CC BY
72
CERTAIN CURVATURE CONDITIONS ON LORENTZIAN PARA-KENMOTSU MANIFOLDS
S. Sunitha Devi1, K. L. Sai Prasad2'* , T. Satyanarayana3
Department of Mathematics 1 Vignan's Institute of Information Technology, Visakhapatnam, 530 049, INDIA 2'* Gayatri Vidya Parishad College of Engineering for Women, Visakhapatnam, 530 048, INDIA 3 Pragati Engineering College, Surampalem, Near Peddapuram, Andhra Pradesh, INDIA sunithamallakula@yahoo.com1 klsprasad@yahoo.com2,* tsn9talluri@gmail.com3
Abstract
We classify Lorentzian para-Kenmotsu manifolds which satisfy the curvature conditions W2.C = 0, Z.C = LCQ(g,C), W2.Z — Z.W2 = 0 and W2.Z + Z.W2 = 0, where W2 is the Weyl- projective tensor, Z is the concircular tensor, and C is the Weyl conformal curvature tensor. We study and have shown that the manifold M is y-Einstein provided that the Weyl-projective curvature tensor W2 meets the condition W2.Z — Z.W2 = 0, and it is an Einstein manifold if W2.Z + Z.W2 = 0. Finally, in this article, we derive the conditions in relation to conformally flatness of the manifold, whenever the LP-Kenmotsu manifold satisfies the condition Z.C = LCQ(g, C).
Keywords: Para-contact metric manifold, LP-Kenmotsu manifold, concircular curvature tensor, conformal curvature tensor, Weyl-projective tensor. 2010 Mathematics Subject Classification: 53C07, 53C25
I. Introduction
In 1989, K. Matsumoto [7] introduced the notion of Lorentzian paracontact and in particular, Lorentzian para-Sasakian (LP-Sasakian) manifolds. Later, these manifolds have been widely studied by many geometers Matsumoto and Mihai [8], Mihai and Rosca [6], Mihai, Shaikh and De [5], Venkatesha and Bagewadi [15], Venkatesha, Pradeep Kumar and Bagewadi [16, 17] and obtained several results of these manifolds.
In 1995, Sinha and Sai Prasad [2] defined a class of almost paracontact metric manifolds namely para-Kenmotsu (briefly P-Kenmotsu) and special para-Kenmotsu (briefly SP-Kenmotsu) manifolds in similar to P-Sasakian and SP-Sasakian manifolds. In 2018, Abdul Haseeb and Rajendra Prasad defined a class of Lorentzian almost paracontact metric manifolds namely Lorentzian para-Kenmotsu (briefly LP-Kenmotsu) manifolds [1] and they studied ^-semisymmetric LP-Kenmotsu manifolds with a quarter-symmetric non-metric connection admitting Ricci solitons [13].
On the other hand, In 1970 [4], Pokhariyal and Mishra introduced new tensor fields, called the Weyl-projective curvature tensor W2 of type (1,3) and the tensor field E on a Riemannian manifold.The Weyl-projective curvature tensor W2 with respect to Riemannian connection on a Riemannian manifold M is given by:
1
W2(X, Y)W = R(X, Y)W + — [g(X, W)QY - g(Y, W)QX], (1)
where QX = (n — 1)X, which plays an important role in the theory of the projective transformations of connections.
Further, Pokhariyal [3] studied the properties of these tensor fields on a Sasakian manifold. Mat-sumoto, Ianus and Mihai extended these concepts to almost para-contact structures and studied para-Sasakian manifolds admitting these tensor fields [9] in 1986 and these results were generalised by De and Sarkar, in 2009 [14]. Sai Prasad and Satyanarayana studied the W2-tensor field in an SP-Kenmotsu manifold [10]. In our earlier work, we consider LP-Kenmotsu manifolds admitting the Weyl-projective curvature tensor W2 and shown that these manifolds admiting a Weyl-flat projective curvature tensor, an irrotational Weyl-projective curvature tensor and a conservative Weyl-projective curvature tensor are an Einstein manifolds of constant scalar curvature [11, 12].
