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Conservative Conformal and Quasi conformal curvature tensors on K-contact manifold with respect to semi-symmetric metric connection.

1. Introduction

In 1924, Friedman and Schouten introduced the notion of semi-symmetric linear connection on a differentiable manifold. Then in 1932, Hayden introduced the idea of metric connection with a torsion on a Riemannain manifold. A systematic study of semi-symmetric metric connection on a Riemannain manifold has been given by Yano in 1970 and later studied by K.S.Amur and S.S.Pujar (1), C.S.Bagewadi (2), U.C.De et al (7), Sharafuddin and Hussain (13) and others. The authors U.C.De et al (6), C.S.Bagewadi and Venkatesha ((15) and C.S.Bagewadi and N.B.Gatti (9)) have obtained results on conservativeness of projective, pseudo projective, conformal, concircular and quasi conformal curvature tensors on K-contact, Kenmotsu and trans-Sasakian manifolds.

In this paper we extend the conservativeness of conformal and Quasi conformal curvature tensor to K-contact manifold admitting semi-symmetric metric connection. The paper is organized as follows: After preliminaries in section 2, we study in section 3 some properties of conservativeness of conformal curvature tensor on K-contact manifold with respect to semi-symmetric metric connection. In the last section we study some properties regarding Quasi conformal curvature tensor with respect to this connection and obtained some interesting results.

2. Preliminaries

Let ([M.sup.n], g) be a contact Riemannian manifold with contact form [eta], associated vector field [xi], (1, 1)-tensor field [phi] and associated Riemannian metric g. If [xi] is a killing vector field, then [M.sup.n] is called a K-contact Riemannian manifold (5). A K-contact Riemannian manifold is called Sasakian (4), if the relation ([[nabla].sub.x][phi])Y = g(X, Y)[xi]-[eta](Y)X holds, where [nabla] denotes the operator of covariant differentiation with respect to g.

For a contact metric manifold, the following relations holds (11), (12)

a) [phi][xi] = 0, b) [eta]([xi]) = 1, c) [[phi].sup.2]X = - X + [eta](X)[xi] (2.1)

a) g([phi]X,[phi]Y) = g(X,Y) - [eta](X)[eta](Y), b)g([xi],X) = [eta](X), c) [eta]([phi]X) = 0. (2.2)

where X and Y are any vector fields.

In addition to above, the following relations holds for K-contact manifold:

[[nabla]x[xi]] = -[phi]X, (2.3)

S(X,[xi]) = g(QX,[xi]) = (n - 1)[eta](X) (2.4)

for any vector fields X, Y. where S and Q denotes the Ricci tensor of type (0, 2) and Ricci operator respectively.

Let ([M.sup.n], g) be an n-dimensional Riemannian manifold of class [C.sup.[infinity]] with metric tensor g and let [nabla] be the Levi-Civita connection on [M.sup.n]. A linear connection [~.[nabla]] on ([M.sup.n], g) is said to be semi symmetric (16) if the torsion tensor T of the connection [~.[nabla]] satisfies

T(X,Y) = [[pi](Y)X-[pi](X)Y], (2.5)

where [pi] is a 1-form on [M.sup.n] with [rho] as associated vector field, i.e., [pi](X) = g(X, [rho]) for any differentiable vector field X on [M.sup.n].

A semi-symmetric connection [~.[nabla]] is called semi-symmetric metric connection (10) if it further satisfies [~.[nabla]]g = 0.

For contact metric manifold the semi-symmetric metric connection is defined by identifying the 1-form [pi] of the above with the contact-form [eta], i.e., by setting (13)

T(X,Y) = [eta](Y)X-[eta](X)Y (2.6)

with [xi] as associated vector field. i.e., g(X, [xi]) = [eta](X).

The relations between the semi-symmetric metric connection [~.[nabla]] and the Levi-Civita connection [nabla] of contact metric manifold ([M.sup.n], g), is given by (16)

[~.[nabla].sub.X]Y = [[nabla].sub.X]Y + [eta](Y)X - g(X,Y)[xi], (2.7)

where [eta](Y) = g(Y, [xi]).

Further, a relation between the curvature tensor R and [~.R] of type (1, 3) of the connections [nabla] and [~.[nabla]] respectively is given by (16)

[~.R](X,Y)Z = R(X,Y)Z-[alpha](Y,Z)X + [alpha](X,Z)Y-g(Y,Z)[bar.L]X + g(X,Z)[bar.L]Y. (2.8)

where [bar.L] is the tensor filed of type (1, 1) and [alpha] is a tensor field of type (0, 2) defined by

[alpha](Y,Z) = g([bar.L]Y,Z) = ([[nabla].sub.Y][eta])(Z)-[eta](Y)[eta](Z) + [1/2][eta]([xi])g(Y,Z) = ([~.[nabla].sub.Y][eta])(Z)-[1/2][eta]([xi])g(Y,Z) (2.9)

for any vector fields X and Y.

