 Open Access
 Total Downloads : 449
 Authors : Naresh Kumar , Mukesh Chandra
 Paper ID : IJERTV1IS4246
 Volume & Issue : Volume 01, Issue 04 (June 2012)
 Published (First Online): 03072012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Note On Conformally Recurrent Kahlerian Manifolds
NARESH KUMAR & MUKESH CHANDRA
ABSTRACT
Present paper delineates to the study of conformally recurrent kahlerian manifolds. In this paper, few interesting results have been obtained. In the last, conformally recurrent kahlerian manifold is flat if its scalar curvature is zero.
Key words: Ricci Tensor, Riemannian Curvature Tensor, Scalar Curvature Tensor, Recurrent Vector, Conformal Curvature Tensor.

INTRODUCTION :
Let g ji
h is a positive definite metric and F i
be the structure tensor of a real 2n
– dimentional Kahlerian space. Then we have the following relations :
i
k F j
= 0,
g = 0
k ji
(1.1) Fr Fi = – i ,
j r j
r
r g t F j
t = g
F
i ji
F
F = g r
ji ri j
F = – F
ij ji
ji jr i F = g F r
ij ji
F = – F
Let R ji
h be the Ricci tensor and R
kji
be the Riemann curvature tensor. Then, we have the
following relations:
g
R = R ji ji
R r
r
kjih = R kji g h
Hij = (1/2) Rijkl Fkl
Then the following relations hold [4]:
(1.2) H = – H ,
ij ji
(1.3) R s
kj
ks F j = H ,
F = – R ,
(1.4) H s
ks j kj
(1.5)
H kj = – R ,
F
kj
(1.6) H + H + H = 0 .
l kj k jl j lk

CONFORMALLY RECURRENT KAHLERIAN MANIFOLDS :
Definition 2.1 :
A 2n – dimensional (n 1,2) Kaehler space which satisfies the relation
C
(2.1) Ch = h
l kji l kji
Wherein is the
h
l is a non – zero vector is called recurrence vector and C kji
conformal curvature tensor and l denotes covariant differentiation with regard to the Riemannian metric of the space. Such a space is called a conformally recurrent Kaehler space [2].
We have the following relation [2]
(2.2) C = C
l kjih l kjih
wherein
ijkh i jk rh C = C r g
(2.3)
ijkh ijkh ih jk jk ih C = R + g L + g L
jh ik ik jh
– g L – g L
(2.4) L = {1/2(n1)} R + {1/4(n1)(2n1)} R g
ji ji ji
Equation (2.2) in covariant form can be written as
(2.5) lRkjih + gkhl Lji – gjhl Lki + gjil Lkh gkil Ljh
= l [Rkjih + gkh Lji – gjh Lki + gji Lkh – gki Ljh] Transvecting equation (2.5) with Fih yields
l
(2.6) [H + {1/(n1)}H ] + {1/2(n1)(2n1)}F R
kj jk jk l
= [H + {1/(n1)} H + {1/2(n1)(2n1)}R F ]
l kj jk jk
kj
Next, transvecting equation (2.6) with F and using the equation (1.5), we get
(2.7) (
l
wherein
R –
l
R) [F
kj
kj + 2 (2n1)(n2)] = 0
l
F
(2.8) R = R
l
Inserting equation (2.8) into equation (2.6), we obtain
(2.9) H = H
l kj l kj
From equation (1.3), we get
l
Hence
s
F j l
R = H
ks l kj
= H kj
j s j
F m F
R = H
j l
ks l kj
F m ,
From this it follows that
(2.10) R = R
l km l km
By virtue of equation (2.4), we obtain
l i i
(2.11) L = L
j l j
From equations (2.11) and (2.5), we obtain
(2.12)
k k
l R jih = l R jih In this regard, we have
R = R
(2.13) R kjih 2 kjih
Remark 2.1 :
It is noteworthy that if we take R = 0, then we get R = 0, i.e. the
kjih
space is flat.
Theorem 2.1 :
In a Kahler space, the scalar curvature is zero and different from zero if a conformally recurrent is flat and a simple recurrent one.
i.e.
m
Taking covariant derivative of equation (2.4) with respect to x , we get
jk jk
m L = m [{1/2(n1)} R
jk
+ {1/4(n1)(2n1)} R g ]
(2.14)
jk jk
m L = {1/2(n1)} m R
jk
+ {1/4(n1)(2n1)} g m R
(2.15)
Inserting equation (2.8) into equation (2.14), we obtain jk jk jk
m L = {1/2(n1)} m [R – {1/2(2n1)} R g ]
DR. NARESH KUMAR Dr. MUKESH CHANDRA
Department of Mathematics, Department of Mathematics,
IFTM University, Moradabad (U.P.) IFTM University, Moradabad (U.P.)
INDIA244001 INDIA244001
REFERENCES
[1]. 
A.G. Walker : 
On Ruses spaces of recurrent curvature, Proc. London Math. Soc., (2) 52, (3664), (1950). 
[2]. 
G. Chuman : 
On the D conformal curvature tensor, Tensor, N.S., 40, (125134), (1983). 
[3]. 
K. Yano : 
Differential geometry on complex and almost complex spaces, Pergamon Press, (7072), (1965). 
[4]. 
T. Adati : 
On a Riemannian space with recurrent and conformal curvature, Tensor, N.S., 18, T. Miyazawa (348354), (1967). 
[5]. 
W. Roter : 
Quelques remarques sur les espaces recurrents et Ricci recurrents. Bull.Acad. Polon. Sci., Ser. Sci. Math Astr. et Phys., 10, (533536), (1962) 
[ 6 ] 
T. S. Chauhan & : I. S. Chauhan 
On Einstein Kaehierian space with Recurrent Bochner curvature tensor, Acta Ciencia Indica , Vol. XXXIV M, No. 1, Page 2326, 2008 
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