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Totally geodesic submanifolds of a trans-Sasakian manifold/ Taielikult geodeetilised trans-Sasaki muutkonna alammuutkonnad.

1. INTRODUCTION

Invariant submanifolds of a contact manifold have been a major area of research for a long time since the concept was borrowed from complex geometry. It helps us to understand several important topics of applied mathematics; for example, in studying non-linear autonomous systems the idea of invariant submanifolds plays an important role [9]. A submanifold of a contact manifold is said to be totally geo- desic if every geodesic in that submanifold is also geodesic in the ambient manifold. In 1985, Oubina [14] introduced a new class of almost contact manifolds, namely, trans-Sasakian manifold of type ([alpha], [beta]), which can be considered as a generalization of Sasakian, Kenmotsu, and cosymplectic manifolds. Trans-Sasakian structures of type (0,0), (0,[beta]), and ([alpha],0) are cosymplectic [2], [beta]-Kenmotsu [10], and [alpha]-Sasakian [10], respectively. Kon [12] proved that invariant submanifolds of a Sasakian manifold are totally geodesic if the second fundamental form of the immersion is covariantly constant. On the other hand, any sub- manifold M of a Kenmotsu manifold is totally geodesic if and only if the second fundamental form of the immersion is covariantly constant, provided [xi] [member of] TM [11]. Recently, Sular and (Ozgur [16] proved some equivalent conditions regarding the submanifolds of a Kenmotsu manifold to be totally geodesic. Several studies ([5,17]) have been done on invariant submanifolds of trans-Sasakian manifolds. Recently, Sarkar and Sen [15] proved some equivalent conditions of an invariant submanifold of trans-Sasakian manifolds to be totally geodesic. In the present paper we rectify proofs of most of the major theorems of [15] and [17], show some theorems of [15] as corollary of our present results, and also introduce some new equivalent conditions for an invariant submanifold of a trans-Sasakian manifold to be totally geodesic.

2. PRELIMINARIES

Let M be a connected almost contact metric manifold with an almost contact metric structure ([phi], [xi], [eta], g), that is, [phi] is a (1,1)-tensor field, [xi] is a vector field, [eta] is a one-form, and g is the compatible Riemannian metric such that

[[phi].sup.2](X) = -X + [eta](X)[xi], [eta]([xi]) = 1, [phi][xi] = 0, [eta]o[phi] = 0, (2.1)

g([phi]X, [phi](Y) = g(X,y) - [eta](x)[eta](Y), (2.2)

g(X,[phi](Y) = -g([phi]X ,Y), g(X,[xi]) = [eta](X), (2.3)

for all X ,Y [member of] TM ([2,18]). The fundamental two-form [PHI] of the manifold is defined by

[PHI](X, Y) = g(X, [phi]Y), (2.4)

for X, Y [member of] TM.

An almost contact metric structure ([phi],[xi],[eta],g) on a connected manifold M is called a trans-Sasakian structure [14] if (M x R, J, G) belongs to the class [W.sub.4] [8], where J is the almost complex structure on M x R defined by

J(X, fd/dt) = ([phi]X - f[xi], [eta](X)d/dt),

for all vector fields X on M and smooth functions f on M x R, and G is the product metric on M x R. This may be expressed by the condition [3]

([bar.[nabla]]x[phi])Y = [alpha](g(X, Y)[xi] - [eta](Y)X) + [beta](g([phi]X,Y)[xi] - [eta](Y)[phi]X) (2.5)

for smooth functions [alpha] and [beta] on M. Here we say that the trans-Sasakian structure is of type ([alpha], [beta]). From the formula (2.5) it follows that

[bar.[nabla]]x[xi] = -[alpha][phi]X + [beta](X - [eta](X)[xi]), (2.6)

([bar.[nabla]]x[eta])Y = -[alpha]g([phi]X, Y) + [beta]g([phi]X, [phi](Y). (2.7)

In a (2n +1)-dimensional trans-Sasakian manifold we also have the following:

S(X,[xi]) = 2n([[alpha].sup.2] - [[beta].sup.2])[eta](X) - (2n - 1)X[beta] - [eta](X)[xi][beta] - ([phi]X)[alpha], (2.8)

