ABSTRACT

Our aim in the present article is to introduce and study new types of retractions of Lobachevsky space. Types of the deformation retracts of Lobachevsky space are presented. The relations between the folding and the deformation retract of Lobachevsky space are deduced .Types of minimal retractions of Lobachevsky space are also presented. Also, the isometric and topological folding in each case and the relation between the deformation retracts after and before folding have been obtained. New types of homotopy maps are deduced. Theorems governing this connection are achieved.

1. Introduction

Lobachevsky space represents one of the most intriguing and emblematic discoveries in the history of geometry. Although if it were introduced for a purely geometrical purpose, they came into prominence in many branches of mathematics and physics. This association with applied science and geometry generated synergistic effect: applied science gave relevance to Lobachevsky space and Lobachevsky allowed formalizing practical problems ElAhmady [1,2].

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. Following the great Soviet geometer, also, used folding to solve difficult problems related to shell structures in civil engineering and aero space design, namely buckling instability El-Ahmady [3,4]. 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 El-Ahmady [5]. For a topological folding the maps do not preserves lengths El-Ahmady [6,7], i.e. A map, where and are Riemannian manifolds of dimension m and n respectively is said to be an isometric folding of M into N, iff for any piecewise geodesic path, the induced path is a piecewise geodesic and of the same length as. If does not preserve length, then  is a topological folding El-Ahmady [8,9].

A subset A of a topological space X is called a retract of X if there exists a continuous map such that where A is closed and X is open El-Ahmady [10-20]. Also, let X be a space and A a subspace. A map such that  is called a retraction of X onto A and A is the called a retract of X Reid [21]. This can be re stated as follows. If is the inclusion map, then is a map such that. If, in addition, , we call a deformation retract and A a deformation retract of X Arkowitz [22], Shick [23] and Storn [24]. The aim of this paper is to describe and study new types of retraction, deformation retract and folding the of Lobachevsky space.

2. Main Results

We start with a metric of the Lobachevsky space in the special spherical Riemann mode Kudryashov [25].

(1)

And is a curvature radius. The spherical coordinates are given by

(2)

Using Lagrangian equations

To find a geodesic which is a subset of spherical Riemann model. Since

Then the Lagrangian equations are expressed as

(3)

(4)

(5)

(6)

From Equation (4) we obtain

= constant say, if, we obtain the following cases, if then; if, then from (2) we obtain

(7)

which is a Riemann sphere in Lobachevsky space with and, which is a retraction and geodesic. Specially if, hence we get the coordinates are defined by

(8)

which is a hypersurface in Lobachevsky space, with, which is a retraction and geodesic. Also if takes the values, , , , , , and we get new types of hypersurface, i = 2 - 9 in Lobachevsky space L⁴ with x₁ = ct. Specially if β = 90˚ hence we get the coordinates are defined by

(9)

Which is a Riemann sphere in Lobachevsky space, it is a geodesic and retraction. Also, if we have a Riemann sphere, it is a geodesic and retraction. Where

(10)

And also, if, we have a Riemann sphere in Lobachevsky space, it is a geodesic and retraction, where

(11)

Now, if Then we get the following coordinates

(12)

Hence, is the great circle, it is a geodesic and retraction.

Also, if, , then the Riemann point in Lobachevsky space is represented by the following coordinates

(13)

it is a minimal retraction in Lobachevsky space.

Now, if, thenthe retraction is represented by the following coordinates

(14)

Which is a Riemann point in Lobachevsky space.. From Equation (3) we obtain, if, then we get the following coordinates

(15)

Hence, is a Riemann hyper sphere, it is a geodesic and retraction.

Theorem 1. The retractions of Lobachevsky space are geodesics Riemann hypersphere, great circles, Riemann point and hyper subspace.

In this position, we present some cases of the deformation retract of Lobachevsky space. The retraction of the open Lobachevsky space is given by

The deformation retract of Lobachevsky space is

where is the open Lobachevsky space and I is the closed interval [0,1], be present as

The deformation retract of the Lobachevsky space into the retraction Riemann sphere is

where

And

The deformation retract of the Lobachevsky space into the retraction hyper surface is

The deformation retract of the Lobachevsky space into the retraction Riemann sphere is defined as

The deformation retract of the Lobachevsky space into the retraction Riemann sphere is

The deformation retract of the Lobachevsky space into the retraction Riemann sphere is defined as

The deformation retract of the Lobachevsky space into the great circle is defined by

The deformation retract of the Lobachevsky space into the retraction Riemann point is given by

The deformation retract of the Lobachevsky space into the retraction Riemann point is defined as

The deformation retract of the Lobachevsky space into the geodesic Riemann hyper sphere is

Now, we are going to discuss the folding of the Lobachevsky space. Let, where

(16)

An isometric folding of the Lobachevsky space into itself may be defined by

The deformation retract of the folded Lobachevsky space into the folded geodesic is

With

The deformation retract of the folded Lobachevsky space into the folded geodesic is

The deformation retract of the folded Lobachevsky space into the folded geodesic is

Then, the following theorem has been proved.

Theorem 2. Under the defined folding, the deformation retract of the folded Lobachevsky space i.e. into the folded geodesic is the same as the deformation retract of the Lobachevsky space into the geodesics.

Now, if the folding is defined by, where

(17)

The isometric folded Lobachevsky space time is defined as

The deformation retract of the folded Lobachevsky space into the folded geodesic is given by

Hence, we can formulate the following theorem .

Theorem 3. Under the defined folding, the deformation retract of the folded Lobachevsky space into the folded geodesic is different from the deformation retract of the Lobachevsky space into the geodesics.

If we let be given by

(18)

Then, the isometric chain folding of Lobachevsky space into itself may be defined by:

,

Then we get

which is hypersurface in Lobachevsky space.

From the above discussion we will arrive to the following theorem.

Theorem 4. The limit folding of the Lobachevsky space into itself, under Condition (18), is different from the retraction of the Lobachevsky space.

If the folding is defined by such that

(19)

Then, the isometric chain folding of Lobachevsky space into itself may be defined by:

,

Then we get which a zero-dimensional hypersphere in Lobachevsky space.

Thus the following theorem is obtained.

Theorem 5. The limit folding of the Lobachevsky space into itself, under Condition (19), is equivalent to the zero-dimensional sphere in Lobachevsky space.

Theorem 6. The end of the limits of the foldings of Lobachevsky space of dimension n is a 0-dimensional Lobachevsky space.

Proof: If we let

then, which is the Lobachevsky space of dimensional.

Also, if we consider

Then

, which is the Lobachevsky space of dimensional n − 2. Consequently,

which is a zerodimensional space.

Proposition 1. Under Condition (19) the retraction of 0-dimensional Lobachevsky space is a 0-dimensional space.

Theorem 7. Under Condition (19) the limit of foldings of Lobachevsky space into itself coincide with minimal retraction.

3. Conclusion

In this paper we achieved the approval of the important of the geodesic retractions of the Lobachevsky space. The relations between folding, retractions, deformation retract, limits of folding and limits of retractions of Lobachevsky space are discussed. Theorems which governs these relations are presented.

REFERENCES