A Note On Conformally Recurrent Kahlerian Manifolds

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.

1. 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

2. 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(n-1)} R + {1/4(n-1)(2n-1)} 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/(n-1)}H ] + {1/2(n-1)(2n-1)}F R

kj jk jk l

= [H + {1/(n-1)} H + {1/2(n-1)(2n-1)}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 (2n-1)(n-2)] = 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 co-variant derivative of equation (2.4) with respect to x , we get

jk jk

m L = m [-{1/2(n-1)} R

jk

+ {1/4(n-1)(2n-1)} R g ]

(2.14)

jk jk

m L = -{1/2(n-1)} m R

jk

+ {1/4(n-1)(2n-1)} g m R

(2.15)

Inserting equation (2.8) into equation (2.14), we obtain jk jk jk

m L = -{1/2(n-1)} m [R – {1/2(2n-1)} R g ]

DR. NARESH KUMAR Dr. MUKESH CHANDRA

Department of Mathematics, Department of Mathematics,

IFTM University, Moradabad (U.P.) IFTM University, Moradabad (U.P.)

INDIA-244001 INDIA-244001

REFERENCES

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