R(X, Y)[xi] = ([[alpha].sup.2] - [[beta].sup.2])([eta](Y)X - [eta](X)Y)+ 2[alpha][beta]([eta](Y)[phi]X - [eta](X)[phi]Y)

-(X[alpha])[phi]Y + (Y[alpha])[phi]X - (X[beta])[[phi].sup.2]X + Y [beta][[phi].sup.2]X, (2.9)

R(X,[xi])[xi] = ([[alpha].sup.2] - [[beta].sup.2])(X - [eta](X)[xi])+ 2[alpha][beta][phi]X + ([xi][alpha])[phi]X + ([xi][beta][[phi].sup.2]X, (2.10)

where S is the Ricci tensor of type (0, 2) and R is the curvature tensor of type (1 , 3).

Let M be a submanifold of a contact manifold [bar.M]. We denote by [nabla] and [bar.[nabla]] the Levi-Civita connections of M and [bar.M], respectively, and by [T.sup.[perpendicular to]](M) the normal bundle of M. Then for vector fields X, Y [member of] TM, the second fundamental form h is given by the formula

h(X,Y)= [bar.[nabla]]xY - [nabla]xY. (2.11)

Furthermore, for N [member of] [T.sup.[perpendicular to]]M

[A.sub.N]X = [[nabla].sup.[perpendicular to].sub.X]N - [bar.[nabla].sub.X]N, (2.12)

where [[nabla].sup.[perpendicular to]] denotes the normal connection of M. The second fundamental form h and [A.sub.N] are related by g(h(X,Y),N)= g([A.sub.N]X,Y) [4].

The submanifold M is totally geodesic if and only if h = 0.

An immersion is said to be parallel and semi-parallel [6] if for all X, Y [member of] TM we get [nabla].h = 0 and R(X, Y).h = 0, respectively.

It is said to be pseudo-parallel [7] if for all X ,Y [member of] TM we get

R(X, Y ).h = fQ(g, h), (2.13)

where f denotes a real function on M and Q(E, T) is defined by

Q(E, T) (X ,Y, Z, W ) = -T ((X [[and].sub.E] Y )Z, W) - T (Z, (X [[and].sub.E] Y )W), (2.14)

where X [[and].sub.E]Y is defined by

(X [[and].sub.E] Y)Z = E(Y,Z)X - E(X, Z)Y.

If f= 0, the immersion is semi-parallel.

Similarly, an immersion is said to be 2-pseudo-parallel if for all X, Y [member of] TM we get R(X, Y).[nabla]h = fQ(g, [nabla]h), and Ricci generalized pseudo-parallel [13] if R(X, Y).h = fQ(S,h), for all X, Y [member of] TM. The second fundamental form h satisfying

([[nabla].sub.z]h)(X, Y )= A(Z)h(X, Y), (2.15)

where A is a nonzero one-form, is said to be recurrent. It is said to be 2-recurrent if h satisfies

([[nabla].sub.X][[nabla].sub.Y]h - [[nabla].sub.[[nabla].sub.x]yh)(Z,W)= B(X,Y)h(Z, W), (2.16)

where B is a nonzero two-form.

Proposition 2.1. [5] An invariant submanifold of a trans-Sasakian manifold is also trans-Sasakian.

Proposition 2.2. [5] Let M be an invariant submanifold of a trans-Sasakian manifold M. Then we have

h(X,[phi]Y) = [phi](h(X,Y)), (2.17)

h([phi]X,[phi]Y) = -(h(X,Y)), (2.18)

h(X,[xi]) = 0, (2.19)

for any vector fields X and Y on M.

For a Riemannian manifold, the concircular curvature tensor Z is defined by

Z(X, Y)V = R(X, Y)V - [tau]/n(n-1) [g(Y, V)X - g(X, V)Y], (2.20)

for vectors X, Y, V [member of] TM, where [tau] is the scalar curvature of M. We also have

(Z(X, Y).h) (U, V) = [R.sup.[perpendicular to]](X, Y)h(U, V) - h(Z(X, Y)U, V) - h(U, Z(X, Y)V). (2.21)

In the next section we consider the submanifold M to be tangent to [xi].

