International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(P): 2249-6955; ISSN(E): 2249-8060 Vol. 5, Issue 6, Dec 2015, 37-44 © TJPRC Pvt. Ltd
FLAT H-CURVATURE TENSORS ON HSU-STRUCTURE MANIFOLD LATA BISHT1 & SANDHANA SHANKER2 1 2
Department of Applied Science, BTKIT, Dwarahat, Almora, Uttarakhand, India
Assistant Professor, Department of Mathematics, REVA University, Bangalore, Karnataka, India
ABSTRACT The main object of this paper is to study the flatness of the Bochner flat, H-Projectively flat, HConharmonically flat, H-Concircurally flat, Conharmonically* flat and Conformally* flat Hsu-structure manifold. KEYWORDS: Riemannian Curvature Tensor, Bochner Curvature Tensor, H-Projective Curvature Tensor, H- ConHarmonic Curvature Tensor, H-Con-Circular Curvature Tensor, Conharmonic* Curvature Tensor, Conformal* Curvature Tensor and Hsu-Structure Manifold
Received: Nov 10, 2015; Accepted: Nov 16, 2015; Published: Nov 21, 2015; Paper Id.: IJMCARDEC20154
1. INTRODUCTION If on an even dimensional manifold Vn, n = 2m of differentiability class C∞, there exists a vector valued real linear function
φ , satisfying
φ 2 = ar I n ,
(1.1a)
φ 2 X = a r X , for arbitrary vector field X.
(1.1b)
where X = φX , 0 ≤ r ≤ n and ' a ' is a real or imaginary number. Then { φ } is said to give to Vn a Hsu-structure defined by the equations (1.1) and the manifold Vn is called a Hsu-structure manifold. Remark (1.1): The equation (1.1) a gives different structure for different values of ' a ' and 'r'. If r = 0, it is an almost product structure, if a = 0, it is an almost tangent structure, if r = ±1 and a = +1, it is an almost product structure, if r = ±1 and a = −1, it is an almost complex structure, if
r = 2 then it is a GF-structure
which includes π-structure for a ≠ 0, an almost complex structure for a = ±i, an almost product structure for a = ±1, an almost tangent structure for a =0. Let the Hsu-structure be endowed with a metric tensor g, such that
g (φX ,φY ) + a r g ( X , Y ) = 0 . Then {φ, g} is said to give to Vn - metric Hsu-structure and Vn is called a metric Hsu-structure manifold. Agreement(1.1): In what follows and the above, the equations containing X,Y,Z., etc. hold for these arbitrary vector in Vn. The curvature tensor K, a vector -valued tri-linear function w.r.t. the connexion D is given by
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Lata Bisht & Sandhana Shanker
R( X , Y ) Z = D X DY Z − DY D X Z − D[ X ,Y ] Z ,
(1.2a)
where [ X , Y ] = D X Y − DY X .
(1.2b)
R (φX , φY )Z = R ( X , Y )Z
(1.3)
The curvature tensor for the manifold of constant curvature tensor
H
is given by
R( X , Y )Z = H {g (Y , Z )X − g ( X , Z )Y }
(1.4)
The Ricci tensor in Vn is given by
Ric(Y , Z ) = (C11R)(Y , Z ).
(1.5)
1 where by (C1 R)(Y , Z ) , we mean the contraction of R ( X , Y ) Z with respect to first slot.
