 Open Access
 Total Downloads : 424
 Authors : Ram Swaroop, Abhishek Singh, Prakash Kumar
 Paper ID : IJERTV1IS9385
 Volume & Issue : Volume 01, Issue 09 (November 2012)
 Published (First Online): 29112012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Linear Connections on Manifold Admitting F (2k + P, P)Structure
Ram Swaroop, Abhishek Singh, Prakash Kumar
Abstract. D. Demetropoulou [2] and others have studied linear connections in the manifold admitting f(2v+3,1)structure. The aim of the present paper is to study some properties of linear connections in a manifold admitting F(2K +P, P)structure. Certain interesting results have been obtained.
Key words. Linear connection, projection, geodesic, parallelism.

Preliminaries
Let F be a nonzero tensor field of the type (1, 1) and of class C1 on an ndimensional manifold Mn such that [5, 8]
F2K+P + FP = 0, (1.1)
where K and P is a fixed positive integer greater than or equal to 1. The rank of (F) = r =constant.
Let us define the operators on M as follows [5, 8]
l = F2K, m = I + F2K (1.2)
where I denotes the identity operator.
We will state the following two theorems [5]
Theorem 1.1. Let Mn be an Fstructure manifold satisfying (1.1), then

l + m = I,
b. l2 = l, (1.3)

m2 = m,

lm = ml = 0.
Thus for (1, 1) tensor field F( 0) satisfying (1.1), there exist complementary distributions Dl and Dm corresponding to the projection operators l and m respectively. Then, dim Dl = r and dim Dm = (nr).
Theorem 1.2. We have
a. lF = Fl = F, mF = Fm = 0.
b. F2Km = 0, F2Kl = l. (1.4)
L
Thus FK acts on D as an almost complex structure and on D
and
XY = l lX(mY )+m mX(lY )+l[lX,mY]+m[mX, lY] (1.6)
Then it is easy to show that and are linear connections on the manifold Mn [2]



Distributions antiparallelism and antihalf parallelism
In this section, first we have the following definitions: Definition 2.1. Let us call the distribution DL as anti parallel if for all TMn denotes the tangent bundle of the manifold Mn. Definition 2.2. The distribution DL will be called antihalf parallel if for all X DL and Y TMn, the vector field YX DM, where
( F)(X,Y ) = F XYF YX FXY+ Y FX (2.1)
F being a (1,1) tensor field on Mn satisfying the equation (1.1). In a similar manner, antihalf parallelism of the distribution DM can also be defined.
Theorem 2.3. In the F(2K+P, P)structure manifold Mn, the distribution DL and DM are antiparallel with respect to connections and .
Proof. Let X TMn and Y DL, therefore mY = 0. Hence in view of equation (1.5), we get
XY = m X(lY ) DM.
l
as a null operator.
m Hence the distribution D
is antihalf parallel with respect to
Let us define the operators and on Mn in terms of an arbitrary connections as under
XY = l X(mY ) + m X (lY ) (1.5)
the linear connection . Similarly, it can also be shown that DM is also is also antiparallel.
Again in view of the equation (1.5), taking mY = 0, we obtain
XY = m mX(lY )+m[mX, lY] DM. (2.2)
Thus the distribution DL is antiparallel with respect to the linear connection . A similar result for DM can also be proved in a similar manner.
Theorem 2.4. In the F(2K+P, P)structure manifold Mn, the distribution DL and DM are antiparallel with respect to connection if and only if and are equal.
Proof. Since the distributions DL and DM are antiparallel with respect
to the , hence
l X(lY ) = m X(mY ) = 0, (2.3)
for the vector fields X, Y TMn.
Since l + m = I, hence in view of equation (2.3), it follows that
X(lY ) = m X(lY ),
X(mY ) = l X(mY ) (2.4)
Thus in view of the equations (1.5) and (2.4), it follows that
XY = XY.
Hence, the connections and are equal. The converse can also be proved easily.
Theorem 2.5. In a F(2K+P, P)structure manifold Mn, the distribution DM is antihalf parallel with respect to connection if
m FX(lY ) = m Y (FX), (2.5)
for arbitrary X DM and Y TMn.
Proof.
Since mF = Fm = 0, hence in view of the equation (2.1), we get for the connection
m( )(X, Y ) = m Y (FX) m FX(Y ). (2.6)
By virtue of the equation (1.5), the above equation (2.6) takes the form
m( F)(X, Y ) = m{l Y (mFX) + m Y (lFX)}
m{l FX(mY ) + m FX(lY )}. (2.7)
Since, ml = lm = 0; Fl = lF = F and m is the projection operator, the above equation (2.7) takes the form,
m( F)(X, Y ) = m Y (FX) m FX(lY ). (2.8)
Since the distribution DM is antihalf parallel so far all X DM, Y TMn,
m( F)(X, Y ) DL.
Thus,
m Y (FX) = m FX(lY ).
Hence, the theorem is proved.
Theorem 2.6. In the manifold Mn equipped with F(2K+P, P) structure, the distribution DL is antihalf parallel with respect to the connection if
F X(lY ) = l FX(mY ),
for arbitrary X DL and Y TMn.
Proof. Proof follows easily in a way similar to that of the theorem 2.5.
Theorem 2.7. In the F(2K+P, P)structure manifold Mn, the distribution DM is antihalf parallel with respect to the connection if for X DM and Y TMn the equation
m mY (FX) + m[mY, FX] = 0
is satisfied.
Proof. For X DM and Y TMn, we have for the connection
( F)(X, Y ) = XYFYXFXY+YFX. (2.9)
As Fm = mF = 0, hence from the above equation (2.9), it follows that
m( F)(X, Y ) = Y FX FXY. (2.10)
In view of the equation (1.4) and (1.6), it is easy to show that
mFXY = 0 (2.11)
and
mYFX = m mY(FX) + m[mY, FX]. (2.12)
Thus, we get
m( F)(X,Y ) = m mY(FX)+m[mY, FX]. (2.13)
The distribution DM will be antihalf parallel if X DM, Y TMn, the vector field ( F)(X,Y ) DL. Thus,
m( F)(X, Y ) = 0
i.e.,
m mY (FX) + m[mY, FX] = 0.
Hence, the theorem is proved.

