6_340_singh.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 5, 2010, 839-852

ISSN 1307-5543 – www.ejpam.com

Some Affine Connexions in a Generalised Structure Manifold-II

R. P. Singh∗and S. D. Singh

Department of Mathematics, Faculty of science, Banaras Hindu University, Varanasi - 221005, (U.
P.) INDIA.

Abstract. In this paper we have studied some affine connexions in a generalised structure manifold.

Certain theorems are also have been proved which are of great geometrical importance.

2000 Mathematics Subject Classifications: 53B05, 57P05, 57R55

Key Words and Phrases: C∞-manifold, Generalised structure manifold, π-structure manifold, Hsu-

structure manifold, F -structure manifold, tangent metric manifold

1. Introduction

We consider a differentiable manifold Vn of differentiability class C∞ and of dimension n.

Let there exist in Vn a tensor field F of the type (1,1), s linearly independent vector fields Ui,

i = 1, 2, . . ., s and s linearly independent 1-forms ui such that for any arbitrary vector field X ,

we have

X = b2X + c ui(X )Ui (1)

Ui = p j
i U j (2)

where F(X )
de f
= X and b2, c are constants.

Then the structure {F,ui , Ui, p j
i ; i, j = 1,2, . . . , s} will be known as generalised structure and

Vn will be known as generalised structure manifold of order s where s < n.

Lemma 1. All the equations which follow hold for arbitrary vector fields X , Y, Z , . . . etc.

Now, replacing X by X in (1),we get

X = b2X + c ui(X )Ui (3)

∗Corresponding author.

Email address: rajabhaia�gmail.
om (R. Singh)

http://www.ejpam.com 839 c© 2010 EJPAM All rights reserved.



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 840

Operating F in (1), we get

X = b2X + c ui(X )Ui

Using (2) in above, we get

X = b2X + c ui(X )p j
i U j (4)

From (3) and (4), we have

ui(X ) = pi
ju

j(X ) (5)

Further, operating F in (2) and using (1), (2) we get

(2)p j
i = b2δ

j
i + c u j(Ui) (6)

where
(r)pi

j =
(r−1) pi

k pk
j

On generalised structure manifold Vn, let us introduce a metric tensor g such that 2-form

′F defined by ′F(X , Y )
de f
= g(X , Y ) is skew-symmetric,then Vn is called generalised metric

structure manifold [8,12].

We have on a generalised metric structure manifold,

g(X , Y ) + g(X , Y ) = 0

Replacing Y by Y in above equation and using (1), we obtain

g(X , Y ) + b2 g(X , Y ) + c ui(X )ui(Y ) = 0 (7)

where

ui(X ) = g(Ui, X ) (8)

Lemma 2. The generalised metric structure manifold always be denoted by Vn.

1.1. Definitions

This section consists of well known definitions required to go through the insuring sections

[1,6].

1. A differentiable manifold Mn on which there a vector valued linear function F , a 2-form

′F defined by ′F(X , Y )
de f
= g(X , Y ) such that

• F2 = 0 and ′F(X , Y ) is skew-symmetric, then Mn is called an almost tangent metric
manifold.

• F2 = −In and ′F(X , Y ) is skew-symmetric, then Mn is called an almost Hermite
manifold.

• F2 = λ2 In, where λ is a non-zero complex constant and ′F(X , Y ) is skew-symmetric,

then Mn is called an metric π-structure manifold [13].



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 841

• F2 = λr In and ′F(X , Y ) is skew-symmetric, then Mn is called an Hsu-structure
metric manifold [4,5].

• F2 = In and ′F(X , Y ) is symmetric, then Mn is said to be an almost product
Riemannian manifold [7].

2. Let us consider a C∞-manifold Mn (n = 2m+ 1). Let there exist in Mn a tensor field F
of the type (1,1), a 1-form u, a vector field U and a Riemannian metric g satisfying

X = −X + u(X )U (9a)

U = 0 (9b)

g(X , Y ) = g(X , Y )− u(X )u(X ) (9c)

where g(X , U) = u(X ) and F(X )
de f
= X

Then Mn is called an almost contact metric manifold or an almost Grayan manifold
[16,17].