For Ricci tensor, we also have
Ric(Y , Z ) = Ric(Z , Y ),
(1.6a)
Ric(Y , Z ) = g (γ (Y ), Z ) = g (Y , γ (Z )),
(1.6b)
(C11γ ) = k
(1.6c)
Let B, P, U, T, L∗ and C ∗ be the bochner, H-Projective, H-Conharmonic, H-Concircular, Conharmonic* and conformal* curvature tensors in Hsu-structure manifold is defined as [7] B( X ,Y )Z = R( X ,Y )Z +
1
(n + 4)
[Ric(X , Z)Y − Ric(Y, Z)X + g( X , Z )γ (Y ) − g(Y, Z)γ ( X ) + Ric(φX , Z)φY − Ric(φY, Z)φX + g(φX , Z )γ (φY )
− g(φY , Z )γ (φX ) + 2Ric(φX ,Y )φZ + 2g(φX ,Y )γ (φZ )] −
k
(n + 2)(n + 4)
[g( X , Z )Y − g(Y , Z )X + g(φX , Z )φY − g(φY , Z )φX
+ 2 g (φ X , Y )φ Z ]
(1.7) 1 [ Ric (Y , Z ) X − Ric ( X , Z )Y − Ric (Y , φZ )φX + Ric ( X , φZ )φY + 2 Ric ( X , φY )φZ ]. ( n + 2)
P( X , Y ) Z = R( X , Y ) Z −
1 [ Ric ( X , Z )Y − Ric(Y , Z ) X + Ric(φX , Z )φY − Ric(φY , Z )φX + 2 Ric(φX , Y )φZ (n + 4) g ( X , Z )γ (Y ) − g (Y , Z )γ ( X ) + g (φX , Z )γ (φY ) − g (φY , Z )γ (φX ) + 2 g (φX , Y )γ (φZ )].
(1.8)
U ( X , Y )Z = R( X , Y ) Z +
T ( X , Y )Z = R( X , Y )Z −
k [ g (Y , Z ) X − g ( X , Z )Y + g (φY , Z )φX − g (φX , Z )φY − 2 g (φX , Y )φZ ]. n ( n + 2)
L∗ ( X , Y ) Z = R ( X , Y ) Z +
Impact Factor (JCC): 4.6257
k [g (Y , Z ) X − g ( X , Z )Y ] ( n − 2)(n − 1)
(1.9) (1.10)
(1.11)
NAAS Rating: 3.80
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Flat H-Curvature Tensors on HSU-Structure Manifold
C ∗ ( X , Y )Z = R( X , Y )Z −
1 k [g (Y , Z )γ ( X ) − g ( X , Z )γ (Y )] + [g (Y , Z ) X − g ( X , Z )Y ] (n − 2) (n − 2)(n − 1)
(1.12)
THEOREM (2.1) A Bochner flat Hsu-structure manifold V n if n 2 − 2na r − n 2 a 2 r − 5na 2 r − 4 − 8a r − 4a 2 r + 3n ≠ 0 is flat. Proof: If the manifold is Bochner flat then from equaton (1.7), we get
(n + 4)R( X ,Y )Z = Ric(Y , Z )X − Ric( X , Z )Y − g( X , Z )γ (Y ) + g(Y, Z )γ ( X ) − Ric(φX , Z )φY + Ric(φY, Z )φX − g(φX , Z )γ (φY ) + g(φY , Z )γ (φX ) − 2Ric(φX ,Y )φZ − 2g(φX ,Y )γ (φZ ) +
k
(n + 2)
[g( X , Z )Y − g(Y , Z )X + g(φX , Z )φY − g(φY , Z )φX + 2g(φX ,Y )φZ ].