3. INVARIANT SUBMANIFOLDS OF A TRANS-SASAKIAN MANIFOLD WITH [alpha], [beta] = CONSTANT

Lemma 3.1. If a non-flat Riemannian manifold has a recurrent second fundamental form, then it is semi-parallel.

Proof. The second fundamental form h is said to be recurrent if

[nabla]h = A [cross product] h,

where A is an everywhere nonzero one-form.

We define a function e on M by

[e.sup.2] = g(h, h). (3.1)

Then we have e(Ye) = [e.sup.2]A(Y). So we obtain Ye = eA(Y), since f is nonzero. This implies that

X(Ye) - Y(Xe) = (XA(Y) - YA(X))e.

Therefore we get

[[bar.[nabla]]x[bar.[nabla]]y - [bar.[nabla]]y[bar.[nabla]]x - [[bar.[nabla]].sub.[X,Y]]]e = [XA(Y) - YA(X) - A([X, Y])]e.

Since the left-hand side of the above equation is identically zero and e is nonzero on M by our assumption, we obtain

dA(X ,Y ) = 0, (3.2)

that is, the one-form A is closed.

Now from ([[nabla].sub.X]h)(U,V) = A(X)h(U,V) we get

([[bar.[nabla]].sub.U][[bar.[nabla]].sub.V]h)(X, Y) - ([[bar.[nabla]].sub.[[bar.[nabla]].sub.U]Vh]] (X, Y) = [([[bar.[nabla]].sub.U]A)V + A(U)A(V)]h(X, Y)= 0.

Using (3.2) we get

(R(X, Y ).h)(U ,V) = [2dA(X ,Y )]h(X ,Y) = 0.

Therefore, for a recurrent second fundamental form we have

R(X,Y).h=0

for any vectors X, Y on M.

If e = 0, then from (3.1) we get h = 0 and thus R(X, Y).h = 0.

Hence the lemma.

Theorem 3.1. An invariant submanifold of a non-cosymplectic trans-Sasakian manifold is totally geodesic if and only if its second fundamental form is parallel.

Proof. Since h is parallel, we have

([[nabla].sub.X]h)(Y,Z) = 0

which implies

[[nabla].sup.[perpendicular to].sub.X]h(Y,Z) - h([[nabla].sub.X]Y,Z) - h(Y,[[nabla].sub.X]Z) = 0.

Putting Z = [xi] in the above equation and applying (2.19) we obtain

h(Y, [[nabla].sub.X][xi])= 0. (3.3)

So from (2.6) and the above equation (3.3) we obtain

[alpha]h(X, Y)= [beta][phi]h(X, Y). (3.4)

Applying [phi] to both sides of (3.4) we get

[alpha][phi]h(X, Y) = -[beta]h(X, Y). (3.5)

From (3.4) and (3.5) we conclude that

([[alpha].sup.2] + [[beta].sup.2])h(X, Y )= 0.

Hence for a non-cosymplectic trans-Sasakian manifold h(X, Y) = 0, for all X, Y [member of] TM.

The converse part is trivial. Hence the result.

Remark 3.1. In Theorem 3.1 [15] the authors proved the same result, but they actually proved h(Y, [[nabla].sub.X][xi]) = 0, and h(Y,[xi]) = 0, [for all]X, Y [member of] TM. Since [[nabla].sub.X][xi] is not an arbitrary vector of TM, hence from this we can not conclude that the submanifold is totally geodesic.

Remark 3.2. Again in the proof of Theorem 4.8 [17] the authors assumed [phi](h(X, Y)) = 0, [for all]X, Y [member of] TM, which is not true in general because this condition directly implies that the submanifold is totally geodesic.

Theorem 3.2. An invariant submanifold of a non-cosymplectic trans-Sasakian manifold is totally geodesic if and only if its second fundamental form is semi-parallel.

Proof. Since h is semi-parallel, we have

(R(X ,Y ).h)(U, V )= 0, (3.6)

which implies

[R.sup.[perpendicular to](X, Y)h(U,V)+ h(R(X,Y)U,V) -h(U,R(X,Y)V)= 0. (3.7)

Putting V = [xi] = Y and applying (2.19) we get from Eq. (3.7)

h(U,R(X, [xi])[xi])= 0.

