On m-projective Recurrent Riemannian Manifold

Transcription

1 Int. Journal of Math. Analysis, Vol. 6, 202, no. 24, On m-projective Recurrent Riemannian Manifold Jay Prakash Singh Department of Mathematics and Computer Science Mizoram University, Aizawl , India Abstract. Recurrent spaces have been studied by many geometers such as Patterson[4], De and Guha[2], Singh and Khan[6], etc. In this paper,we have investigated some properties of the m-projective recurrent curvature tensor in Riemannian manifold. Mathematics Subject Classification: 53C5, 35B05 Keywords: Riemannian manifold,recurrent parameter, m-projective curvature tensor, Einstein manifold. introduction Let M be an n-dimensional Riemannian manifold with Riemannian metric g. Let K and D denotes the Riemannian curvature and Riemannian connection respectively. A Riemannian manifold is recurrent if (.) (D U K)X, Y, Z) =α(u)k(x, Y, Z) where α is a non zero -form known as recurrence parameter. If -form α is zero in (.),then the manifold reduces to symmetric manifold. Contracting (.) with respect to X, we get (.2) From(.2),we have (.3) (D U Ric)(Y,Z) =α(u)ric(y,z). (D U Q)(Y )=α(u)q(y ) where Q is Ricci operator of type (,), defined as (.4) Ric(Y,Z) =g(q(y ),Z). Contracting (.3) with respect to Y, we get (.5) where r is the scalar curvature. Ur = α(u)r

2 74 J. P. Singh 2. m-projective curvature tensor In 97,G.P. Pokhariyal and R.S. Mishra [5] defined a tensor field W on a Riemannian manifold as W (X, Y )Z = R(X, Y )Z [Ric(Y,Z)X Ric(X, Z)Y 2(n ) (2.) + g(y,z)qx g(x, Z)QY ]. R.H.Ojha [3] defined and studied the properties of the m-projective curvature in Sasakian manifolds. He has also shown that it bridges the gap between the conformal curvature tensor,coharmonic curvature tensor and Concircular curvature tensors. In this paper,we have considered a non flat n-dimensional smooth Riemannian manifold in which the M-projective curvature tensor W satisfies the following condition (2.2) (D U W )(X, Y )Z = α(u)w (X, Y )Z. If the -form α is zero,then manifold reduces to the m-projectively symmetric manifold. The Projective curvature tensor P is given by (2.3) P (X, Y )Z = R(X, Y )Z [Ric(Y,Z)X Ric(X, Z)Y ]. n A Riemannian manifold is said to be projectively recurrent manifold if (2.4) (D U P )(X, Y )Z = α(u)p (X, Y )Z A manifolds is said to be an Einstein manifold if (2.5) Ric(X, Y )=kg(x, Y ) where k is a constant. From (2.5),we have (2.6) Q(X) =kx. Contracting the above equation with respect to X,we have (2.7) r = nk. 3. m-projectively recurrent manifold Theorem 3.. In an n-dimensional m-projectively recurrent Riemannian manifold,the constant curvature tensor r is given by (3.) (n 3)(Ur) + 2(2 n)α(u)r 2nα(QU) =0.

3 On m-projective recurrent Riemannian manifold 75 Proof. Let M n be an n-dimensional M-projectively recurrent manifold. Then from equations (2.) and (2.2), it follows that (D U R)(X, Y )Z = α(u)r(x, Y )Z + 2(n ) [(D URic(Y,Z)X (D U Ric)(X, Z)Y + g(y,z)(d U Q)X g(x, Z)(D U Q)Y α(u){ric(y,z)x Ric(X, Z)Y (3.2) + g(y,z)qx g(y,z)qy }]. Permuting equation (3.) twice with respect to U, X, Y ; adding the three equations and using Bianchi s second identity,we have (3.3) α(u)r(x, Y )Z + α(x)r(y,u)z + α(y )R(U, X)Z + 2(n ) [ (D U Ric)(Y,Z)X (D U Ric)(X, Z)Y + g(y,z)(d U Q)(X) g(x, Z)(D U Q)(Y )+(D X Ric)(U, Z)Y (D X Ric)(Y,Z)U + g(u, Z)(D X Q)(Y ) g(y,z)(d X Q)(U)+(D Y Ric)(X, Z)U (D Y Ric)(U, Z)X + g(x, Z)(D Y Q)(U) g(u, Z)(D Y Q)(X) α(u){ric(y,z)x Ric(X, Z)Y + g(y,z)qx g(x, Z)QY } α(x){ric(u, Z)Y Ric(Y,Z)U + g(u, Z)QY g(y,z)qu} α(y ){Ric(X, Z)U Ric(U, Z)X + g(x, Z)QU g(u, Z)QX}] = 0. Contracting the above equation with respect to X, we obtain + α(u)ric(y,z) α(y )Ric(U, Z)+R (Y, U, Z, ρ) 2(n ) [(n )(D URic(Y,Z)+g(Y,Z)(Ur) g((d U Q)Y,Z)+(D Y Ric)(U, Z) (D U Ric)(Y,Z) (3.4) + 2 g(u, Z)(Yr) 2 g(y,z)(ur)+( n)(d Y Ric)(U, Z) + g((d Y Q)U, Z) g(u, Z)(Yr)+( n)α(u)ric(y,z) + α(u)g(y,z)r + α(u)g(q(y ),Z) α(y )Ric(U, Z) + α(u)ric(y, Z) α(q(y )g(u, Z)+α(Q(U)g(Y, Z) α(y )g(r(u),z)+(n )α(y )Ric(U, Z)+α(Y )g(u, Z)r] = 0. where ρ is a vector field defined by (3.5) g(x, ρ) =α(x).