(2.1)
Now applying φ on X and Y in equation (2.1) and using equation (1.3), we get
(n + 4)R( X ,Y )Z = Ric(φY, Z )φX − Ric(φX , Z )φY − g(φX , Z )γ (φY ) + g(φY, Z )γ (φX ) − Ric(φ 2 X , Z )φ 2Y + Ric(φ 2Y, Z )φ 2 X − g(φ 2 X , Z )γ (φ 2Y )
(
)( )
(
)
(
)
+ g φ 2Y, Z γ φ 2 X − 2Ricφ 2 X ,φY φZ − 2g φ 2 X ,φY γ (φZ ) +
(
k
(n + 2)
[g(φX, Z)φY − g(φY, Z)φX + g(φ X, Z)φ Y − g(φ Y, Z)φ X 2
2
2
2
) ]
+ 2g φ 2 X ,φY φZ
(2.2)
Comparing the equation (2.1) and (2.2) then we get Ric(Y , Z )X − Ric( X , Z )Y − g( X , Z )γ (Y ) + g (Y , Z )γ ( X ) − Ric(φX , Z )φY + Ric(φY , Z )φX − g (φX , Z )γ (φY ) + g (φY , Z )γ (φX ) − 2Ric(φX , Y )φZ − 2g(φX ,Y )γ (φZ ) +
k
(n + 2)
[g( X , Z )Y − g(Y , Z )X + g(φX , Z )φY − g(φY , Z )φX + 2g(φX ,Y )φZ ]
(
)
(
)
( )( ) k [g(φX , Z )φY − g(φY , Z )φX + g(φ X , Z )φ Y − g(φ Y , Z )φ + g (φ Y , Z )γ (φ X ) − 2Ric(φ X ,φY )φZ − 2g (φ X , φY )γ (φZ ) + (n + 2) = Ric(φY , Z )φX − Ric(φX , Z )φY − g(φX , Z )γ (φY ) + g (φY , Z )γ (φX ) − Ric φ 2 X , Z φ 2Y + Ric φ 2Y , Z φ 2 X − g φ 2 X , Z γ φ 2Y 2
2
(
) ]
2
2
2
+ 2g φ 2 X ,φY φZ
2
2
2
X
(2.3)
Ric(Y , Z )X − Ric( X , Z )Y − g ( X , Z )γ (Y ) + g (Y , Z )γ ( X ) − Ric(φX , Z )φY + Ric(φY , Z )φX − g (φX , Z )γ (φY ) + g (φY , Z )γ (φX ) − 2Ric(φX , Y )φZ − 2g (φX , Y )γ (φZ ) +
k [g ( X , Z )Y − g(Y , Z )X + g(φX , Z )φY − g(φY , Z )φX + 2g(φX ,Y )φZ ] (n + 2)
= Ric(φY, Z )φX − Ric(φX, Z )φY − g(φX, Z )γ (φY ) + g(φY, Z)γ (φX ) − a2r Ric( X, Z)Y + a2r Ric(Y, Z)X − a2r g( X, Z )γ (Y ) + a2r g(Y, Z)γ ( X ) − 2ar Ric( X,φY )φZ − 2ar g( X,φY )γ (φZ ) +
[
k g(φX, Z )φY − g(φY, Z)φX + a2r g( X, Z )Y − a2r g(Y, Z)X + 2ar g( X,φY )φZ (n + 2)
]
(2.4) Contracting of equation (2.4) w.r.t X, we get
(n
2
− 4na r − n 2 a 2r − 4na 2r − 4 − 8a r − 4a 2r Ric(Y , Z ) = k a 2r − 2a r − 3 g (Y , Z )
)
(n
2
− 4na r − n 2 a 2r − 4na 2r − 4 − 8a r − 4a 2r RY − k a 2r − 2a r − 3 Y = 0
)
(
(
)
)
(2.5a) (2.5b)
Again contracting equation (2.5b), we get
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Lata Bisht & Sandhana Shanker
(n
2
)
− 2na r − n 2 a 2r − 5na 2r − 4 − 8a r − 4a 2r + 3n k = 0
If n 2 − 2na r − n 2 a 2 r − 5na 2 r − 4 − 8a r − 4a 2 r + 3n ≠ 0 Then k = 0 . Putting k = 0 in equation (2.5a), we get Ric=0, since n 2 − 2na r − n 2 a 2 r − 5na 2 r − 4 − 8a r − 4a 2 r + 3n ≠ 0 . Putting k = 0 and Ric=0 in equation (2.1), we get R=0, which proves the statement.