So from (2.10) and (2.19) we get

([[alpha].sup.2] - [[beta].sup.2] )h(U, X) = 2[alpha][beta][phi]h(U, X). (3.8)

Applying [phi] to both sides of Eq. (3.8) we obtain

([[alpha].sup.2] - [[beta].sup.2])[phi]h(U, X) = -2[alpha][beta]h(U, X). (3.9)

So from (3.8) and (3.9) we conclude that

[([[alpha].sup.2] + [[beta].sup.2]).sup.2]h(U, X ) = 0.

Hence as in the previous case, for non-cosymplectic trans-Sasakian manifolds the invariant submanifold is totally geodesic. The converse part follows trivially. ?

Now, by Lemma 3.1 we get that if a second fundamental form is recurrent, then it is semi-parallel. Also, the second fundamental form of a totally geodesic submanifold is trivially recurrent, so from Theorem 3.2 we obtain the following:

Corollary 3.1. An invariant submanifold of a non-cosymplectic trans-Sasakian manifold is totally geodesic if and only if its second fundamental form is recurrent.

Remark 3.3. In Theorem 3.2 [15] the authors proved the above corollary, but they just showed that h(Y, [[nabla].sub.X][xi]) = 0, and h(Y, [xi]) = 0, [for all]X, Y [member of] TM. Since [[nabla].sub.X][xi] is not an arbitrary vector of TM, we can not conclude from this that the submanifold is totally geodesic.

In [1] Aikawa and Matsuyama proved that if a tensor field T is 2-recurrent, then R(X, Y).T = 0. Also it can be easily seen that in a totally geodesic submanifold the second fundamental form is 2-recurrent. Therefore by Theorem 3.2 we also obtain the following:

Corollary 3.2. An invariant submanifold ofa non-cosymplectic trans-Sasakian manifold is totally geodesic if and only if its second fundamental form is 2-recurrent.

Remark 3.4. In Theorem 3.4 [15] the authors proved the above corollary, but they considered [[nabla].sub.X][xi] as an arbitrary vector of TM, and actually proved h(Y, [[nabla].sub.X][xi]) = 0, [for all]X, Y [member of] TM, hence the proof of Theorem 3.4 [15] is incorrect.

Theorem 3.3. An invariant submanifold of a trans-Sasakian manifold is totally geodesic if and only if its second fundamental form is 2-semi-parallel, provided [[alpha].sup.2] [([[alpha].sup.2] - 3[[beta].sup.2]).sup.2] + [[beta].sup.2] [([[beta].sup.2] - 3[[alpha].sup.2]).sup.2] [not equal to] 0.

Proof. Since, the second fundamental form is 2-semi-parallel, we have

(R(X, Y).( [[nabla].sub.U]h))(Z, W) = 0,

which implies

([R.sup.[perpendicular to]] (X,Y)([[nabla].sub.U]h))(Z,W) - ([[nabla].sub.U]h)(R(X,Y)Z, W) - ([[nabla].sub.U]h)(Z,R(X,Y)W)= 0.

Now,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly,

([[nabla].sub.U]h)([xi],R(X,[xi])[xi]) = [alpha]([[alpha].sup.2] - 3[[beta].sup.2])[phi]h(X,U) + [beta]([[beta].sup.2] - 3[[alpha].sup.2])h(X,U). (3.10)

So putting Y = Z = W = [xi] in (3.10) we obtain

[alpha]([[alpha].sup.2] - 3[[beta].sup.2])[phi]h(X,U) + [beta]([[beta].sup.2] - 3[[alpha].sup.2])h(X,U) = 0. (3.11)

Applying [phi] on both sides of (3.11) we get

[alpha]([[alpha].sup.2] - 3[[beta].sup.2])[phi]h(X,U) + [beta]([[beta].sup.2] - 3[[alpha].sup.2])[phi]h(X,U). (3.12)

From (3.11) and (3.12) we conclude that

[[[alpha].sup.2][([[alpha].sup.2] - 3[[beta].sup.2]).sup.2] + [[beta].sup.2][([[beta].sup.2] - 3[[alpha].sup.2]).sup.2]]h(X,U) = 0.

Hence the submanifold is totally geodesic. The converse holds trivially.

