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We introduce the deformation retract of the Eguchi-Hanson space using Lagrangian equations. The retraction of this space into itself and into geodesics has been presented. The deformation retract of the Eguchi-Hanson space into itself and after the isometric folding has been discussed. Theorems concerning these relations have been deduced.

The real revolution in mathematical physics in the second half of twentieth century (and in pure mathematics itself) was algebraic topology and algebraic geometry [

Today, the concepts and methods of topology and geometry have become an indispensable part of theoretical physics. They have led to a deeper understanding of many crucial aspects in condensed matter physics, cosmology, gravity, and particle physics. Moreover, several intriguing connections between only apparently disconnected phenomena have been revealed based on these mathematical tools [

Topology enters general relativity through the fundamental assumption that spacetime exists and is organized as a manifold. This means that spacetime has a well-defined dimension, but it also carries with it the inherent possibility of modified patterns of global connectivity, such as distinguishing a sphere from a plane or a torus from a surface of higher genus. Such modifications can be present in the spatial topology without affecting the time direction, but they can also have a genuinely spacetime character in which case the spatial topology changes with time [

In general relativity, boundaries that are

The simplest example of nontrivial bundles arises in quantum cosmology in which the boundary is a compact

The four-dimensional Riemannian manifolds for gravitational instantons can be asymptotically flat, asymptotically locally Euclidean, asymptotically locally flat, or compact without boundary [

In order to remove the apparent singularity, we take

The theory of deformation retract is very interesting topic in Euclidean and non-Euclidean spaces. It has been investigated from different points of view in many branches of topology and differential geometry. A retraction is a continuous mapping from the entire space into a subspace which preserves the position of all points in that subspace [

Let

A subset

A subset

The deformation retract is a particular case of homotopy equivalence, and two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger space.

Deformation retracts of Stein spaces have been studied in [

Most of the studies on deformation retract and folding, if not all, are pure mathematical studies. The authors believe that these two concepts should be given more attention in modern mathematical physics. Topological studies of some famous metrics of mathematical physics could be a nice topological exploration start.

A four-dimensional flat metric can be written as

In general relativity, the geodesic equation is equivalent to the Euler-Lagrange equations

To find a geodesic which is a subset of the 6D Schwarzchild space, the Lagrangian could be written as

The retraction of the Eguchi-Hanson spacetime is the great circle in spacetime geodesic in Eguchi-Hanson space. The deformation retract of the Eguchi-Hanson space

The deformation retract of the Eguchi-Hanson space

Now we are going to discuss the folding

The folding of the Eguchi-Hanson space (

Now let the folding be defined by

The deformation retract of the folded Eguchi-Hanson space

The deformation retract of the isometric folding of Eguchi-Hanson space and any folding homeomorphic to this type of folding is different from the deformation retract of Eguchi-Hanson space under condition (

The deformation retract of the Eguchi-Hanson space has been investigated by making use of Lagrangian equations. The retraction of this space into itself and into geodesics has been presented. The deformation retraction of the Eguchi-Hanson space is a geodesic which is found to be a great circle. The folding of the Eguchi-Hanson space has been discussed and it was found that this folding and any folding homeomorphic to that folding have the same deformation retract of the Eguchi-Hanson space onto a geodesic. Also, the deformation retract of the isometric folding of Eguchi-Hanson space and any folding homeomorphic to this type of folding is found to be different from the deformation retract of Eguchi-Hanson space under condition (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are 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 financially supported by Grant no. 1435/6164 from the deanship of the scientific research at Taibah University, Al-Madinah Al-Munawwarah, Saudi Arabia.