[(
]
)
THEOREM (2.2): A H-Projectively flat Hsu-structure manifold V n , n 1 − a 2r + a 2r − 4a r − 1 ≠ 0 is flat. Proof: If the manifold is H-Projectively flat then from equaton (1.8), we get
(n + 2 )R ( X , Y )Z = Ric (Y , Z )X − Ric ( X , Z )Y − Ric (Y , φZ )φX + Ric ( X , φZ )φY + 2 Ric ( X , φY )φZ
(2.6)
Now applying φ on X and Y in equation (2.6) and using equation (1.3), we get
(n + 2 )R ( X , Y )Z
(
)
= Ric (φY , Z )φX − Ric (φX , Z )φY − Ric (φY , φZ )φ 2 X + Ric (φX , φZ )φ 2 Y + 2 Ric φX , φ 2 Y φZ
(2.7)
Comparing the equation (2.6) and (2.7) then we get Ric(Y , Z )X − Ric( X , Z )Y − Ric(Y ,φZ )φX + Ric( X ,φZ )φY + 2Ric( X ,φY )φZ = Ric(φY , Z )φX − Ric(φX , Z )φY − Ric(φY , φZ )φ 2 X
(
)
+ Ric(φX , φZ )φ 2Y + 2Ric φX , φ 2Y φZ .
(2.8)
Ric(Y , Z )X − Ric( X , Z )Y − Ric(Y , φZ )φX + Ric( X , φZ )φY + 2Ric( X , φY )φZ = Ric(φY , Z )φX − Ric(φX , Z )φY − a r Ric(φY ,φZ )X
+ a r Ric(φX , φZ )Y + 2a r Ric(φX , Y )φZ .
(2.9)
Contracting of equation (2.9) w.r.t X, we get
{n(1 − a ) − 4a 2r
r
}
+ a 2r − 1 Ric(Y , Z ) = 0 .
{(
)
}
From theorem if n 1 − a 2r − 4a r + a 2r − 1 ≠ 0 Hence Ric (Y , Z ) = 0 . Now putting Ric = 0 in equation (2.6), we get R = 0 . Hence the theorem. THEOREM (2.3): A H-Conharmonically flat Hsu-structure manifold V n , n − 1 − 2a r − na 2 r − a 2 r ≠ 0 is flat. Proof: If the manifold is H-Conharmonically flat then from equation (1.9), we get
(n + 4)R( X ,Y )Z = Ric(Y, Z)X − Ric( X , Z)Y − Ric(φX, Z)φY + Ric(φY, Z)φX − 2Ric(φX ,Y )φZ − g( X, Z)γ (Y) + g(Y, Z)γ ( X ) − g (φX , Z )γ (φY ) + g (φY , Z )γ (φX ) − 2 g (φX , Y )γ (φZ )
(2.10)
Now applying φ on X and Y in equation (2.10) and using equation (1.3), we get
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Flat H-Curvature Tensors on HSU-Structure Manifold
(n + 4)R( X,Y)Z = Ric(φY, Z)φX − Ric(φX, Z)φY − Ric(φ 2 X, Z)φ 2Y + Ric(φ2Y, Z)φ 2 X − 2Ric(φ 2 X,φY )φZ − g(φX, Z)γ (φY)
(
)
(
)
(
)
+ g (φY , Z )γ (φX ) - g φ 2 X , Z γ (φ 2Y ) + g φ 2Y , Z γ (φ 2 X ) − 2g φ 2 X ,φY γ (φZ )
(2.11)
Comparing the equation (2.10) and (2.11) then we get Ric(Y, Z )X − Ric( X , Z )Y − Ric(φX , Z )φY + Ric(φY, Z )φX − 2Ric(φX ,Y )φZ − g( X , Z)γ (Y) + g(Y, Z)γ ( X ) − g(φX , Z )γ (φY) + g(φY, Z )γ (φX )
(
)
(
)
(
)
− 2g(φX ,Y )γ (φZ ) = Ric(φY , Z )φX − Ric(φX , Z )φY − Ric φ 2 X , Z φ 2Y + Ric φ 2Y , Z φ 2 X − 2Ric φ 2 X ,φY φZ − g(φX , Z )γ (φY )
(
)
(
)
(
)
+ g (φY , Z )γ (φX ) - g φ 2 X , Z γ (φ 2Y ) + g φ 2Y , Z γ (φ 2 X ) − 2g φ 2 X ,φY γ (φZ )
(2.12)
Ric(Y, Z )X − Ric( X , Z )Y − Ric(φX , Z )φY + Ric(φY, Z )φX − 2Ric(φX ,Y )φZ − g( X , Z)γ (Y ) + g(Y, Z)γ ( X ) − g(φX , Z )γ (φY ) + g(φY, Z )γ (φX )
− 2g (φX , Y )γ (φZ ) = Ric(φY , Z )φX − Ric(φX , Z )φY − a 2r Ric( X , Z )Y + a 2r Ric(Y , Z )X − 2a r Ric( X ,φY )φZ − g (φX , Z )γ (φY )
+ g (φY , Z )γ (φX ) - a 2r g ( X , Z )γ (Y ) + a 2r g (Y , Z )γ ( X ) − 2a r g ( X ,φY )γ (φZ )
(2.13)
Contracting the equation (2.13) w.r.t X, we get
(n − 2 − 6a
r
+ 2a r − na2r − 2a 2r Ric(Y , Z ) − k a 2r − 1 g (Y , Z ) = 0
)
(
)
(2.14)
(n − 2 − 6a
r
+ 2a r − na 2r − 2a 2r RY − k a 2r − 1 Y = 0
(2.15)
)
(
)
Contracting the equation (2.15), we get
{
}
{
}
2 n − 1 − 2a r − na 2r − a 2r k = 0
If n − 1 − 2a r − na 2r − a 2r ≠ 0 , then k = 0 .