Theorem 3.4. An invariant submanifold of a trans-Sasakian manifold is totally geodesic if and only if its second fundamental form is pseudo-parallel, provided [[([[alpha].sup.2] - [[beta].sup.2] - f).sup.2] + 4[[alpha].sup.2] [[beta].sup.2]] [not equal to] 0.

Proof. Since the second fundamental form is pseudo-parallel, we have

(R(X,Y).h)(U,V) = fQ(g,h)(X,Y,U,V),

which implies

([R.sup.[perpendicular to]](X,Y))h(U,V) -h(R(X,Y)U,V) -h(U,R(X,Y)V)

= f (-g(V,X)h(U,Y) + g(U,X)h(V,Y) -g(V,Y)h(U,X) + g(U,Y)h(V,X)). (3.13)

Putting V = [xi] = Y in Eq. (3.13) and applying (2.19) and (2.10) we obtain

-h(U,([[alpha].sup.2] - [[beta].sup.2])X + 2[alpha][beta][phi]X) = f(-h(U,X)). (3.14)

Applying [phi] to both sides of (3.14) we obtain

([[alpha].sup.2] - [[beta].sup.2] - f)[phi]h(U,X) = 2[alpha][beta]h(U,X). (3.15)

From (3.14) and (3.15) we conclude that

[[([[alpha].sup.2] - [[beta].sup.2] - f).sup.2] + 4[[alpha].sup.2][[beta].sup.2]]h(U,X)= 0.

Hence the submanifold is totally geodesic. The converse holds trivially. []

Theorem 3.5. An invariant submanifold of a trans-Sasakian manifold is totally geodesic if and only if its second fundamental form is 2-pseudo-parallel.

Proof. Since, the second fundamental form is 2-pseudo-parallel, we have

(R(X,Y).[[nabla].sub.Z]h)(U,V)= fQ(g,[[nabla].sub.Z]h)(X,Y,U,V). (3.16)

Now,

(R(X,Y).[[nabla].sub.Z]h)(U,V) = [R.sup.[perpendicular to]](X,Y)([[nabla].sub.Z]h)(U,V) - ([[nabla].sub.Z]h)(R(X, Y)U,V) - ([[nabla].sub.Z]h)(U,R(X, Y)V). (3.17)

From (2.10) and (2.19) we have

([[nabla].sub.Z]h)([xi],[xi])= 0 (3.18)

and

([[nabla].sub.Z]h)(R(X,[xi])[xi],[xi]) = -h(R(X,[xi])[xi],[[nabla].sub.Z][xi])

= [alpha]([[alpha].sup.2] - [[beta].sup.2])h(X,[phi]Z) + [beta]([[alpha].sup.2] - [[beta].sup.2])h(X,[[phi].sup.2]Z) - 2[[alpha].sup.2][beta]h([phi]X,[phi]Z) - 2[alpha][[beta].sup.2]h([phi]X,[[phi].sup.2]Z)

= ([[alpha].sup.2] + [[beta].sup.2])([alpha][phi]h(X,Z) + [beta]h(X,Z)). (3.19)

So, putting Y = u = V = [xi] in (3.16) we obtain

2([[alpha].sup.2] + [[beta].sup.2])([alpha][phi]h(X,Z) + [beta]h(X,Z)) = 0, (3.20)

which implies

[alpha][phi]h(X,Z) + [beta]h(X,Z)= 0. (3.21)

Applying [phi] on both sides of Eq. (3.21) we get

[alpha]h(X,Z) = [beta][phi]h(X,Z). (3.22)

Combining (3.21) and (3.22) we conclude that

[[[alpha].sup.2] + [[beta].sup.2]]h(X,Z) = 0. (3.23)

Hence the submanifold is totally geodesic. The converse holds trivially.

Theorem 3.6. An invariant submanifold of a trans-Sasakian manifold is totally geodesic if and only if its second fundamental form is Ricci generalized pseudo-parallel, provided [[([[alpha].sup.2] - [[beta].sup.2]).sup.2][(1 - 2nf).sup.2] + 4[[alpha].sup.2][[beta].sup.2]] [not equal to] 0.