Inspired by these studies, in the present work, we explore a class of Lorentzian para-Kenmotsu manifolds that admits certain curvature conditions. The current study is arranged as follows: Section 2 has certain prerequisites. In section 3, it is illustrated that the manifold M is ^-Einstein provided that the Weyl-projective curvature tensor W2 meets the condition W2.Z — Z.W2 = 0, and it is an Einstein manifold if W2.Z + Z.W2 = 0. Finally, we derive the conditions in relation to conformally flatness of the manifold, whenever the LP-Kenmotsu manifold satisfying the condition Z.C = LCQ(g, C), where the concircular curvature tensor Z(X, Y) is given by:
r
Z(X, Y)W = R(X, Y)W — ^n^—^ [g(Y, W)X — g(X, W)Y]. (2)
II. Preliminaries
An n-dimensional differentiable manifold M admitting a (1, 1) tensor field Q, contravariant vector field £, a 1-form n and the Lorentzian metric g(X, Y) satisfying
Q2X = X + n(X)£, g(QX,QY) = g(X,Y) + n(X)n(Y) (3)
and
n(£) = —1, Q£ = 0, n(QX) = 0, g(X,£) = n(X), rankty = n — 1. (4)
is called Lorentzian almost paracontact manifold [7].
In a Lorentzian almost paracontact manifold, we have
$(X, Y) = &(Y, X), (5)
where 0(X, Y) = g(X, QY).
A Lorentzian almost paracontact manifold M is called Lorentzian para-Kenmotsu manifold if [1]
(Vx Q)Y = —g(QX, Y)£ — n (Y)QX, (6)
for any vector fields X and Y on M and V is the operator of covariant differentiation with respect to the Lorentzian metric g.
In the Lorentzian para-Kenmotsu manifold, the following relations hold good:
Vx £ = —Q2 X = —X — n(X)£ (7)
and
(Vx n)Y = —g(X, Y)£ — n(X)n(Y). (8)
Further, on a Lorentzian para-Kenmotsu manifold M, the following relations hold [1]:
g(R(X, Y)Z, £) = n(R(X, Y)Z)= g(Y, Z)n(X) - g(X, Z)n(Y), (9)
R(£, X)Y = g(X, Y)£ - n(Y)X, (10)
R(X, Y)£ = n(Y)X - n(X)Y; when X is orthogonal to £, (11)
R(£, X)£ = X + n(X% (12)
S(X,£) = (n - 1)n(X), (13)
S($X,$Y) = S(X, Y) + (n - 1)n(X)n(Y). (14)
A Lorentzian para-Kenmotsu manifold M is said to be an n-Einstein manifold if its Ricci tensor S(X, Y) is of the form
S(X, Y) = ag(X, Y) + bn(X)n(Y), (15)
where a and b are scalar functions on M.
Next we define endomorphisms R(X, Y) and X AaY by
R(X, Y)W = VX -VY VXW X, Y] W, (16)
(X AaY)W = A(Y, W)X - A(X, W)Y, (17)
A is the symmetric (0,2)- tensor.
For a (0, k)-tensor field T, K > 1, on (Mn,g) we define W2.T, Z.T and Q(g, T) by
(W2(X, Y).T(Xlr X2,..., Xk) = - T(W2(X, Y)X1, X2,..., Xk)
- T (X1, W2 (X, Y)X2,..., Xk ) (18)
- ... - T(Xi, X2,..., W2(X, Y)Xk),
(Z(X, Y).T(X1,X2.....Xk) = - T(Z(X, Y)X1,X2.....Xk)
- T(X1, Z(X, Y)X2.....Xk) (19)
- ... - T(X1,X2.....Z(X, Y)Xk),
Q(g, T) (X1, X2.....Xk ; X, Y) = - T((X A Y)X1, X2.....Xk )
- T(X1, (X A Y)X2.....Xk) (20)
- ... - T(X1, X2.....(X A Y)Xk),
respectively.
By definition the Weyl Conformal curvature tensor C is given by
1
C(X, Y)Z =R(X, Y)Z - — [S(Y, Z)X - S(X, Z)Y + g(Y, Z)QX - g(X, Z)QY] ' [g(Y, Z)X - g(X, Z)Y],
(n - 1)(n - 2)
(21)
where Q denotes the Ricci operator, i.e., S(X, Y) = g(QX, Y) and r is scalar curvature. The Weyl conformal curvature tensor C is defined by C(X, Y, Z, W) = g(C(X, Y)Z, W). If C = 0, n > 4, then M is conformally flat.
