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
, then
the 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.