Proof. Since the submanifold is Ricci generalized pseudo-parallel, we have

(R(X,Y).h)(U,V)= fQ(S,h)(X,Y,U,V). (3.24)

So,

R(X,Y)h(U,V) - h(R(X,Y)U,V) - h(U,R(X,Y)V)

= f(-S(V,X)h(U,Y) + S(U,X)h(V,Y) - S(V,Y)h(X,U) + S(U,Y)h(X,V)). (3.25)

Putting Y = V = [xi] and applying (2.19) we obtain

-h(U,R(X,[xi])[xi]) = -fS([xi],[xi])h(X,U).

Since [alpha] and [beta] are constants, from (2.19), (2.10), and (2.8) we can write

([[alpha].sup.2] - [[beta].sup.2])(1 -2nf)h(X, u) = 2[alpha][beta][phi]h(X,U). (3.26)

Applying [phi] on both sides of (3.26) we obtain

([[alpha].sup.2] - [[beta].sup.2])(1 - 2nf)[phi]h(X,U) = -2[alpha][beta]h(X,U). (3.27)

From (3.26) and (3.27) we conclude that

[[([[alpha].sup.2] - [[beta].sup.2]).sup.2][(1 - 2nf).sup.2] + 4[[alpha].sup.2][[beta].sup.2]]h(X,U) = 0.

Hence the submanifold is totally geodesic. The converse holds trivially.

Theorem 3.7. An invariant submanifold of a trans-Sasakian manifold is totally geodesic if and only if it satisfies Z(X, Y).h = 0, provided [([[alpha].sup.2] - [[beta].sup.2] - [tau]/2n(2n+1)).sup.2] + 4[[alpha].sup.2][[beta].sup.2] [not equal to] 0.

Proof. We have

(Z(X,Y).h)(U,V) = 0.

So from (2.21) we can write

[R.sup.[perpendicular to]](X,Y)h(UV) -h(Z(X,Y)U,V) -h(Z(X,Y)U,V) = 0.

Putting Y = V = [xi] in the above equation and applying (2.19) we obtain

h(U,Z(X,[xi])[xi]) = 0,

which implies that

h(U,([[alpha].sup.2] - [[beta].sup.2])X + 2[alpha][beta][phi]X - [tau]/2n(2n+1)X) = 0, since h(X,[xi]) = 0.

Simplifying we get

[([[alpha].sup.2] - [[beta].sup.2]) - [tau]/2n(2n+1)]h(U,X) + 2[alpha][beta][phi]h(U,X) = 0. (3.28)

Applying [phi] on both sides of the above equation we get

[([[alpha].sup.2] - [[beta].sup.2]) - [tau]/2n(2n+1)][phi]h(U,X) = 2[alpha][beta]h(U,X). (3.29)

From (3.28) and (3.29) we conclude

[[([[alpha].sup.2] - [[beta].sup.2] - [tau]/2n(2n+1)).sup.2] + 4[[alpha].sup.2][[beta].sup.2]]h(U,X) = 0.

The converse part follows trivially. Hence the result.

4. CONCLUSION

A trans-Sasakian manifold can be regarded as a generalization of Sasakian, Kenmotsu, and cosymplectic structures. For an invariant submanifold of a trans-Sasakian manifold with constant coefficients the following conditions are equivalent under certain conditions:

* the submanifold is totally geodesic,

* the second fundamental form of the submanifold is parallel,

* the second fundamental form of the submanifold is semi-parallel,

* the second fundamental form of the submanifold is recurrent,

* the second fundamental form of the submanifold is 2-recurrent,

* the second fundamental form of the submanifold is 2-semi-parallel,

* the second fundamental form of the submanifold is pseudo-parallel,

* the second fundamental form of the submanifold is 2-pseudo-parallel,

* the second fundamental form of the submanifold is Ricci generalized pseudo-parallel,

* the second fundamental form of the submanifold satisfies Z(X ,Y ).h = 0.

doi: 10.3176/proc.2013.4.05

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Avik De

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia; de.math@gmail.com

Received 24 September 2012, revised 10 January 2013, accepted 14 January 2013, available online 19 November 2013
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Title Annotation:MATHEMATICS
Author:De, Avik
Publication:Proceedings of the Estonian Academy of Sciences
Article Type:Report
Date:Dec 1, 2013
Words:4061
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