III. MAIN RESULTS
In the present section we consider the LP-Kenmotsu manifold satisfying the curvature conditions W2.C = 0, Z.C = LCQ(g, C), W2.Z - Z.W2 = 0, and W2.Z + Z.W2 = 0.
First we give the following proposition.
Proposition 1. Let M be an n-dimensional (n > 3) LP-Kenmotsu manifold. If the condition W2.C = 0 holds on M, then
S2(X, U) = (n - 1)(r - 2)n(X)n(U) + (r + n - 2)S(U, X) - (n - l)g(X, U)
is satisfied on M, where S2(X, U) = S(QX, U).
Proof: Assume that M is an n-dimensional, n > 3, LP-Kenmotsu manifold satisfying the condition W2.C = 0. From (18) we have
(W2(V,X).C)(Y, U)W = - W2(V,X)C(Y, U)W
- C(W2(V, X)Y, U)W - C(Y, W2(V, X)U)W (22)
- C(Y, U)W2(V, X)W = 0,
where X, Y, U, V, W e x(M). Taking V = Ç in (22), we have
(W2(£, X).C)(Y, U)W = - W2(Z, X)C(Y, U)W
- C(W2(Ç, X)Y, U)W - C(Y, W2(Ç, X)U)W (23)
- C(Y, U)W2(Ç, X)W = 0,
Furthermore, substituting (1), (9), (13), (21) into (23) and multiplying with we get.
- g(X, C(Y, U)W) - g(X, Y)n(C(Ç, U)W + q(Y)q(C(X, U)W)
- g(X, U)V(C(Y, I)W) + n(U)n(C(Y, X)W) - g(X, W)V(C(Y, U)£)
1[
+ n(W)n(C(Y, U)X) + ~1 [n(C(Y, U)W) - n(Y)n(C(QX, U)W) (24)
+ g(X, Y)n(C(QÇ, U)W) + g(X, U)n(C(Y, QÇ)W) - n(U)n(C(Y, QX)W)
- n(W)n(C(Y, U)QX) + g(X, W)n(C(Y, U)QÇ)] = 0. Thus replacing W with £in (24), we have
1
- g(X, C(Y, U)Ç) - n(C(Y, U)X) + — [n(C(Y, U)QX)] = 0. (25)
Again taking Y = Ç in (25)and after some calculations, since n > 3, we get
S2(U, X) = (n - 1)(r - 2)n(X)n(U) + (r + n - 2)S(U, X) - (n - 1)g(X, U).
Theorem 2. Let M be an n-dimensional (n > 3) LP-Kenmotsu manifold. If the condition Z.C = LCQ(g, C) holds on M, then either M is conformally flat or Lc = n(tW-1) - 1.
Proof. Let M be an LP-Kenmotsu manifold. So we have
(Z(V, X).C)(Y, U)W = LcQ(g, C)(Y, U, W; V, X). Then from (19) and (20) we can write,
Z(V, X)C(Y, U)W - C(Z(V, X)Y, U)W - C(Y, Z(V, X)U)W
- C(Y, U)Z(V, X)W
= LC[(V A X)C(Y, U)W - C((V A X)Y, U)W
- C(Y, (V A X)U)W - C(Y, U)(V A X)W].
Therefore, replacing v with £ in (26), we have
Z(£, X)C(Y, U)W - C(Z(£, X)Y, U)W - C(Y, Z(£, X)U)W
- C(Y, U)Z(£, X)W
= LC [(£ A X)C(Y, U)W - C((£ A X)Y, U)W
- C(Y, (£ A X)U)W - C(Y, U)(£ A X)W].
Using (20), (9) and taking the inner product of (27) with £, we get
r
[! - nn^Tj - Lc] [-g(X, C(Y, U)W) - n (X)n (C(Y, U)W)
- g(X, Y)n(C(£, U)W) + n(Y)n(C(X, U)W) (28)
- g(X, U)n(C(Y, £)W) + n(U)n(C(Y, X)W) + n(W)n(C(Y, U)X)] = 0.
Putting X = Y in (28), we have
r
[1 - n^TTy - Ld [-g(Y, C(Y, U)W) + n(W)n(C(Y, U)Y)
- g(Y, Y)n(C(£, U)W) - g(Y, U)n(C(Y, £)W)] = 0.