From (2.8), it follows that

[~.S](Y,Z) = S(Y,Z)-(n-2)[alpha](Y,Z)-A.g(Y,Z) (2.10)

where [~.S] denotes the Ricci tensor with respect to [~.[nabla]], A = Tr.[alpha]. Then differentiating (2.10) covariantly with respect to X, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

From this, it follows that

[~.[nabla].sub.X][~.r] = [[nabla].sub.X]r-(n-2)([[nabla].sub.X]A) (2.12)

Here, [~.r] and r are respectively the scalar curvatures of [~.[nabla]] and [nabla].

Further, since [xi] is a killing vector. S, [alpha], r and A are invariant under it, i.e.,

[L.sub.[xi]]S = 0, [L.sub.[xi]]r = 0, (2.13)

([[nabla].sub.[xi]]S)(Y,Z) = -S([[nabla].sub.Y][xi],Z) -S(Y,[[nabla].sub.Z][xi]) and dr([xi]) = [[nabla].sub.[xi]]r = 0, (2.14)

[L.sub.[xi]][alpha] = 0, [L.sub.[xi]]A = 0, (2.15)

([[nabla].sub.[xi]][alpha])(Y,Z) = -[alpha]([[nabla].sub.Y][xi],Z)-[alpha](Y,[[nabla].sub.Z][xi]) and dA([xi]) = [[nabla].sub.[xi]]A = 0, (2.16)

where L is the Lie derivation (9).

Definition 1. A K-contact manifold [M.sup.n] is said to be [eta]-Einstein if its Ricci tensor S is of the form S(X, Y) = [bar.P]g(X, Y )+[bar.Q][eta](X)[eta](Y) for any vector fields X, Y where [bar.P], [bar.Q] are functions on M,

If in particular [bar.Q]=0, then manifold is said to be an Einstein.

Definition 2. If a Riemannian manifold satisfies condition

([[nabla].sub.X]S)(Y,Z) + ([[nabla].sub.Y]S)(Z,X) + ([[nabla].sub.Z]S)(X,Y) = 0, (2.17)

then we say that the manifold has a cyclic Ricci tensor.

3. K-contact manifold admitting a semi-symmetric metric connection

The conformal curvature tensor [~.C] with respect to semi-symmetric metric connection is given by

[~.C](X,Y)Z = [~.R](X,Y)Z + [1/(n - 2)][[~.S](X,Z)Y-[~.S](Y,Z)X + g(X,Z)[~.Q]Y-g(Y,Z)[~.Q]X]-[[~.r]/[(n-1)(n-2)]][g(X,Z)Y-g(Y,Z)X], (3.1)

where [~.Q] and [~.r] are respectively the Ricci operator and scalar curvature with respect to [~.[nabla]].

First differentiate (3.1) covariantly and then contracting and after simplifications one gets div[~.C]. By virtue of conservativeness of [~.C] i.e.,div[~.C] = 0 and (2.11) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

For a K-contact manifold, we have

[alpha](Y,Z) = g(Y,[phi]Z) + [1/2]g(Y,Z) - [eta](Y)[eta](Z) (3.3)

[alpha](Y,[xi]) = -[1/2][eta](Y) (3.4)

Putting X = [xi] in (3.2) and then using (2.4), (3.4) and (2.13) to (2.16), we obtain

[[n - 3]/[n - 2]][S([[nabla].sub.Y][xi],Z) + S(Y,[[nabla].sub.Z][xi])-([[nabla].sub.Y]S)([xi],Z)] = [[n - 1]/[n -2]]S(Y,Z) - [[n - 1]/[n - 2]] + [1/2] + [1/2] + A]g(Y,Z)-[1/[n-1]][eta](Z)[dA(Y) + dr(Y)][[(1 - n).sup.2]/[n - 2]] + [[n - 5]/[2(n - 2)]] + (n - 1) + [1/2] + A][eta](Y)[eta](Z). (3.5)

Again using (2.3), (2.4) and (3.4) in above, we have

[[n - 3]/[n - 2]]S(Y,Z) = [[[n - 1]/[n - 2]] + [1/2] + A] g(Y,Z) + [[n - 3]/[n - 2]](n - 1)g([phi]Y,Z)-[[(1 - n).sup.2]/[n - 2]] + [[n-5]/[2(n - 2)]] + (n - 1) + [1/2] + A][eta][(Y).sub.[eta]](Z) + [1/[n - 1]][eta](Z)[dA(Y) + dr(Y)]-[[n - 3]/[n - 2]]S([phi]Y,Z).

