On nearly Ricci recurrent manifolds

. The object of the present paper is to introduce a new type of Ricci recurrent manifold called nearly Ricci recurrent manifold . Some geometric properties of nearly Ricci recurrent manifold have been studied. Finally we give an example of nearly Ricci recurrent manifold.


Introduction
Let (M n , g) be an n-dimensional Riemannian manifold with the matric g. A tensor field T of type (0, q) is said to recurrent [1] if the relation (D X T )(Y 1 , Y 2 , ..., Y q )T (Z 1 , Z 2 , ..., Z q ) − T (Y 1 , Y 2 , ..., Y q )(D X T )(Z 1 , Z 2 , ..., Z q ) = 0 holds on (M n , g). From definition it follows that if at a point x ∈ M ; T (X) = 0, then on some neighbourhood of x, there exits a unique 1-form A satisfying (D X T )(Y 1 , Y 2 , ..., Y q ) = A(X)T (Y 1 , Y 2 , ..., Y q ) B. Prasad and R.P.S. Yadav In 1952, Patterson [2] introduced a Ricci recurrent manifolds. According to him, a manifold (M n , g) of dimension n, was called Ricci recurrent if (D X S)(Y, Z) = A(X)S(Y, Z) for some 1-form A. He denoted such a manifold by R n . Ricci recurrent manifolds have been studied by several authors ( [3], [4], [1], [5] ) and many others. In a recent paper De, Guha and Kamilya [6] introduced the notion of generalized Ricci recurrent manifold as follows: A non-flat Riemannian manifold (M n , g)(n > 2) is called generalized Ricci recurrent if the Ricci tensor S is non-zero and satisfies the condition: where A and B non-zero 1-forms. Such a manifold where denoted by them as GR n . If the associated 1-form B becomes zero, then the manifold GR n reduces to a Ricci recurrent manifold R n . This justifies the name generalized Ricci recurrent manifold and the symbols GR n for it. Also in a paper De, and Guha [7] introduced a non flat Riemannian (M n , g)(n > 2) called a generalized recurrent manifold if its curvature tensor R(X, Y )Z of type (1,3) satisfies the condition: where A and B are two non-zero 1-forms and D denotes the operator of covariant differentiation with respect to metric tensor g. Such a manifold has been denoted by GK n . If the associated 1-form B becomes zero, then the manifold GK n reduces to recurrent manifold introduced by Ruse [8] and Waker [9] which was denoted by K n . In recent papers Arslan etal [10], Shaikh and Patra [11], Mallick, De and De [12], Khairnar [Kh], Shaikh, Prakasha and Ahmad [14], Kumar, Singh and Chowdhary [15], Hui [16], Singh and Mayanglambam [17], Singh and Kishor [18] etc. explored various geometrical propertis by using generlaized recurrent and generlaized Ricci recurrent manifold on Riemannian manifolds , Lorentzian Trans-Sasakian manifolds, LP-Sasakian manifolds, (k − µ) contact metric manifolds.
Further the authors Prasad and Yadav [19] considered a non-flat Riemannian manifold (M n , g)(n > 3) whose curvature tensor R satisfies the following condition: where A and B are two non-zero 1-forms and D has the meaning already mentioned. Such a manifold where called by them as nearly recurrent Riemannian manifold and denoted by (N R) n .
The motivation of the above studies, we define a new type of non flat Riemannian manifold is called nearly Ricci recurrent manifolds if the Ricci tensor S is non zero and satisfies the condition: where A and B non-zero 1-forms, P and Q be two vector fields such that Such a manifold shall be called as a nearly Ricci recurrent manifold and 1-forms A and B shall be called its associated 1-form and n dimensional nearly Ricci recurrent manifold of this kind shall be denoted by N {R(R n )}.
The name nearly Ricci recurrent Riemannian manifold was chosen because if B = 0 in (1.1) then the manifold reduces to a Ricci recurrent manifold which is very close to Ricci recurrent space. This justifies the name N early Ricci recurrent manif old for the manifold defined by (1.1) and the use of the symbol N {R(R n )} for it.