4 76 J. P. Singh Factoring off Z in above,we have α(u)qy α(y )QU R(Y, U, ρ)+ 2(n ) [(D Y Q)U + (n )(D U Q)Y + Y (Ur) (D U Q)Y + 2 U(Yr) (D U Q)Y 2 Y (Ur)+( n)(d Y Q)U +(D Y Q)U U(Yr) (n )α(u)qy α(u)(yr)+α(u)qy α(y )QU + α(u)qu α(q(y ))U + α(y )Ur + α(q(u))y +(n )α(y )QU] =0. Or R(Y, U, ρ) = 2(n ) [(n 3)(D UQ)Y + α(u)qy (n 3)(D Y Q)U α(y )QU + α(q(u))y (3.6) α(q(y ))U α(u)yr+ α(y )Ur]. Contracting (3.6) with respect to Y, we get (n 3) Ric(Y,ρ) = [(n 3)(Ur) (Ur) 2(n ) 2 (3.7) + (2 n)α(u)r +(n 2)α(QU)]. Using(.4) and (3.4) in equation (3.7),we get (n 3)(Ur) + 2(2 n)α(u)r 2nα(QU) =0. This completes the proof of the theorem. Theorem 3.2. The necessary and sufficient condition for an n-dimensional Ricci-recurrent Riemannian manifold to be a recurrent manifold is that it is m-projectively recurrent manifold for the same recurrence parameter. Proof. Taking the covariant derivative of (2.) with respect to U, we get (D U W )(X, Y )Z = (D U R)(X, Y )Z 2(n ) [(D URic)(Y,Z)X (D U Ric)(X, Z)Y + g(y,z)(d U Q)(X) (3.8) g(x, Z)(D U Q)(Y )]. Let M n be a Ricci recurrent Riemannian manifold,then from (.2),(.3) and (3.8),we have (D U W )(X, Y )Z = (D U R)(X, Y )Z α(u) 2(n ) [Ric(Y,Z)X (Ric(X, Z)Y + g(y,z)qx (3.9) g(x, Z)QY ].

5 On m-projective recurrent Riemannian manifold 77 From(3.5) it is evident that if any one of the equations (.) and (2.5) hold then the second also hold. Theorem 3.3. In an Einstein manifold M n the m-projective curvature tensor satisfies the following identity (3.0) (D U W )(X, Y )Z +(D X W )Y,U)Z +(D Y W )(U, X)Z =0. Proof. Using(2.7) and (2.8) in (2.),it folows that (3.) W (X, Y )Z = R(X, Y )Z k [g(y,z)x g(x, Z)Y ]. n Taking covariant derivative of the above with respect to U,we get (3.2) (D U W )(X, Y )Z =(D U R)(X, Y )Z. permuting equation(3.0) twice withrespect to U, X, Y ;adding the three equations and using Bianchi s second identity,we get the required result. Theorem 3.4. LetM n be an n-dimensional Ricci recurrent Riemannian manifold. Then M n is m-projective recurrent if and only if it is a projectively recurrent manifold of the same recurence parameter. Proof. We have the following relation in projective curvature tensor and m- Projective curvature tensor W (X, Y )Z = P (X, Y )Z + [Ric(Y,Z)X Ric(X, Z)Y 2(n ) (3.3) g(y,z)qx + g(x, Z)QY ]. Taking the covariant derivative of (3.0) with respect to U,we get (D U W )(X, Y )Z = (D U P )(X, Y )Z + 2(n ) [(D URic)(Y,Z)X (D U Ric)(X, Z)Y g(y,z)(d U QX) (3.4) + g(x, Z)(D U QY )]. let M n be a Ricci-recurrent Riemannian manifold,then from(.2),(.3) and (3.2) it follows that (D U W )(X, Y )Z = (D U P )(X, Y )Z + α(u) 2(n ) [Ric(Y,Z)X Ric(X, Z)Y g(y,z)qx (3.5) + g(x, Z)QY ]. With the help of equations (3.2) and (3.3) it follows that if any one of the equations (2.6) and (2.2) hold then the second equation also holds.

6 78 J. P. Singh References [] Blair, D. E.: Contact manifolds in Riemannian geometry, Lecture Notes in Math. No Springer (976). [2] De, U. C. and Guha, N.:On generalized recurrent manifolds, Proc. Math. Soc. 7 (99), 7-. [3] Ojha,R.H.:m-projectvely flat Saskian manifold,indian J.Pure Appl.Math. 4(985), [4] Patterson,E.M.:Some theorems on Ricci -recurrent spaces,j. London math. soc. 27(952), [5] Pokhariyal,G.P. and Mishra,R.S.:Curvature tensor and their relativistic significance II,Yokohama math. Journal 9 (97), [6] Singh,H. and Khan,Q.: On symmetric Riemannian manifolds, Novi Sad J. math.,29(3)(999), [7] Singh,J.P.: On an Einstein M-projectve P-Sasakian amnifolds,bull. Cal. Math. Soc. 0(2)(2009), Received: December, 20