Putting Z = [phi]Z in (3.6), and after simplification, we get

[[n - 3]/[n - 2]]S([phi]Y,[phi]Z) = [[n - 1]/[n - 2]]S([phi]Y,Z) + [[[n - 1]/[n - 2]] + [1/2] + A]g(Y,[phi]Z) + [[n - 3]/[n - 2]](n-1)g([phi]Y,[phi]Z).

Interchanging Y and Z in above and then using the symmetric property of S, we get

[[n - 3]/[n - 2]]S([phi]Y,[phi]Z) = [[n - 1]/[n - 2]]S([phi]Z,Y) + [[[n - 1]/[n - 2]] + [1/2] + A] g(Z,[phi]Y) + [[n - 3]/[n - 2]](n-1)g([phi]Y,[phi]Z). (3.8)

Adding (3.7) with (3.8) and then by using the skew-symmetric property of [phi], we get

S([phi]Y,[phi]Z) = (n - 1)g([phi]Y,[phi]Z). (3.9)

This implies

S(Y,Z) = (n - 1)g(Y,Z). (3.10)

From this, it follows that

QY = (n - 1)Y. (3.11)

Also from (3.10), we get

r = n(n - 1). (3.12)

Hence we have the following theorem.

Theorem 1. In a K-contact manifold [M.sup.n] admitting a semi-symmetric metric connection the Conformal curvature tensor with respect to this connection is Conservative, then the manifold is Einstein with respect to Levi-Civita connection and the scalar curvature of such a manifold is r = n(n - 1).

4. K-contact manifold admitting a semi-symmetric metric connection with Div.[~.W] = 0

The quasi-conformal curvature tensor [~.W] with respect to semi-symmetric metric connection is given by

[~.W](X,Y)Z = a[~.R](X,Y)Z + b[[~.S](X,Z)Y-[~.S](Y,Z)X + g(X,Z)[~.Q]Y-g(Y,Z)[~.Q]X]-[[a + 2b(n-1)]/[n(n-1)]][~.r][g(Y,Z)X-g(X,Z)Y] (4.1)

where a and b are arbitrary constants such that ab [not equal to] = 0.

First differentiate (4.1) covariantly and then contracting and after simplifications one gets div[~.W]. By virtue of conservativeness of [~.W] i.e., div[~.W] = 0 and (2.11) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

Putting X = [xi] in above and then using (2.4), (3.4) and (2.13) to (2.16), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Again, by using (2.3), (2.4) and (3.4) in above, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

Putting Z = [phi]Z in (4.5), and after simplification, we get

(a + b)S([phi]Y,[phi]Z) = [(a + b)(n-1) + 2a + b(n-2)]g([phi]Y,[phi]Z)-[b(2n - 3)-a[A - [1/2]] - a + b(n-2)]g(Y,[phi]Z). (4.5)

Interchanging Y and Z in above and then using the symmetric property of S, we get

(a + b)S([phi]Y,[phi]Z) = [(a + b)(n-1) + 2a + b(n-2)]g([phi]Y,[phi]Z) - [b(2n - 3) - a[A - [1/2]] - a + b(n - 2)]g([phi]Y,Z). (4.6)

Adding (4.5) with (4.6) and then by using the skew-symmetric property of [phi], we get

(a + b)S([phi]Y,[phi]Z) = [(a + b)(n-1) + 2a + b(n-2)]g([phi]Y,[phi]Z). (4.7)

This implies

S(Y,Z) = [P.sub.3]g(Y,Z) + [Q.sub.3][eta](Y)[eta](Z) (4.8)

where [P.sub.3] = [(n - 1) + [2/[a + b]][a + b(n - 2)]] and [Q.sub.3] = [-[2/[a + b]][a + b(n - 2)]]

which means that the manifold is [eta]-Einstein. Thus we can state the following:

Theorem 2. In a K-contact manifold [M.sup.n] admitting a semi-symmetric metric connection the Quasi Conformal curvature tensor with respect to this connection is conservative, then the manifold is [eta]-Einstein with respect to Levi-Civita connection.