{
}
Putting k = 0 in equation (2.14), we get Ric=0, since n − 1 − 2a r − na 2r − a 2r ≠ 0 . Putting k = 0 and Ric=0 in equation (2.10), we get R=0, which proves the statement. THEOREM (2.4): A H-Concircularlly flat Hsu-structure manifold V n , n ≠ 1 + 2a r + na 2 r + a 2 r is flat. Proof: If the manifold is H-Concircularlly flat then from equation (1.10), we get n(n + 2 )R ( X , Y ) Z = k[ g (Y , Z ) X − g ( X , Z )Y + g (φY , Z )φX − g (φX , Z )φY − 2 g (φX , Y )φZ ].
(2.16)
Now applying φ on X and Y in equation (2.16) and using equation (1.3), we get
(
)
(
)
(
)
n(n + 2 )R ( X , Y ) Z = k [ g (φY , Z )φX − g (φX , Z )φY + g φ 2Y , Z φ 2 X − g φ 2 X , Z φ 2Y − 2 g φ 2 X , φY φZ ].
(2.17)
Comparing equation (2.16) and (2.17) and using equation (1.3), we get g (Y , Z ) X − g ( X , Z )Y + g (φY , Z )φX − g (φX , Z )φY − 2 g (φX , Y )φZ == g (φY , Z )φX − g (φX , Z )φY
(
)
(
)
(
)
+ g φ 2Y , Z φ 2 X − g φ 2 X , Z φ 2Y − 2g φ 2 X , φY φZ
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(2.18)
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42
Lata Bisht & Sandhana Shanker
g (Y , Z ) X − g ( X , Z )Y + g (φY , Z )φX − g (φX , Z )φY − 2 g (φX , Y )φZ == g (φY , Z )φX − g (φX , Z )φY
+ a 2r g (Y , Z )X − a 2r g ( X , Z )Y − 2a r g ( X , φY )φZ
(2.19)
Contracting the equation (2.19) w.r.t X, we get
{n −1 − 2a
r
}
− na2r − a 2r g (Y , Z ) = 0
{
}
If n − 1 − 2a r − na 2r − a 2r ≠ 0 then g (Y , Z ) = 0 . Putting g (Y , Z ) = 0 in equation (2.16), we get R=0. Which proves the theorem. THEOREM (2.5): A Conharmonically* flat Hsu-structure manifold V n , n ≠ 1 − a r is flat. Proof: If the manifold is Conharmonically* flat then from equation (1.11), we get
(n − 1)(n − 2)R( X , Y ) Z = k [g ( X , Z )Y − g (Y , Z )X ] Applying
(2.20)
φ on X and Y in (2.20) and using equation (1.3), we get
(n − 1)(n − 2)R( X , Y )Z = k [g (φX , Z )φY − g (φY , Z )φX ]
(2.21)
Comparing the equation (2.20) and (2.21) then we get g ( X , Z )Y − g (Y , Z )X = g (φX , Z )φY − g (φY , Z )φX
Contracting of this equation w.r.t X, we get
(1 − n − a )g(Y , Z ) = 0 r
(2.22)
If n ≠ 1 − a r then g (Y , Z ) = 0 Putting g (Y , Z ) = 0 in equation (2.20), we get R=0, this proves the theorem. THEOREM (2.6): A Conformally* flat Hsu-structure manifold V n , n ≠ 1 − a r is flat. Proof: If the manifold is Conharmonically* flat then from equation (1.12), we get
(n − 2)R( X , Y )Z = g (Y , Z )γ ( X ) − g ( X , Z )γ (Y ) + Applying
k
(n − 1)
[g ( X , Z )Y − g (Y , Z )X ]
(2.23)
φ on X and Y in (2.23) and using equation (1.3), we get
(n − 2)R( X , Y )Z = g (φY , Z )γ (φX ) − g (φX , Z )γ (φY ) +
k [g (φX , Z )φY − g (φY , Z )φX ] (n − 1)
(2.24)