3. We consider a manifold Mn of differentiability class C∞. Let there exist in Mn, a tensor

field F of the type (1,1) and rank r (1≤ r ≤ n) satisfying

F3 + F = 0 (10)

then {F} is called F -structure and Mn satisfying (10) is called F -structure manifold [2].

If we consider ′F(X , Y )
de f
= g(X , Y ) where g is a Riemannian metric and ′F is skew-

symmetric then, F -structure manifold Mn is called a metric F -structure manifold.

4. The tensor K of the type (1,3) defined by [14]

K(X , Y, Z)
de f
= DX DY Z − DY DX Z − D[X ,Y]Z (11)

is called the curvature tensor of the connexion D.

5. The vector field Ui in generalised structure metric manifold Vn is called a Killing vector

if it satisfies [11]

DX ui)(Y ) + (DY ui)(X ) = 0

6. A connection D which satisfies

(DX F)(Y ) = 0, (DX ui)(Y ) = 0, DX Ui = 0 (12)

is called an (F, Ui ,u
i)-connexion.

7. A connection D is called an F -connexion if it satisfies

(DX F)(Y ) = 0 i.e., DX Y = DX Y (13)



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 842

8. Nijenhuis tensor is a vector valued bilinear function N , given by

N(X , Y )
de f
= [ X , Y ] + [X , Y ]− [X , Y ]− [X , Y ] (14)

9. A vector valued, skew-symmetric, bilinear function S defined by

S(X , Y )
de f
= DX Y − DY X − [X , Y ] (15)

is called torsion tensor of a connexion D.

For symmetric or torsion free connexion D, the torsion tensor vanishes, i.e.

DX Y − DY X = [X , Y ] (16)

10. Lie derivative along any vector V in a C∞-manifold Mn is a type preserving mapping

such that [3]

LV f = V f ; f is a C∞–function (17a)

LV X = [V, X ] (17b)

LV B(X ) = V (B(X ))− B([V, X ]) (17c)

where B is an arbitrary 1-form.

Remark 1. It may be noted that Vn gives an almost tangent metric manifold, an almost Hermite
manifold, metric π-structure manifold, Hsu-structure manifold, F-structure manifold, an almost
product Riemannian manifold, an almost Grayan manifold and {F, g,u1,u2, U1, U2} structure
manifold according as (b2 = 0, c = 0) ; (b2 = −1, c = 0) ; (c = 0) ; (b2 = λr , c = 0) ;
(b2 = −1, p j

i = 0) ; (b2 = 1, c = 0) ; (b2 = −1, c = 1, p1
1 = 0 : i, j = 1) ; and (b2 = −1, c =

1, p j
i + pi

j = 0 : i, j = 1,2) respectively.

1.2. Some Basic Results

1. If we put, ρF ′ = Fρ , U ′i =
−1 ρUi and u′i = ui ◦ ρ, where ρ is a non-singular tensor of

the type (1,1), then it can be easily seen that {F ′,u′i , U ′i , p j
i ; i, j = 1,2, . . . , s} is also a

generalised structure.

2. The eigen values of F are given by b,−b,
p

Ai,−
p

Ai where Ai are the roots of the

equation |λ2δ
j
i −
(2) p j

i | = 0. The multiplicity of the eigen values depends on rank((F)),

on p j
i and the nature of b2, c.



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 843

2. An Affine Connexion I

In this section an affine connexion in a generalised structure manifold Vn is defined and

its properties have been studied [9,10,15].

Let us define an affine connexion D such that

ui(Y )(DX Ui) + (DX ui)(Y )Ui = 0 (18)

where D is an F -connexion given by (13). It can be easily seen that,

ui(DX Ui) = −(DX ui)(Y ) (19a)

((2)pi
j − b2δi

j)(DX Ui) = ui(DX Ui)U j (19b)

(DX ui)(Y )(Ui) = −pi
ju

j(Y )(DX Ui) (19c)

((2)pi
j − b2δi

j) div U j = c u j(DU j
Ui) (19d)

where div(X )
de f
= (C1

1∇X ) and (∇X )(Y ) = (DY X ).