Again, differentiating (4.8) covariantly along X, we have

([[nabla].sub.X]S)(Y,Z) = [Q.sub.3][([[nabla].sub.X][eta])(Y)[eta](Z) + ([[nabla].sub.X][eta])(Z)[eta](Y)]. (4.9)

From (4.9), we have

([[nabla].sub.Y]S)(Z, X) = [Q.sub.3][([[nabla].sub.Y][eta])(Z)[eta](X) + ([[nabla].sub.Y][eta])(X)[eta](Z)] (4.10)

([[nabla].sub.Z]S)(X,Y) = [Q.sub.3][([[nabla].sub.Z][eta])(X)[eta](Y) + ([[nabla].sub.Z][eta])(Y)[eta](X)]. (4.11)

Adding the equations (4.9) (4.10) and (4.11) and by virtue of skew-symmetry of [phi], we obtain

([[nabla].sub.X]S)(Y,Z) + ([[nabla].sub.Y]S)(Z,X) + ([[nabla].sub.Z]S)(X,Y) = 0. (4.12)

Taking an orthonormal frame field and contracting (4.9) over X and Z we obtain

dr(Y) = [-[2/[(a + b)]][a + b(n - 2)]][psi][eta](Y) (4.13)

where [psi] = Tr.[phis]. From (4.13), it follows that

dr(Y) = 0 if and only if [[a + b(n - 2)]/[(a + b)]] = 0 or [psi] = 0. (4.14)

From the above results we able to state the following:

Theorem 3. In a K-contact manifold [M.sup.n] admitting a semi-symmetric metric connection [M.sup.n] the Quasi conformal curvature tensor with respect to this connection is Conservative, then the manifold has a cyclic-Ricci tensor with respect to Levi-Civita connection; and moreover, the scalar curvature of the manifold is constant if and only if a = b(2 - n) or the vector field [xi] is harmonic.

References

(1) K. Amur and S. S. Pujar, On Submanifolds of a Riemannian manifold admitting a metric semi-symmetric connection,, Tensor, N.S., 32 (1978), 35-38.

(2) C. S. Bagewadi, On totally real submanifolds of a Kahlerian manifold admitting Semi symmetric metric F-connection, Indian. J. Pure. Appl. Math., 13(5) (1982), 528-536.

(3) C. S. Bagewadi and Venkatesha, Some curvature tensors on trans-Sasakian manifolds, Turk.j.Math, 30 (2007), 1-11.

(4) D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics 509, Springer-Verlag, Berlin, 1976.

(5) M. C. Chaki and M. Tarafdar, On a type of Sasakian manifold, Soochow J. Math., 16 (1990), 23-28.

(6) U. C. De and Absos Ali Shaikh, K-contact and Sasakian manifolds with conservative quasi-conformal curvature tensor, Bull. Cal. Math. Soc., 89 (1997), 349-354

(7) U. C. De and Joydeep Sengupta, On a type of semi-symmetric metric connection on an almost contact metric manifold, Filomat 14 (2000), 33-42.

(8) A. Friedmann and J. A. Schouten, Uber die geometric der holbsym-metrischen Ubertragurgen, Math. Zeitschr. 21 (1924), 211-233.

(9) N. B. Gatti, A study on conservtive and irrotational curvature tensors of contact manifolds, Ph. D Thesis, Kuvempu University, 2003.

(10) H. A. Hayden, Subspaces of space with torsion, Proc. Lond. Math. Soc., 34 (1932), 27-50.

(11) S. Sasaki, Lecture notes on almost contact manifolds, Part I, Tohoku University, 1965.

(12) S. Sasaki, Lecture notes on almost contact manifolds, Part II, Tohoku University, 1967.

(13) A. Sharafuddin and S. I. Hussain, Semi-symmetric metric connections in almost contact manifolds, Tensor. N. S., 30 (1976), 133-139.

(14) Venkatesha and C. S. Bagewadi, On 3-dimensional trans-Sasakian manifolds, Journal of Modeling, Measurement & Control (Series-A), 42 (5) (2005), 63-72.

(15) Venkatesha, A study on contact manifolds, Ph. D Thesis, Kuvempu University, 2006.

(16) K. Yano, On semi-symmetric metric connections, Revue Roumaine de Math. Pures et Appliques 15 (1970), 1579-1586.

Prakasha D. G. [dagger]

Department of Mathematics, PES Institute of Technology and Managemtnt, N. H. 206, Sagar Rood, Shimoga, Karnataka, India

Bagewadi C. S., [double dagger] and Venkatesha [dagger]

Department of Mathematics, Kuvempu University, Jnana Sahyadri-577 451, Shimoga, Karnataka, India

Received March 6, 2007, Accepted September 5, 2007.

* 2000 Mathematics Subject Classification. 53D15, 53b15.

[dagger] E-mail: prakashadg@gmail.com

[double dagger] E-mail: profbagewadi@yahoo.co.in.
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Author:D.G., Prakasha; C.S., Bagewadi; Venkatesha
Publication:Tamsui Oxford Journal of Mathematical Sciences
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Date:May 1, 2009
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