Linear Connections on Manifold Admitting F (2k + P, P)-Structure

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.

1. Preliminaries

   Let F be a non-zero tensor field of the type (1, 1) and of class C1 on an n-dimensional 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 F-structure manifold satisfying (1.1), then

   1. l + m = I,

      b. l2 = l, (1.3)

      1. m2 = m,

      2. 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]

2. Distributions anti-parallelism and anti-half 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 anti-half 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, anti-half 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 anti-parallel 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 anti-half 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 anti-parallel.

   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 anti-parallel 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 anti-parallel with respect to connection if and only if and are equal.

   Proof. Since the distributions DL and DM are anti-parallel 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 anti-half 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 anti-half 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 anti-half 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 anti-half 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 anti-half 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.

3. 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

1. Demetropoulou-Psomopoulou, D. and Gouli-Andreou, F., On necessary and sufficient conditions for an n-dimensional manifold to admit a tensor field f( 0) of type (1, 1) satisfying f2_+3+f = 0, Tensor, N.S., vol. 42, 252-257 1985).

2. Demetropoulou-Psomopoulou, D., Linear connections on manifold admitting f(2v + 3, 1)-structure, Tensor, N.S., vol. 47, 235-239 (1988).

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

4. Yano, K., On structure defined by a tensor field f of type (1,1) satisfying f3 + f = 0. Tensor, N. S., 14, 99-109 (1963).

5. Singh, A., On CR-structures and F-structure satisfying F2K+P + FP = 0, Int. J. Contemp. Math. Sciences, Vol. 4, no. 21, 2009.

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

7. Goldberg, S.I., On the existence of manifold with an f-structure, Tensor, N.S. 26, 323-329 (1972).

8. Nikkie, J., F(2k+1,1)-structure on the Lagrangian space FILOMAT (Nis), 161-167 (1995).

9. Verma, G., On some structures in a differentiable manifold and its tangent, cotangent bundles, (Ph.D. Thesis), Lucknow, India (2001).

   Author

   1. 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, Lucknow-226004.

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

   3. 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, Bihar-843109.