Theorem 1. In Vn, let us put

M(X , Y )
de f
= DX Y + DX Y − DX Y − DX Y (20)

then,
M(X , Y ) = 0 (21)

Proof. Using (13) in (20), we get (21).

Theorem 2. If connexion D is torsion free in Vn, then we have

N(X , Y ) = 0 (22)

where N(X , Y ) is Nijenhuis tensor.

Proof. Using (16) and (13) in (14), we get (22).

Now, corresponding to the Nijenhuis tensor of an almost complex manifold, we have three

tensors µ,ν and σ given by

µ(X , Y )
de f
= (DY ui)(X )− (DX ui)(Y ) + (DY ui)(X )− (DX ui)(Y ) (23)

ν(X )
de f
= (DUi

F)(X )− (DX F)(Ui)− DX Ui (24)

σ(X )
de f
= (DX u j)(Ui)− (DUi

u j)(X ) (25)

respectively.



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 844

Theorem 3. If connexion D is torsion free in Vn, then we have

µ(X , Y )Ui = (DX Ui) [u
i(Y )+ ui(Y )] − (DY Ui) [u

i(X )+ ui(X )] (26)

ν(X ) = −(DX Ui) (27)

σ(X ) = −u j(DX Ui)− (DUi
u j)(X ) (28)

Proof. Using (5), (18) and (19c) in (23), we get (26). The equation (24) yields (27) on

use of (13). (28) is obtained on the use of (19a) in (25).

Theorem 4. In Vn, let us put

C(X , Y ) = (DX ui)(Y )− (DY ui)(X ) (29)

Then, we have

C(X , Y ) + C(X , Y ) = −µ(X , Y ) (30)

C(X , Y )Ui = ui(Y )ν(X )− ui(X )ν(Y ) (31)

Proof. Replacing Y by Y and X by X in (29) separately and adding resulting these two

equations, we get (30). Further, replacing X by X , Y by Y and multiplying with Ui in (29),

we get (31) on use of (5), (19c) and (27).

Corollary 1. In Vn, we have

C(X , Ui) = σ(X ) (32)

C(X , Ui) = (DX u j)(X )− u j(ν(X )) (33)

Proof. By replacing i by j and Y by Ui in (29), we get (32). Further, by replacing X by X
in (32) and using (19c) & (27), we obtain (33).

Theorem 5. In Vn, with Ui as a killing vector, we have

C(X , Ui) = −2ui(DX Ui) (34)

Proof. Considering Ui as a killing vector with respect to connexion D and using (19a) in

(29) after putting Ui for Y , we get (34).

Theorem 6. In Vn, we have

(LX ui)(Y )− (LY ui)(X ) = C(X , Y )− ui(LX Y ) (35)

Proof. Lie derivative of ui is given by

(LX ui)(Y ) = (DX ui)(Y ) + ui(DY X ) (36)

Interchanging X and Y in the above equation and subtracting the resulting equation from

above equation, we get (35) on the use of (29) and (17b).



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 845

Corollary 2. In Vn, we have

(LX ui)(Ui)− (LUi
ui)(X ) = σ(X )− ui(LX Ui) (37)

(LX ui)(Ui)− (LUi
ui)(X ) = (DUi

u j)(X )− u j(ν(X ))− ui(LX Ui) (38)

Theorem 7. In Vn, we have

(LX ui)(Y )− (LY ui)(X ) +µ(X , Y ) = (DY ui)(X )− (DX ui)(Y ) + pi
ju

j([X , Y ]) (39)

Proof. Replacing X by X in (36) and using (13) & (5), we get

(LX ui)(Y ) = (DX ui)(Y ) + pi
ju

j(DY X ) (40a)

Similarly, we can get

(LY ui)(X ) = (DY ui)(X )+ pi
ju

j(DX Y ) (40b)

Subtracting (40b) from (40a) and using (16) and (23), we get the required result.

3. An Affine Connexion II

In this section an affine connexion E has been defined in terms of another affine connex-

ion D such that their torsions are equal but opposite in sign. The properties of this affine

connexion E have been studied in a generalised structure manifold Vn [18].

