U.C. DE and AVIJIT SARKAR
The object of the present paper is to study a P-Sasakian manifold admitting a W2-curvature tensor.
AMS 2000 Subject Classification: 53C15, 53C40.
Key words: P-Sasakian manifold,W2 curvature tensor,W2semisymmetric.
1. INTRODUCTION
Pokhariyal and Mishra [5] have introduced new tensor fields, called W2 and E-tensor fields, in a Riemannian manifold, and studied their properties.
Next, Pokhariyal [4] has studied some properties of these tensor fields in a Sasakian manifold. Recently, Matsumoto, Ianus and Mihai [3] have studied P-Sasakian manifolds admitting W2 andE-tensor fields.
On the other hand, Sato [6, 7] introduced the notion of an (almost) paracontact structure, either P-Sasakian or SP-Sasakian, and gave a lot of very interesting results about such manifolds. In this paper we generalize some results of Matsumoto, Ianus and Mihai [3]. A Riemannian manifold is called locally symmetric if its Riemannian curvature tensor R satiesfies ∇R = 0, where ∇ denotes the operator of covariant differentiation. This notion of locally symmetric manifold has been weakened by many authors in several ways. As a proper generalization of locally symmetric manifolds, the notion of semisymmetric manifold was defined by R(X, Y) ·R = 0 [8]. A locally symmetric manifold is semisymmetric but the converse is not true in general.
In Section 2 we recall the notions of P-Sasakian and SP-Sasakian manifolds and the essential properties of such manifolds. In Section 3, we prove that a W2-symmetric P-Sasakian manifold is of constant curvature, hence it is an SP-Sasakian manifold. Next, we prove that a W2-recurrent manifold is W2-symmetric, hence a W2-recurrent P-Sasakian manifold is an SP-Sasakian manifold. Finally, we prove that a P-Sasakian manifold is Ricci-semisymmetric if and only if it is an Einstein manifold.
MATH. REPORTS11(61),2 (2009), 139–144
2. PRELIMINARIES
Let (M, g) be an n-dimensional manifold admitting a 1-form η which satiesfies the conditions
(∇Xη)(Y)−(∇Yη)(X) = 0, (2.1)
(∇X∇Yη)(Z) =−g(X, Z)η(Y)−g(X, Y)η(Z) + 2η(X)η(Y)η(Z), (2.2)
where ∇denotes the operator of covariant differentiation with respect to the metric tensorg. If, moreover, (M, g) admits a vector fieldξ and a (1,1) tensor field φsuch that
g(X, ξ) =η(X), (2.3)
η(ξ) = 1, (2.4)
∇Xξ=φX, (2.5)
then such a manifold is called a Para Sasakian manifold, a P-Sasakian for short, by Adati and Matsumoto [1]. This is a special case of an almost paracontact manifold introduced by Sat¯o.
It is known that in a P-Sasakian manifold there hold [1, 6] the relations φ2X =X−η(X)ξ,
(2.6)
η(R(X, Y)Z) =g(X, Z)η(Y)−g(Y, Z)η(X), (2.7)
R(ξ, X)Y =η(Y)X−g(X, Y)ξ, (2.8)
where R denotes the curvature tensor,
R(ξ, X)ξ=X−η(X)ξ, (2.9)
η(φX) = 0, (2.10)
S(X, ξ) =−(n−1)ηX.
(2.11)
The above results will be used in the next sections.
3. W2-SEMISYMMETRIC P-SASAKIAN MANIFOLDS
Pokhariyal and Mishra [5] have defined a new curvature tensor (3.1)
W2(X, Y, Z, U) =R(X, Y, Z, U) + 1
n−1[g(X, Z)S(Y, U)−g(Y, Z)S(X, U)], where S is a Ricci tensor of type (0,2). In this section we shall study W2- semisymmetric P-Sasakian manifold.
