure of a Class of Generalized
Sasakian-Space-Forms
Pradip Majhi, U.C. De
Department of Pure Mathematics, University of Calcutta,
35 Ballygaunge Circular Road, Kolkata 700019, West Bengal, India
mpradipmajhi@gmail.com, uc de@yahoo.com
Presented by Manuel de Leon Received February 14, 2012
Abstract : The object of the present paper is to study -conformally at and -conformally
at generalized Sasakian-space-forms.
Key words: -conformally at, -conformally at, Sasakian manifold, Einstein manifold,
-Einstein manifold, conformally at.
AMS Subject Class. (2010): 53C15, 53C25.
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 curvatures of a
manifold determine curvature tensor completely. A Riemannian manifold with
constant sectional curvature c is known as real space-form and its curvature
tensor is given by
R(X;Y )Z = c

g(Y; Z)X   g(X;Z)Y 	:
A Sasakian manifold with constant -sectional curvature is a Sasakian-
space-form and it has a specic form of its curvature tensor. Similar notion
also holds for Kenmotsu and cosymplectic space-forms. In order to generalize
such space-forms in a common frame P. Alegre, D. E. Blair and A. Carri-
azo introduced the notion of generalized Sasakian-space-forms in 2004 [1].
In this connection it should be mentioned that in 1989 Z. Olszak [13] stud-
ied generalized complex-space-forms and proved its existence. A generalized
Sasakian-space-forms are dened as follows:
Given an almost contact metric manifold M(; ; ; g), we say that M is a
generalized Sasakian-space-form if there exist three fuctions f1, f2, f3 on M
such that the curvature tensor R is given by
301
302 p. majhi, u.c. de
R(X;Y )Z = f1

g(Y; Z)X   g(X;Z)Y 	
+ f2

g(X;Z)Y   g(Y; Z)X + 2g(X;Y )Z	
+ f3

(X)(Z)Y   (Y )(Z)X
+ g(X;Z)(Y )   g(Y; Z)(X)	:
(1.1)
for any vector elds X, Y , Z on M . In such a case we denote the mani-
fold as M(f1; f2; f3). In [1] the authors cited several examples of generalized
Sasakian-space-forms. If f1 =
c+3
4 , f2 =
c 1
4 and f3 =
c 1
4 , then a generalized
Sasakian-space-form with Sasakian structure becomes Sasakian-space-form.
In [12], U.K. Kim studied conformally at generalized Sasakian-space-forms
and locally symmetric generalized Sasakian-space-forms. He proved that some
geometric properties of generalized Sasakian-space-form depend on the nature
of the fuctions f1, f2 and f3. Generalized Sasakian-space-forms have also been
studied in the papers [2], [3], [4], [5], [9], [10], [12] and [13].
Let M be an almost contact metric manifold equipped with an almost
contact metric structure (; ; ; g). At each point p 2 M , decompose the
tangent space TpM into direct sum TpM = (TpM) fpg, where fpg is the
1-dimensional linear subspace of TpM generated by p. Thus the conformal
curvature tensor C is a map
C : TpM  TpM  TpM  ! (TpM) fpg; p 2M:
It may be natural to consider the following particular cases:
(1) C : Tp(M)  Tp(M)  Tp(M)  ! fpg; i.e, the projection of the image
of C in (Tp(M)) is zero.
(2) C : Tp(M)  Tp(M)  Tp(M)  ! (Tp(M)); i.e, the projection of the
image of C in fpg is zero. This condition is equivalent to
C(X;Y ) = 0: (1.2)
(3) C : (Tp(M))(Tp(M))(Tp(M))  ! fpg; i.e, when C is restricted
to (Tp(M)) (Tp(M)) (Tp(M)); the projection of the image of C
in (Tp(M)) is zero. This condition is equivalent to
2C(X; Y )Z = 0: (1.3)
a class of generalized sasakian-space-forms 303
A generalized Sasakian-space-form satisfying (1:2) and (1:3) is called -
conformally at and -conformally at respectively. A K-contact manifold
satisfying the cases (1), (2) and (3) is considered in [14], [15] and [16] respec-
tively.
Definition 1.1. A (2n+1)-dimensional generalized Sasakian-space-form
is said to be -conformally at if
C(X;Y ) = 0; (1.4)
where X;Y 2 T (M).
Definition 1.2. A (2n+1)-dimensional generalized Sasakian-space-form
is said to be -conformally at if
g
 