Let us define an affine connexion E in Vn by the relation

EX Y
de f
= −DX Y + [X , Y ] (41)

where D is an (F, Ui ,u
i)-connexion given by (12) and the torsions of E and D are equal but

opposite in sign.

We shall study
◦
N ,
◦

M and curvature tensors of connexion E.

Remark 2. Since the torsions of the connexions D and E are equal and opposite to each other,
therefore, if D is half symmetric, semi-symmetric and almost symmetric, E is also half symmetric,
semi-symmetric and almost symmetric respectively.

Theorem 8. In Vn, we have

EX Y − EY X = DY X − DX Y + 2[X , Y ] (42a)

EX Ui − EUi
X = DUi

X + 2[X , Ui] (42b)

EX Y − EY X = DY X − DX Y + 2[X , Y ] (42c)

EX Y − EY X = b2(DY X − DX Y ) + cui(DY X − DX Y )Ui + [X , Y ]− [Y, X ] (42d)

EX Y − EY X = DY X − DX Y + [X , Y ]− [Y , X ] (42e)



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 846

Proof. The equation (41) yields (42a). Putting Ui for Y in (42a) and using (12), we get

(42b). Replacing X by X and Y by Y in (42a), we get (42c) on the use of (12). The value

of (EX Y − EY X ) is obtained by making use of (41) and operating F on the resulting equation

and using (1) & (12), we get (42d). Similarly, we can get (42e).

Theorem 9. In Vn, we have

(EX F)(Y ) = [X , Y ]− [X , Y ] (43a)

(EX F)(Ui) = pi
j[X , U j]− [X , Ui] (43b)

Proof. Replacing Y by Y in (41), we have

EX Y = −DX Y + [X , Y ] (44)

which on use of (41) yields (43a) and by replacing Y by Ui and using (5), we get (43b).

Theorem 10. In Vn, we have

c{(EX ui)(Y ) + ui([X , Y ])}= 0 (45)

Proof. Replacing Y by Y in (41) and using (1), we get

EX (b
2Y + cui(Y )Ui) = −DX (b

2Y + cui(Y )Ui) + b2[X , Y ] + cui(Y )[X , Ui] (46a)

Using (1.12) and (3.1) in above, we get

c Ui(EX ui)(Y ) = −c Uiu
i([X , Y ]) (46b)

which implies(45).

Now, let us consider Nijenhuis tensor N(X , Y ) in Vn, which is given by (14). For the

symmetric connexion D, it takes the following form :

N(X , Y ) = DX Y − DY X + DX Y − DY X − DX Y + DY X − DX Y + DY X (47)

Using (1) in above, we get

N(X , Y ) = DX Y − DY X + b2(DX Y − DY X ) + c ui(DX Y − DY X )Ui

−DX Y + DY X − DX Y + DY X
(48)

Similar to Nijenhuis tensor for connexion D, let us introduce a tensor
◦
N (X , Y ) for the

connexion E, given by

◦
N (X , Y )

de f
= EX Y − EY X + b2(EX Y − EY X )+ c ui(EX Y − EY X )Ui

−EX Y + EY X − EX Y + EY X
(49)



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 847

Theorem 11. In Vn, we have

◦
N (X , Y ) = 2([ X , Y ] + [X , Y ]− [X , Y ]− [X , Y ]) = 2N(X , Y ) (50)

Proof. Using (42a), (42c), (42d) and (42e) in (49), we obtain

◦
N (X , Y ) = (DY X − DX Y + 2[ X , Y ]) + b2(DY X − DX Y + 2[X , Y ])

+ c ui(DY X − DX Y + 2[X , Y ])Ui − b2(DY X − DX Y )

− c ui(DY X − DX Y )Ui − [X , Y ] + [Y, X ]

− DY X + DX Y − [X , Y ] + [Y , X ]

= 2([ X , Y ] + b2[X , Y ] + c ui([X , Y ])Ui − [X , Y ]− [X , Y ])

which on use of (1) and (14) yields (50).