Definition. Ann-dimensional P-Sasakian manifold is calledW2-semisym- metric if it satisfies
(3.2) R(X, Y)·W2 = 0,
whereR(X, Y) is to be considered as a derivation of the tensor algebra at each point of the manifold for tangent vectorsX, Y. It can be easily shown that in a P-Sasakian manifold the W2-curvature tensor satisfies the condition
(3.3) W2(X, Y, Z, ξ) = 0.
Theorem 3.1. A W2-semisymmetric P-Sasakian manifold is an SP- Sasakian manifold.
Proof. From (3.2) we have
R(X, Y)W2(Z, U)V −W2(R(X, Y)Z, U)V−
−W2(Z, R(X, Y)U)V −W2(Z, U)R(X, Y)V = 0.
This equation implies
g(R(X, Y)W2(Z, U)V.ξ)−g(W2(R(X, Y)Z, U)V, ξ)−
(3.4)
−g(W2(Z, R(X, Y)U)V, ξ)−g(W2(Z, U)R(X, Y)V, ξ) = 0.
Putting X=ξ in (3.4) we obtain
R(ξ, Y, W2(Z, U)V, ξ)−W2(R(ξ, Y)Z, U, V, ξ)−
−W2(Z, R(ξ, Y)U, V, ξ)−W2(Z, U, R(ξ, Y)V, ξ) = 0.
Using (2.8) and (3.3), from the above equation we obtain η(Y)η(W2(Z, U)V)−g(Y, W2(Z, U)V) = 0.
Using again (3.3) we get
W2(Z, U, V, Y) = 0.
Then it follows from (3.1) that (3.5) R(X, Y, Z, V) = 1
n−1[g(Y, Z)S(X, V)−g(X, Z)S(Y, V)].
Contracting (3.5) we have
(3.6) S(Y, Z) = r
ng(Y, Z).
From (3.5) and (3.6) we get (3.7) R(X, Y, Z, V) = r
n(n−1)[g(Y, Z)g(X, V)−g(X, Z)g(Y, V)].
Hence aW2-semisymmetric P-Sasakian manifold is of constant curvature. But it is known ([2]) that if a P-Sasakian manifold is of constant curvature, then it is an SP-Sasakian manifold. The proof is complete.
AW2-semisymmetric Sasakian manifold satisfies the condition∇W2= 0.
Since ∇W2 = 0 impliesR(X, Y)·W2 = 0, we can state the result below.
Corollary. A W2-symmetric P-Sasakian manifold is an SP-Sasakian manifold.
The above corollary has been proved by Matsumoto, Ianus and Mihai [3]
in another way.
4. W2-RECURRENT P-SASAKIAN MANIFOLDS
Definition. Ann-dimensional P-Sasakian manifold is calledW2-recurrent if it satisfies
(4.1) (∇UW2)(X, Y)Z =A(U)W2(X, Y)Z, for some nonzero 1-form A.
Theorem 4.1. AW2-recurrent P-Sasakian manifold is an SP-Sasakian manifold.
Proof. We first prove that aW2-recurrent manifold isW2-semisymmetric.
Let us suppose that W2 6= 0.We now define a function by
(4.2) f2 =g(W2, W2).
Using the fact that ∇Ug= 0, it follows from (4.2) that 2f(U f) = 2f2(A(U)).
Since f 6= 0,from get
(4.3) U f =f(A(U)).
From (4.3) we have
X(U f) = 1
f(Xf)(U f) + (XA(U))f, hence
X(U f)−U(Xf) = [XA(U)−U A(X)]f.
Therefore,
(∇X∇U− ∇U∇X− ∇[X,U])f = [XA(U)−U A(X)−A([X, U])]f (4.4)
= 2[dA(X, U)]f.
Since the left hand side of (4.4) is zero andf6= 0, we deduce thatdA(X, Y) = 0.
