DEFORMATION RETRACTS OF THE REISSNER-NORDSTROM SPACETIME AND ITS FOLDINGS

Our aim in the present article is to introduce and study the relation between the deformation retract of the Reissner-Nordstrom spacetime N4 and the deformation retract of the tangent space Tp (N4). Also, this relation discussed after and before the isometric and topological folding of N4 into itself. New types of conditional folding are presented. Some commutative diagrams are obtained.


INTRODUCTION
As is well known, the theory of foldings is always one of interesting topics in Euclidian and Non-Euclidian space and it has been investigated from the various viewpoints by many branches of topology and differential geometry (El-Ahmady, 2013a;2013b;2013c;2013d;2013e).
Most folding problems are attractive from a pure mathematical standpoint, for the beauty of the problems themselves.The folding problems have close connections to important industrial applications Linkage folding has applications in robotics and hydraulic tube bending.Paper folding has application in sheet-metal bending, packaging and air-bag folding (El-Ahmady, 2012a;2012b;2011).Following the great Soviet geometer (El-Ahmady and Al-Rdade, 2013), also, used folding to solve difficult problems related to shell structures in civil engineering and aero space design, namely buckling instability (El-Ahmady and Al-Hazmi, 2013).Isometric folding between two Riemannian manifold may be characterized as maps that send piecewise geodesic segments to a piecewise geodesic segments of the same length.For a topological folding the maps do not preserves lengths i.e., A map ℑ: M→N, where M and N are C ∞ -Riemannian manifolds of dimension m,n respectively is said to be an isometric folding of M into N, iff for any piecewise geodesic path γ: J→M, the induced path ℑ 0 γ: J→N is a piecewise geodesic and of the same length as γ (El-Ahmady and El-Araby, 2010).If ℑ does not preserve length, then ℑ is a topological folding.A subset A of a topological space X is called a retract of X if there exists a continuous map r: X→A such that r(α) = α, ∀α∈ A where A is closed and X is open (Arkowitz, 2011;Banchoff and Lovett, 2010;El-Ahmady, 2007a;2007b, El-Ahmady, 2006;2004a;2004b).Also, let X be a space and A a subspace.A map r: X→A such that r (α) =α, for all α∈A, is called a retraction of X onto A and A is the called a retract of X (El-Ahmady and Shamara, 2001).This can be re stated as follows.If i: A→X is the inclusion map, then r: X→A is a map such that ri = id A .If, in addition, ri x ri id ≃ we call r a deformation retract and A a deformation retract of X (El-Ahmady, 1994).Another simple-but extremely useful-idea is that of a retract.If A, X ⊂ M, then A is a retract of X if there is a commutative equation:

AJAS
If f: A→B and g: X→Y, then f is a retract of g if ri = id A and js = id B (Naber, 2011;Reid and Szendroi, 2011;Arkowitz, 2011;Strom, 2011;Shick, 2007).At each point p of a complete Riemannian manifold M, we define a mapping of the tangent space T p (M) at p onto M in the following manner.If X is a tangent vector at P we draw a geodesic g(t) starting at P in the direction of X.If X has length α, then we map X into the point g(α) of the geodesic.We denote this mapping by exp p : Tp (M)→M, the map exp p is everywhere C ∞ and in a neighborhood of p in M, it is a diffeomorphism (Kuhnel, 2006;Banchoff and Lovett, 2010).

Main Results
The Reissner-Nordström spacetime N 4 is given by the following metric (El-Ahmady and Al-Rdade, 2013;Hartle, 2003;Griffiths and Podolsky, 2009;Straumann, 2003) where, m represents the gravitational mass and e the electric charge of the body.The coordinates of Reissner-Nordstr O ɺɺ m spacetime N 4 are given by Equation 2: where, C 1 ,C 2 ,C 3 and C 4 are the constant of integration .The Reissner-Nordström space time N 4 geodesic equations for the metric (1) are given by the following Equation 3-6: where, τ is an affine parameter.Suppose that for all τ where 2 π φ = .Then Equation 7: Under the condition u 3 = 0 the above equations become Equation 8-11: Integrating Equation ( 9), we get Equation 12:

AJAS
where, 1 ω and 2 ω are the constant of integrations.

Theorem 1
Types of the geodesic retraction of Reissner-Nordstrom spacetime N 4 are hypersphere retraction and curves retraction.

Theorem 2
The deformation retract of (N 4 -(p 1 ,q 1 )) onto under the exponential map is an induced deformation retract of T p1 (N 4 ) onto . Any isometric folding 4 4 F : N N → such that F (x 1 ,x 2 ,x 3 ,x 4 ) = (x 1 ,x 2 ,|x 3 |,x 4 ) induces the same deformation retract of T p1 (N 4 ) under the condition x 3 = 0, which makes the equation: Commutative, where ( ) is an open ball of radius π and center at p 1 .

Proof
The parametric equation of the Reissner-Nordström space time N 4 is given: C r ( )sin ( ) ( )) By using lagrangian equations: where, we obtain the deformation retract of (N 4 -(p 1 , q 1 )) given by: .If F 1 is a deformation retract of (N 4 -(p 1 ,q 1 )) onto a geodesic retraction and the following equation is commutative: If F 1 : N 4 →N 4 is an isometric folding and any folding homeomorphic to this type of folding: ( ) Let F µ is the set of all types of folding homeomorphic to F 1 under the condition: ( ) Then the deformation retract of any F∈F µ (N 4 ) is invariant, i.e., = , the induced invariant deformations retract:

Theorem 3
Under the condition t = e = m = 0, the deformation retract of 2 , under the exponential map is an induced deformation retract of induces the same deformation retract of 2 p1 1 T (S ) , which makes the equation: Commutative, where 2 p1 D ( ) π is an open ball of radius π and of center at p1.

Theorem 4
Any isometric folding .There is an induced isomtric folding of the tangent space 3 p1 1 T (S ) such that the following equation is commutative:

Proof
Since q 1 is a conjugate point to p 1 , then exp -1 : F : (S q ) (S q ) − → − such that F(p 1 ) = p 1 be an isometric folding, then there is an induced isometric folding F such that: Let γ be any curve in 3 1 1 there is no conjugate point to p 1 on 3 1 1 (S q ) − , then exp -1 (γ) = β, then p 1 ∈β, p 1 is the beginning of β, also . There is an induced isometric folding such that: q 1 is the conjugate point of Is commutative and F 1 (F 1 ) =F 1 , then the following equation is commutative: F exp oF o exp , F (F ) F ,F (F ) F We get:

CONCLUSION
In this study we achieved the approval of the important of the curves and surface in Reissner-Nordström spacetime N 4 by using some geometrical transformations.The relations between folding, retractions, deformation retracts, limits of folding and limits of retractions of the curves and surface in the Reissner-Nordström spacetime N 4 are discussed.New types of the tangent space T p (N 4 ) in Reissner-Nordström spacetime N 4 are deduced.

ACKNOWLEDGMENT
The author is deeply indebted to the team work at the deanship of the scientific research, Taibah University for their valuable help and critical guidance and for facilitating many administrative procedures.This research work was financed supported by Grant no.3066/1434 from the deanship of the scientific research at Taibah University, Al-Madinah Al-Munawwarah, Saudi Arabia.
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