Corollary 3. In Vn, we have

◦
N (X , Ui) = 2 pi

j([ X , U j ]− [X , U j]) + 2([X , Ui]− [X , Ui]) (51)

Let us define
◦
µ,
◦
ν ,
◦
σ analogues to µ, ν , σ for connexion E.

◦
µ (X , Y )

de f
= (EY ui)(X )− (EX ui)(Y ) + (EY ui)(X )− (EX ui)(Y ) (52)

◦
ν (X )

de f
= (EUi

F(X )− (EX F)(Ui)− EX Ui (53)

◦
σ (X )

de f
= (EX u j)(Ui)− (EUi

u j)(X ) (54)

Theorem 12. In Vn, we have

c
◦
µ (X , Y )Ui = 2c{ui([X , Y ])+ ui([X , Y ])} (55a)

◦
ν (X ) = {[X , Ui] + 2[X , Ui]− [X , Ui] + EUi

X + DUi
X } (55b)

c
◦
σ (X ) = 2c{u j([Ui, X ])} (55c)

Proof. On account of (45) and (52), we get(55a). Due to (42b) and (43b), we obtain

(55b). Finally, (55c) is obtained by using (45) in (54).

Corollary 4.
◦
µ (X , Y ) is skew-symmetric in both the slots X and Y , i.e.

◦
µ (X , Y )+

◦
µ (Y, X ) = 0 (56)

Let us define a vector valued, bilinear function
◦

M by

◦
M (X , Y )

de f
= EX Y + EX Y − EX Y − EX Y (57)



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 848

Theorem 13. In Vn, we have

◦
M (X , Y )− [ X , Y ]− [X , Y ] + [X , Y ] + [X , Y ] = 0 (58a)

◦
M (X , Y )−N(X , Y ) = 0 (58b)

Proof. Using (12), (14) and (41) in (57), we get (58a) and (58b).

Corollary 5.
◦

M (X , Y ) is skew-symmetric in both the slots X and Y , i.e.

◦
M (X , Y )+

◦
M (Y, X ) = 0 (59)

Corollary 6. In, Vn, we have

◦
M (X , Ui) = [X , Ui]− [X , Ui] + pi

j{[X , U j]− [X , U j]} (60)

It can be obtained that,

K(X , Y, Ui) = 0 (61a)

K(X , Y, Z) = K(X , Y, Z) (61b)

where K is the curvature tensor of (F, Ui ,u
i)-connexion.

Let us define a curvature tensor
◦
K with respect to connexion E, by

◦
K (X , Y, Z)

de f
= EX EY Z − EY EX Z − E[X ,Y]Z (62)

Theorem 14. In, Vn, we have

◦
K (X , Y, Z) = K(X , Y, Z) + 2D[X ,Y]Z − [X , DY Z] + [Y, DX Z]− DX ([Y, Z]) + DY ([X , Z]) (63)

Proof. From (41), we have

EX EY Z = DX DY Z − [X , DY Z]− DX ([Y, Z])+ [X , [Y, Z]] (64)

−EY EX Z = −DY DX Z + [Y, DX Z] + DY ([X , Z])− [Y, [X , Z]] (65)

−E[X ,Y]Z = −D[X ,Y]Z − [[X , Y ], Z] (66)

Adding (64), (65) and (66) and using (11), Jacobi Identity
([X , [Y, Z]] + [Y, [Z , X ]] + [Z , [X , Y ]] = 0) and (62), we get (63).

Corollary 7. In, Vn, we have

◦
K (X , Y, Ui) = DY ([X , Ui])− DX ([Y, Ui]) (67a)

◦
K (X , Ui, Ui) = DUi

([X , Ui]) (67b)

Proof. Replacing Ui for Z in (63) and using (12) and (61a), we get (67a). (67b) is obtained

by putting Ui for Y in (67a).



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 849

4. An Affine Connexion III

In this section an affine connexion is considered which is different one from section 2 and

section 3. Its properties are also have been studied.