This means that the 1-form A is closed. Now, from (4.1) we have (∇V∇UW2)(X, Y)Z = [V A(U) +A(V)A(U)]W2(X, Y)Z.
Hence
(∇V∇UW2)(X, Y)Z−(∇U∇VW2)(X, Y)Z−(∇[U,V]W2)(X, Y)Z (4.5)
= 2dA(V, U)W2(X, Y)Z = 0.
Therefore, we have R(V, U)·W2 = 0, where R(V, U) is to be considered as a derivation of tensor algebra at each point of the manifold for tangent vec- tors V, U. Thus, a W2-recurrent manifold is W2-semisymmetric. Hence Theo- rem 3.1 completes the proof.
5. RICCI-SEMISYMMETRIC P-SASAKIAN MANIFOLDS
Definition 5.1. An n-dimensional (n > 2) Riemannian manifold is said to be an Einstein manifold if its Ricci tensor satisfies the condition
(5.1) S(X, Y) =λg(X, Y),
where λis a constant.
Definition5.2. Ann-dimensional Riemannian manifold is said to be Ricci- semisymmetric if
(5.2) R(X, Y)·S= 0.
Theorem 5.1. An n-dimensional(n >2)P-Sasakian manifold is Ricci- semisymmetric if and only if it is an Einstein manifold.
Proof. By the above definition, every Einstein manifold is Ricci-semi- symmetric but the converse is not true in general. Here, we prove that in a P-Sasakian manifold R(X.Y)·S= 0 implies that the manifold is an Einstein manifold. It follows from (5.2) that
(5.3) S(R(X, Y)U, V) +S(U, R(X, Y)V) = 0.
Putting X=ξ in (5.3) we get
(5.4) S(R(ξ, Y)U, V) +S(U, R(ξ, Y)V) = 0.
Using (2.8) and (2.11), from (5.4) we get
η(U)S(Y, V) + (n−1)η(V)g(Y, U)+
(5.5)
+S(U, Y)η(V) + (n−1)η(U)g(Y, V) = 0.
Now, puttingU =ξ in (5.3), we obtain
S(Y, V) =−(n−1)g(Y, V).
Therefore, M is an Einstein manifold.
Remark. We conclude from (3.6) that a W2-semisymmetric P-Sasakian manifold is an Einstein manifold. Hence, by Theorem 5.1, aW2-semisymmetric manifold is Ricci-semisymmetric.
Acknowledgement. The authors express their sincere thanks to the referee for his valuable suggestions in the improvement of the paper.
REFERENCES
[1] T. Adati and K. Matsumoto, On conformally recurrent and conformally symmetric P- Sasakian manifolds.TRU Math.13(1977), 25–32.
[2] T. Adati and T. Miyazawa,On P-Sasakian manifolds admitting some parallel and recur- rent tensors. Tensor (N.S.)33(1979), 287–292.
[3] K. Matsumoto, S. Ianus and Ion Mihai, On P-Sasakian manifolds which admit certain tensor fields. Publ. Math. Debrecen 33(1986), 61–65.
[4] G.P. Pokhariyal,Study of a new curvature tensor in a Sasakian manifold. Tensor (N.S.) 36(1982), 222–225.
[5] G.P. Pokhariyal and R.S. Mishra,,The curvature tensor and their relativistic significance.
Yokohoma Math. J.18(1970), 105–108.
[6] I. Sat¯o, On a structure similar to almost contact structures.Tensor (N.S.) 30 (1976), 219–224.
[7] I. Sat¯o,On a structure similar to almost contact structures, II. Tensor (N.S.)31(1977), 199–205.
[8] Z.I. Szabo,Structure theorems on Riemannian spaces satisfyingR(X, Y)·R= 0, I.The local version. J. Diff. Geom.17(1982), 531–582.
Received 18 June 2008 University of Kalyani
Department of Mathematics Kalyani, Nadia, W.B.
India, Pin 741235 avjaj@yahoo.co.in