A contraction of (29) with respect to Y gives us
r
[! - - lc\n(C(£, U)W) = 0. (30)
If LC = 1 - n(nr-T), then eq.(30) is reduced to
n(C(£, U)W) = 0, (31)
which gives
r r
S(U, W) = -1)g(U, W) + - n)n(U)n(W). (32)
Therefore, M is a n-Einstein manifold. So, using (31) and (32), we have eq. (28) in the form
C(Y, U, W, X) = 0, which means that M is conformally flat.
If Lc = 0 and n(C(£, U)W) = 0, then 1 - - Lc = 0, which gives Lc = 1 - ^^.
This completes the proof of the theorem.
Corollary 3. Every n-dimensional (n > 3) nonconformally flat LP-Kenmotsu manifold satisfies Z.C = a - nn-T))QgC).
Theorem 4. Let M be an n-dimensional (n > 3) LP-Kenmotsu manifold. M satisfies the condition
W2.Z - Z.W2 = 0 if and only if M is a n-Einstein manifold.
Proof. Let M satisfy the condition W2.Z — Z.W2 = 0. Then we can write
i
W2.Z — Z.W2 =R(V, X)R(Y, U)W + [g(V, R(Y, U)W)QX — g(X, R(Y, U)W)QV]
r i
-g(U, W) [R(V, X)Y + — {g(V, Y)QX — g(X, Y)QV)]
n(n — 1) /L V ' n — 1
+ g(Y, W) [R(V, X)U + (g(V, U)QX — g(X, U)QV)]
r[
— R(V, X)R(Y, U)W + -^n^T) [g(X, R(Y, U)W)V — g(V, R(Y, U)W)X]
1 r
— g(Y, W) R(V, X)QU — ^—j) (g(X, QU)V — g(V, QU)X)]
+ g(U, W) [R(V, X)QY — -(^ (g(X, QY)V — g(V, QY)X)] = 0.
Therefore, replacing V with £ in (33), we have
1 [ ] W2.Z — Z.W2 = — [g(£, R(Y, U)W)QX — g(X, R(Y, U)W)Q£]
r 1
— g(U, W) [R(£, X)Y + — (g(£, Y)QX — g(X, Y)Q£)]
+ g(Y, W) [R(£, X)U + (g(£, U)QX — g(X, U)Q£)]
r [ ]
— R(£, X)R(Y, U)W + [g(X, R(Y, U)W)£ — g(£, R(Y, U)W)X]
i r
— g(Y, W) [R(£, X)QU — (g(X, QU)£ — g(£, QU)X)]
+ ig(U, W) [R(£, X)QY — (g(X, QY)£ — g(£, QY)X)] = 0.
Using (10), (13), we get
1 [ ] W2.Z — Z.W2 = — [g(£, R(Y, U)W)QX — g(X, R(Y, U)W)Q£]
r r
-g(U, W) [g(X, Y)£ — V(Y)X] — g(U, W)n(Y)X
n(n — 1) ' /L6V 7 J n(n — 1)c
r r
+ gU W )gX Y)£+g(Y, W) [gX U)£—n (U)X
rr
-g(Y, W)n(U)X — -7-—g(Y, W)g(X, U)£
n(n — 1) n(n — 1)'
+ tin—) [g(X, R(Y, U)W)£ — g(£, R(Y, U)W)X
(33)
(34)
(35)
1 [ ] r
\g(Y, W) [g(X, QU)£ — n(QU)X] + 2g(Y, W)g(X, QU)£
(n — 1)^ ' 'L5V ' /J n(n — 1)2<:
1[] Tg(Y, W)n(QU)X + -—■ g(U, W) [g(X, QY)£ — V(QY)X]
n(n — 1)2 (n — 1)'
r r
nin—W g(U,W )g(X,QY)£ — W—W g(U,W )n(QY)X =
Again, taking U = £ in (35), we get 1
n — 1
[g(t,g(Y, W)t - n(W)Y)(n - 1)X - g(X,g(Y, W)t - n(W)Y)(n - 1)£]
Y Y
-n(W) [g(X, Y)t - n(Y)X] - n(Y)n(W)X
n(n -1) L 1 n(n -1)
Y Y
+ nn-ï) g(X, Y)n(W )t+g(Y,W) [n(X)t+X
Y -g(Y, W )n(U )X -—*—g(Y, W )g(X, U)t
n(n - 1) n(n - 1)'
r r
+ nn-1)g(Y,W)X - njn-ï)g(Y,W)n(X)t (36)
+ [g(X,g(Y, W)t - n(W)Y)t - g(t,g(Y, W)t - n(W)Y)x
1[]r
-g(Y, W) [(n - 1)n(X)t - (n - 1)X] + g(Y, W)n(X)t
(n -1) ' L J n(n -1)'
Y 1
2g(Y, W)X + -—n(W) [(n - 1)g(X, Y)t - (n - 1)n(Y)X]
n(n - 1)26V ' (n - 1)
r r
n(W )S(X, Y)t -, u n(W )n (Y)X = 0.