Geodesic in the manifold Mn
Let C be a curve in Mn, T a tangent field and arbitrary connection on Mn. Then, we have
Definition 3.1. The curve C is a geodesic with respect to the connection if TT = 0 along C.
Applying the definition for the connection and , we have the following results in the F(2K + P, P)structure manifold Mn.
Theorem 3.2. A curve C is a geodesic in the manifold Mn with respect to the connection if the vector fields
TT T (lT) DM and T (lT) DL.
Proof. The curve C will be geodesic if TT = 0.
In view of the equation (1.5), the above equation takes the form
l T (I l)T + m T (lT) = 0
or equivalently
l TT l T (lT) + m T (lT) = 0,
which implies that
T T T (lT) DM and T (lT) DL.
This proves the theorem.
Theorem 3.3. A curve C is a geodesic in the manifold Mn with respect to the connection if
lTT lT (lT) + [lT,mT] DM And mT (lT) + [mT, lT] DL.
Proof. Using definition of from the equation (1.6), proof follows easily as of theorem 3.2.
Theorem 3.4. The (1, 1) tensor field l is covariant constant with respect to the connection if
m X(lY ) = l X(mY ) (3.1)
but the tensor field m is always covariant constant.
Proof. We have
( Xl)Y = X(lY ) l XY (3.2)
In view of equation (1.5), the above equation takes the form ( Xl)Y = l X(mlY ) + m X(lY )
l{l X(mY ) + m X(lY )}. (3.3)
Since l2 = l and lm = ml = 0, the equation (3.3) takes the form ( Xl)Y = m X(lY ) l X(mY ). (3.4)
The (1, 1) tensor field l is covariant with respect to the
connection if
References

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Mishra, R.S., Structures on a differentiable manifold and their applications, Chandrama Prakashan, 50 A, Balrampur House, Allahabad, India (1984).

Yano, K., On structure defined by a tensor field f of type (1,1) satisfying f3 + f = 0. Tensor, N. S., 14, 99109 (1963).

Singh, A., On CRstructures and Fstructure satisfying F2K+P + FP = 0, Int. J. Contemp. Math. Sciences, Vol. 4, no. 21, 2009.

Yano, K. and Kon, M., Structures on manifold, World Scintific Press 1984.

Goldberg, S.I., On the existence of manifold with an fstructure, Tensor, N.S. 26, 323329 (1972).

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Author

Dr. Ram Swaroop : He had done PhD in Mathematics from Lucknow University in year 2010. Currently he is working as Associate Professor & HOD in Department of Mathematics in NIMS University, Jaipur, Rajasthan.
Mailing Address: H.No.10B, Khasbag Colony, Aishbagh, Lucknow226004.

Dr. Abhishek Singh: He is working in Department of Mathematics and Astronomy, University of Lucknow, Lucknow, Uttar Pradesh, India.

Mr. Prakash Kumar: He has done M.Tech

( Xl)Y = 0. (3.5)
Hence in view of the equation (3.4) and (3.5), we get m X(lY ) = l X(mY ).
This proves the first part of the theorem.
Again it can be easily shown that ( Xm)Y = 0,
for all vector fields X,Y TMn. Thus, the tensor field m is always covariant constant.
from NIMS University, Jaipur. Currently he is working as Assistant Professor in Department of Electrical Engineering in ITM University, Gwalior, Madhya Pradesh, India.
Mailing Address: East of Government Hospital Kanti, NH 28, Prakash Market, Kanti, Muzaffarpur, Bihar843109.