C(X; Y )Z; W

= 0: (1.5)
In [15], it is proved that a K-contact manifold is -conformally at if
and only if it is an -Einstein Sasakian manifold. In [11], De and Biswas
-conformally at N(k)-contact metric manifolds. A compact -conformally
at K-contact manifold with regular contact vector eld has been studied in
[16]. Moreover, in [6], Arslan, Murathan and Ozgur studied -conformally at
(k; )-contact metric manifold. Motivated by the above studies, in this paper
we study -conformally at and -conformally at generalized Sasakian-space-
forms.
The present paper is organized as follows:
After preliminaries in Section 3, we study -conformally at generalized
Sasakian-space-forms and prove that a generalized Sasakian-space-form is al-
ways -conformally at. Section 4 deals with -conformally at generalized
Sasakian-space-forms and we prove that the space-form is -conformally at
if and only if f2 = 0: As a consequence of the result we obtain some important
corollaries.
2. preliminaries
In an almost contact metric manifold we have ([7], [8])
2(X) =  X + (X);  = 0; (2.1)
() = 1; g(X; ) = (X); (X) = 0; (2.2)
304 p. majhi, u.c. de
g(X; Y ) = g(X;Y )  (X)(Y ); (2.3)
g(X; Y ) =  g(X;Y ); g(X;X) = 0; (2.4)
g(X; ) = 0: (2.5)
Again for a (2n+1)-dimensional generalized Sasakian-space-form we have ([1])
S(X;Y ) = (2nf1 + 3f2   f3)g(X;Y )  (3f2 + (2n  1)f3)(X)(Y ); (2.6)
QX = (2nf1 + 3f2   f3)X   (3f2 + (2n  1)f3)(X); (2.7)
R(X;Y ) = (f1   f3)[(Y )X   (X)Y ]; (2.8)
R(;X)Y = (f1   f3)[g(X;Y )   (Y )X)]; (2.9)
S(X; ) = 2n(f1   f3)(X); (2.10)
S(; ) = 2n(f1   f3); (2.11)
Q = 2n(f1   f3); (2.12)
r = 2n(2n+ 1)f1 + 6nf2   4nf3; (2.13)
where R, S and r are the curvature tensor, Ricci tensor and scalar curvature of
the space-form respectively and Q is the Ricci operator dened by g(QX;Y ) =
S(X;Y ). We know (see [1]) that the -sectional curvature of a generalized
Sasakian-space-form M(f1; f2; f3) is f1 + 3f2.
In a Riemannian manifold of dimension (2n+ 1) the Weyl conformal cur-
vature tensor is given by
C(X;Y )Z = R(X;Y )Z   1
(2n  1)

S(Y; Z)X   S(X;Z)Y
+ g(Y; Z)QX   g(X;Z)QY 
+
r
2n(2n  1)

g(Y; Z)X   g(X;Z)Y ;
(2.14)
for any X, Y and Z 2 T (M). A generalized Sasakian-space-form of dimension
greater than three is said to be conformally at if its Weyl conformal curvature
tensor vanishes. It is known (see [12]) that a (2n + 1)-dimensional (n > 1)
generalized Sasakian-space-form is conformally at if and only if f2 = 0.
a class of generalized sasakian-space-forms 305
3. -conformally flat generalized Sasakian-space-forms
In this section we study -conformally at generalized Sasakian-space-
form. LetM(f1; f2; f3) be a generalized Sasakian-space-forms. Putting Z = 
in (2:15), we obtain
C(X;Y ) = R(X;Y )   1
(2n  1)

S(Y; )X   S(X; )Y
+ (Y )QX   (X)QY 
+
r
2n(2n+ 1)