Nearly Ricci recurrent manifolds
In this paper, after preliminaries, the existence of a N {R(R n )} is first established and then it proved that the scalar curvature of N {R(R n )} cannot be zero. In section 4, the necessary and sufficient condition for constant scalar curvature of N {R(R n )} is obtained. Here it is established if A is closed then B is also closed and conversely in section 5. In section 6, it is shown that if the scalar curvature is constant in N {R(R n )} then the eigen value of the Ricci tensor S corresponding to the given eigen vector not exist. In section 7, it is proved that in In section 8, a necessary and sufficient condition for N {R(R n )} to be a (N R) n is obtained. Finally the existence of nearly Ricci recurrent manifold N {R(R n )} is ensured by a non trivial example.

Preliminaries
Let L denotes the symmetric endomorphism of the tangent space at each point of the manifold corresponding to the Ricci tensor S that is g(LX, Y ) = S(X, Y ) for every vector field X, Y . Therefore, In this section, it show that there exist a Riemannian manifold (M n , g)(n ≥ 2) whose Ricci tensor S of type (0,2) satisfies the condition and for which (D X S)(Y, Z) = A(X)S(Y, Z). For this we consider a Riemannian manifold (M n , g) which admits a linear connection D defined by where B is non zero 1-form L is a symmetric endomorphism of the tangent space at each point (M n , g) corresponding to the Ricci tensor S defined by g(LX, Y ) = S(X, Y ) and L 2 X = X and which satisfies the condition From this, we get The

Nature of the 1-forms A and B
We have Now in virtue of (2.2), we get from (5.1) or rdA(X, Y ) + (n + r)dB(X, Y ) = 0 Since B is closed then rdA(X, Y ) = 0. But r = 0, A is closed.
Conversely if A is closed then B is closed. Hence we have the following theorem:

2[A(LX) + B(LX)] = [A(X) + B(X)]r + (n − 2)B(X)
In view of (2.2) and (6.3), we get Now if r is constant then S(X, P ) + S(X, Q) = 1 1 + n r g(X, P ) (6.4) Hence we can state the following theorem: Theorem 6.1. In a N {R(R n )}, none of P and Q can be an eigen vector corresponding to any eigen values.

Conformally flat N {R(R n )} with constant scalar curvature
In Conformally flat (M n , g) it known [20] (  Now from (1.1),we get From above, we have Suppose the 1-form A is closed. Then in virtue of theorem (5.1) and (7.3) we get from (7.4) Hence we have the following theorem: 8. Necessary and sufficient condition for a N {R(R n )} to be a (N R) n It is known that the Conformal curvature tensor C of type (0, 4) of a Riemannian manifold (M n , g)(n > 3) is given by Conversely if (8.2) holds, then putting Y = Z = e j in (8.2) where {e j } , j = 1, 2, 3, ..., n is orthonormal basis of the tangent space at each point of the manifold and l is summed for l ≤ j ≤ n, we get But in view of C(X, W ) = 0, we get from (8.3) that In particular, if the M n Conformal to a flat space or if n = 3 then C = 0. In the first case it follows (8.2) that the N {R(R n )} is a (N R) n . In the second case it follows that N {R(R 3 )} is a (N R) 3 . Thus we can state the following theorem:

Example
Let us consider the 3-dimensional manifold M = (x, y, z) ∈ R 3 , z = 0 , where (x, y, z) are standard coordinate of R 3 .
We choose the vector fields e 1 = e iy ∂ ∂x , e 2 = ∂ ∂y , e 3 = e −iy ∂ ∂z The Riemannian connection D of the metric g is given by  The curvature tensor is given by  The Ricci tensor is given by From (9.7) and (9.8), we get S(e 1 , e 1 ) = 0, S(e 2 , e 2 ) = 2, S(e 3 , e 3 ) = 0 (9.9) and the scalar curvature is r = 2.

B. Prasad and R.P.S. Yadav
Since {e 1 , e 2 , e 3 } forms a basis of Riemannian manifold any vector field X, Y, Z ∈ χ(M ) can be written as where a j , b j , c j ∈ + ( the set of all positive real numbers), j = 1, 2, 3. Hence S(X, Y ) = b 1 b 2 (9.10) g(X, Y ) = a 1 a 2 + b 1 b 2 + c 1 c 2 (9.11) By view of (9.10), we get It can be easily seen that the Riemannian manifold with 1-forms satisfies relation (9.13). Hence the manifold under consideration is a nearly Ricci recurrent manifold (M 3 , g), which is neither Ricci recurrent nor Ricci symmetric. Thus we have the following theorem: Theorem 9.1. There exist a nearly Ricci recurrent manifold (M 3 , g), which is neither Ricci recurrent nor Ricci symmetric.