Comparing the equation (2.23) and (2.24) then we get g(Y , Z )γ ( X ) − g( X , Z )γ (Y ) +
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k
(n −1)
[g( X , Z )Y − g(Y , Z )X ] = g(φY , Z )γ (φX ) − g(φX , Z )γ (φY ) +
k
(n −1)
[g(φX , Z )φY − g(φY , Z )φX ]
NAAS Rating: 3.80
43
Flat H-Curvature Tensors on HSU-Structure Manifold
Contracting of this equation w.r.t X, we get
(n − 1)(a r − 1)Ric(Y , Z ) − kar g(Y , Z ) = 0
(2.25a)
Or
(n − 1)(a r − 1)RY − ka r Y = 0
(2.25b)
Again contracting the equation (2.25b), we get
(1 − n − a )k = 0 r
(2.25)
If n ≠ 1 − a r then k = 0
CONCLUSIONS Putting k = 0 in equation (2.25b), we get Ric=0, since n ≠ 1 − a r . Putting k = 0 and Ric=0 in equation (2.23), we get R=0, which proves the statement. REFERENCES 1.
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Duggal, K. L., (1971). On differentiable structures defined by algebraic equations II, F-connexions, Tensor, N. S., 22, 255260.
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Habeeb M. Abood., (2010.) Almost Hermatian manifold with flat Bochner tensor, European Journal of pure and applied mathematics, 3(4), 730-736.
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Mishra, R. S. and Pandey, S. B., (1974). On H-Conharmonic curvature tensor in Kähler manifolds, Ind. J. Pure and Appl. Math, 7(11), 1277-1287.
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Mishra, R. S. and Singh, S. D., (1975). Some properties of Pseudo H-Projective curvature tensor in a differentiable manifolds equipped with KH-structures, Ind. J. Pure. And Appl. Math., 6, 444-450.
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Musha Jawarneh and Mohammad Tashtoush., (2011). M-Projective curvature tensor on kaehlar manifold, Int. J. Contemp. Math. Sciences, Vol. 6, no. 33, 1607 – 1617.
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Zhen, G. Cabrerizo, J. L., Fernandez, L. M. and Fernandez, M., (1997). On conformally flat at contact metric manifolds, Indian J. Pure Appl. Math., 28, 725-734.
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Eisenhart, L. P., (1966). Riemannian Geometry, Princeton University Press.
10. Kirichenko V. F., (1977). Differential geometry of K-spaces, Itogi, Nauki I Tehniki., VINITI, Moscow, problem geometric, 8,139-161. 11. Kobayashi S., and Nomizu K., (1969). Foundations of differential Geometry, V.II, John Wiley and Sons. 12. Mishra, R. S., (1973). A course in tensors with applications to Riemannian Geometry, II edition Pothishala Pvt. Ltd., Allahabad. 13. Mishra, R. S., (1984). Structure on a differentiable manifold and their applications, Chandrama Prakashan, 50-A, Balrampur House, Allahabad. www.tjprc.org
editor@tjprc.org.