Let us consider an affine connexion D in a generalised structure manifold Vn with torsion

tensor S such that

(DX ui)(Y ) + (DY ui)(X ) = 0 (68a)

(DX F)(Y ) + (DY F)(X ) = 0 (68b)

(DX Ui) = 0 (68c)

Theorem 15. In, Vn, we have

DUi
Y = DUi

Y (69a)

DY X + b2DX Y − (DX Y + DY X ) = −c X (ui(Y ))Ui (69b)

DX Y + DY X − b2(DX Y + DY X ) = c ui(DX Y + DY X )Ui (69c)

b2(DX Y + DY X )− (DX Y + DY X ) = −c {X (ui(Y )) + Y (ui(X ))} (69d)

DX Y + DY X − DX Y − DY X = 0 (69e)

Proof. Taking covariant derivative of F(Y ) = Y with respect to Ui and using (68c), we get

(69a). Now, (68b) is equivalent to

DX Y + DY X = DX Y + DY X (70)

Replacing Y by Y in (70) and using (1), we get (69b). By operating F on both sides of (70)

and using (1), (69c) can be obtained. Replacing X by X in (69b), we get (69d). Replacing X
by X and Y by Y in (70) separately and adding the resulting equations, we get

DX Y + DY X − DX Y − DY X = DY X + DX Y − DX Y − DY X (71)

Using (1), (68a) and (69c) in (71), we get (69e).

Let us define a tensor H of the type (1,2) by

H(X , Y )
de f
= DX Y − DX Y (72)

Theorem 16. In, Vn, we have the following relations :

H(X , Y ) +H(Y, X ) = 0 (73a)

H(X , Y ) +H(Y, X ) = 0 (73b)

H(X , Y )− b2H(X , Y ) = −c[{p j
i Y (u j(X ))Ui}+ Y (ui(X ))Ui] (73c)



R. Singh and S. Singh / Eur. J. Pure Appl. Math, 3 (2010), 839-852 850

Proof. (73a) follows from (69e) and (72). Now replacing Y by Y in (72) and using (1),

we get

H(X , Y ) = b2DX Y + c X (ui(Y ))Ui − DX Y (74)

Interchanging X and Y in (74) and adding the resulting equation with (74), we get (73b), by

making use of (69d). Replacing X by X in (73b) and using (1) and (5), we obtain (73c).

Theorem 17. In, Vn, we have

M(X , Y ) = 2H(X , Y )− B(X , Y )Ui (75)

where,

B(X , Y ) = (DX ui)(Y )

Proof. Adding (69b) and (69e), we have

DX Y − DX Y − DX Y + b2DX Y + c X (ui(Y ))Ui = 0 (76)

Using (1) in (20), we get

M(X , Y ) = DX Y + b2DX Y + c ui(DX Y )Ui − DX Y − DX Y (77)

Due to (76) and (77) yields (75).

Theorem 18. The connexion D is an F-connexion if and only if

H(X , Y ) = 0 (78)

Proof. Let D be an F -connexion, then (13) & (72), gives H(X , Y ) = 0. Conversely, if (78)

is true, then DX Y = DX Y . Replacing X by X in this relation, we get

b2DX Y + c ui(X )DUi
Y = b2 DX Y + c ui(X )DUi

Y (79)

Due to (69a), (79) yields (DX F)(Y ) = 0, which implies that D is an F -connexion.

Theorem 19. When D is an F-connexion ,any one of the following holds if remaining two hold :

(a) H(X , Y ) = 0

(b) B(X , Y ) = 0

(c) M(X , Y ) = 0

Remark 3. All the results discussed in section 2, section 3 and section 4 are true in an almost tan-
gent metric manifold, an almost Hermite manifold, metric π-structure manifold, Hsu-structure
manifold, F-structure manifold, an almost product Riemannian manifold, an almost Grayan
manifold and
{F, g,u1,u2, U1, U2} structure manifold if (b2 = 0, c = 0); (b2 = −1, c = 0); (c = 0);
(b2 = λr , c = 0); (b2 = −1, p j

i = 0); (b2 = 1, c = 0); (b2 = −1, c = 1, p1
1 = 0 : i, j = 1); and

(b2 = −1, c = 1, p j
i + pi

j = 0 : i, j = 1,2) respectively.



REFERENCES 851

ACKNOWLEDGEMENTS The first author is very thankful to University Grants Commission

for partial financial assistance without which the present work was not possible.

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