n(n - 1)2 n(n - 1)
Taking the inner product of (36) with £, we find
r 2r
-2n(W)n(Y)n(X) - 2n(W)g(X,Y) + n(W)g(X,Y) + n(W)n(Y)n(X)
2r r (37)
+ nn-T) n (X)g(Y,W) + nn-W n(W )S(X,Y) = °
Again, taking W = £ and using (4) in (37), we get
S(X, Y) = [fci)n(X)n(Y) + (n - r)(n - 1)g(X, Y) (38)
So, M is a n-Einstein manifold.
Conversely, if M is a n-Einstein manifold, then it is easy to show that W2.Z - Z.W2 = 0. Our theorem is thus proved.
Theorem 5. Let M be an n-dimensional (n > 3) LP-Kenmotsu manifold. M satisfies the condition
W2.Z + Z.W2 = 0 if and only if M is an Einstein manifold.
Proof. Let M satisfy the condition W2.Z + Z.W2 = 0. Then from (33) and (34) we can write
1[] 2R(V, X)R(Y, U)W + —j [g( V, R(Y, U)W)QX - g(X, R(Y, U)W)QV]
r 1
-g(U, W) [R(V, X)Y + -— (g(V, Y)QX - g(X, Y)QV)]
n(n - 1) ' n n - 1
+ g(Y, W) [R(V, X)U + ^ (g(V, U)QX - g(X, U)QV)]
- ^t-j [g(X, R(Y, U)W)V - g(V, R(Y, U)W)X]
1 Y + ^g(Y, W) [R(V, X)QU - ^—j) (g(X, QU)V - g(V, QU)X)]
1 - g(U, W )[R(V, X)QY - (g(X, QY)V - g(V, QY)X)] = 0.
n - ' L V n(n - 1)
Therefore, replacing V with £ in (39), we have
1
2R(£, X)R(Y, U)W + — [g(£, R(Y, U)W)QX - g(X, R(Y, U)W)Q£]
r 1
g(U, W) [R(£, X)Y + — (g(£, Y)QX - g(X, Y)Q£)]
+ g(Y, W) [R(£, X)U + ^ (g(£, U)QX - g(X, U)Q£)]
[g(X, R(Y, U)W)£ - g(£, R(Y, U)W)X]
n(n - 1)
r
n(n - 1)
r
n(n - 1)
1 r
+ g(Y, W) [R(£, X)QU - (g(X, QU)£ - g(^ QU )X)]
1 g(U, W )[R(£, X)QY - r (g(X, QY)£ - g(£, QY)X)] = 0.
n - 1 n(n - 1)
Again, taking Y = £ in (40), we get
1 [ ] 2R(£, X)R(£, U)W + — [g(£, R(£, U)W)QX - g(X, R(£, U)W)Q£]
r 1 1
-g(U, W) [R(£, X)£ - — QX --— n(X)Q£]
n(n -1) ' 1 n -1 n -1
r 1 1 + nn-1) n(W) [R(£,X)U + n-1 n(U)QX - n-1 g(X,U)Q£]
r [g(X, R(£, U)W)£ - g(£, R(£, U)W)X]
n(n -1)
1 r r
+ n-1 n(W) [R(£,X)QU - mn-T)S(X,U )£+nt-)(n - 1)n(U)n(X)]
1 r r
g(U, W) [(n - 1)R(£, X)£ - —— (n - 1)n(X)£ - —— (n - 1)X] = 0.