(Y )X   (X)Y :
(3.1)
Using (2:7), (2:8) and (2:10) in (3:1), yields
C(X;Y ) =

(f1   f3) + r
2n(2n  1)
  4nf1 + 3f2   (2n+ 1)f3
(2n+ 1)

(Y )X   (X)Y : (3.2)
Putting the value of r from (2:13) in (3:2), we get
C(X;Y ) = 0: (3.3)
Therefore we conclude that the generalized Sasakian-space-form is -conformally
at. Thus we can state the following:
Theorem 3.1. A (2n + 1)-dimensional generalized Sasakian-space-form
M(f1; f2; f3) is always -conformally at.
4. -conformally flat generalized Sasakian-space-forms
In this section we consider -conformally at generalized Sasakian-space-
forms. Let the generalized Sasakian-space-formM(f1; f2; f3) be -conformally
at, i.e.,
g
 
C(X; Y )Z; W

= 0: (4.1)
306 p. majhi, u.c. de
Now,
g
 
C(X; Y )Z; W ) = g
 
R(X; Y )Z; W

  1
2n  1

S(Y; Z)g(X; W )
  S(X; Z)g(Y; W )
+ g(Y; Z)S(X; W )
  S(Y; W )g(X; Z)
+
r
2n(2n  1)

g(Y; Z)g(X; W )
  g(Y; W )g(X; Z):
(4.2)
Using (1:1), (2:6) and (2:13) in (4:2), we obtain
g
 
C(X; Y )Z; W

= f2
h
2g(X; Y )g(Z; W )
+ g(X;Z)g(Y; W )
  g(Y;Z)g(X;W )	
  3
(2n  1)

g(Y; Z)g(X; W )
  g(X; Z)g(Y; W )	i:
(4.3)
Therefore from (4:1) and (4:3) we obtain either f2 = 0, or
2g(X; Y )g(Z; W ) + g(X;Z)g(Y; W )  g(Y;Z)g(X;W )	
  3
(2n  1)

g(Y; Z)g(X; W )  g(X; Z)g(Y; W )	 = 0: (4.4)
Putting W = W in (4:5) and using (2:1), we obtain  2g(X; Y )g(Z;W )  g(X;Z)g(Y;W ) + g(Y;Z)g(X;W )	
  3
(2n  1)
  g(Y; Z)g(X;W ) + g(X; Z)g(Y;W )	 = 0: (4.5)
Putting Z =  in (4:5), yields
g(X; Y )(W ) = 0; (4.6)
which implies either  = 0 or g(X; Y ) = 0. Both the cases do not occur.
Therefore -conformally at generalized Sasakian-space-form implies f2 = 0:
a class of generalized sasakian-space-forms 307
Conversely, if f2 = 0, then from (4:3) we obtain g(C(X; Y )Z); W ) =
0 for all X, Y , Z and W 2 T (M). Thus f2 = 0 implies the generalized
Sasakian-space-form is -conformally at.
Therefore we can state the following:
Theorem 4.1. A (2n + 1)-dimensional generalized Sasakian-space-form
M(f1; f2; f3) is -conformally at if and only if f2 = 0.
In [12] U.K. Kim proved that for a (2n+1)-dimensional generalized Sasakian-
space-form the following holds:
(i) If n > 1, then M is conformally at if and only if f2 = 0.
(ii) If M is conformally at and  is Killing vector eld, then M is locally
symmetric and has constant -sectional curvature.
Since conformally atness implies -conformally atness, hence in the view
of the rst part of the above theorem we have the following:
Corollary 4.1. A (2n + 1)-dimensional (n > 1) generalized Sasakian-
space-form M(f1; f2; f3) is -conformally at if and only if it is conformally
at.
Again, in the view of the second part of the above theorem we have the
following:
Corollary 4.2. For a (2n + 1)-dimensional (n > 1) -conformally at
generalized Sasakian-space-form M(f1; f2; f3) with  as a Killing vector eld,
the space-form is locally symmetric and has constant -sectional curvature.
Acknowledgements
The authors are thankful to the referee for his valuable suggestions
in the improvement of the paper.
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The structure of a class of generalized Sasakian-space-forms

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