(40)
(41)
n -1 n(n -1) n(n -1)
Taking the inner product of (41) with £ and using (7), (10), we get
r 1 r
eta(W)g(X,U) - ^n^Y) n(W)gX U) - ^n(W)SX U) + n(n - 1)2S(X, U) = 0. (42)
Again, taking W = £ and using (4) in (42), we get
r 1 r
- g(X,U) - nn-1) g(X,U) + n-1S(X,U) + Mn-wS(X,U) = a (43)
Thus, from (43), we have
S(X, U) = (n - 1)g(X, U) (44)
So, M is an Einstein manifold.
Conversely, if M is an Einstein manifold, then it is easy to show that W2.Z + Z.W2 = 0. Our theorem is thus proved.
Acknowledgements: The authors acknowledge Dr. A. Kameswara Rao, Assistant Professor of G. V. P. College of Engineering for Women for his valuable suggestions in preparation of the manuscript.
Declarations of conflicting interests: The authors declares that there is no conflict of interest.
References
[1] Abdul Haseeb and Rajendra Prasad, Certain results on Lorentzian para-Kenmotsu manifolds, Bulletin of Parana's Mathematical Society, (2018) doi.10.5269/bspm.40607.
[2] B. B. Sinha and K. L. Sai Prasad, A class of almost para contact metric Manifold, Bulletin of the Calcutta Mathematical Society, 87(1995), 307-312.
[3] G. P. Pokharial, Study of a new curvature tensor in a Sasakian manifold, Tensor (N.S.), 36(1982), 222-225.
[4] G. P. Pokhariyal and R. S. Mishra, The curvature tensors and their relativistic significance, Yokohoma Math. J., 18(1970), 105-108.
[5] I Mihai, A. A. Shaikh and U. C. De, On Lorentzian para-Sasakian manifolds, Rendiconti del Seminario Matematico di Messina, (1999),Serie II.
[6] I Mihai and R. Rosca, On Lorentzian P-Sasakian manifolds, Classical Analysis, World Scientific Publ., Singapore, (1992), 155-169.
[7] K. Matsumoto, On Lorentzian Paracontact manifolds, Bulletin of the Yamagata University Natural Science, 12(2) (1989), 151-156.
[8] K. Matsumoto and I Mihai, On a certain transformation in a Lorentzian para-Sasakian manifold, Tensor, N.S., 47(1988), 189-197.
[9] K. Matsumoto, S. Ianus and Ion Mihai, On P-Sasakian manifolds which admit certain tensor fileds, Publ. Math. Debrece, 33(1986), 61-65.
[10] K. L. Sai Prasad and T. Satyanarayana, Some curvature properties on a Special paracontact Kenmotsu manifold with respect to Semi-symmetric connection, Turkish Journal of Analysis and Number Theory, 3(2015), no.4, 94-96.
[11] K. L. Sai Prasad, S. Sunitha Devi and G. V. S. R. Deekshitulu, On a class of Lorentzian para-Kenmotsu manifolds admitting the Weyl-projective curvature tensor of type (1,3), Italian Journal of Pure and Applied Mathematics, 45(2021), 990-1001.
[12] K. L. Sai Prasad, S. Sunitha Devi and G. V. S. R. Deekshitulu, On a class of Lorentzian paracontact metric manifolds, Italian Journal of Pure and Applied Mathematics, Accepted for publication in 2021, to appear in 2022 issue.
[13] Rajendra Prasad, Abdul Haseeb and Umesh Kumar Gautam, On ^-semisymmetric LP-Kenmotsu manifolds with a QSNM-connection admitting Ricci solitons, Kragujevac Journal of Mathematics, 45(5),(2021), 815-827.
[14] U. C. De and Avijit Sarkar, On a type of P-Sasakian manifolds, Math. Reports, 11(2009), no.2, 139-144.
[15] Venkatesha and C. S. Bagewadi, On concircular ^-recurrent LP-Sasakian manifolds, Differ. Geom. Dyn. Syst., 10 (2008), 312-319.
[16] Venkatesha and C. S. Bagewadi and K. T. Pradeep Kumar, Some results on Lorentzian para-Sasakian manifolds, ISRN Geometry, Vol. 2011, Article ID 161523, 9 pages, (2011). doi: 10.5402/2011/161523.
[17] Venkatesha, K. T. Pradeep Kumar and C. S. Bagewadi, On Lorentzian Para-Sasakian manifolds satisfying W2 curvature tensor, IOSR J. of Mathematics, 9(6), (2014), 124-127.
