Fuzzy Retraction of Fuzzy Space Time

A E El-Ahmady

doi:10.17485/ijst/2013/v6i6.5

Abstract

Our aim in the present article is to introduce and study a new type of metric, namely fuzzy space time metric. The geodesics of fuzzy space time metric will be obtained from the view point of Lagrangian equations. Types of the fuzzy retraction of fuzzy space time metric are presented. Types of the fuzzy folding of fuzzy space time metric are defined and discussed. The deformation retraction is also discussed. Some applications concerning these relations are presented. Keywords: Fuzzy Space Time Metric, Fuzzy Retraction, Fuzzy Deformation Retraction, Fuzzy Geodesics, Fuzzy Folding. 2000 Mathematics Subject Classification: 53A35, 51H05, 58C05.

Highlights

The theory of retraction 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
An n-dimensional topological manifold M is a Hausdorff topological space with a countable basis for the topology which is locally homeomorphic to Rn
A differentiable structure on M is a differentiable atlas and a differentiable manifolds is a topological manifolds with a differentiable structure [28, 30, 31, 32, 33]

Summary

Introduction and Definitions

The theory of retraction 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 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27, 28, 30,31,32,33]. The number of structures may be infinite In this case the manifold is said to be a chaotic manifold [17, 23]. A map I : M 4 → M 4 , is said to be an isometric folding of fuzzy Space Time M 4 into itself iff for any piecewise fuzzy geodesic path g : J → M 4 the induced path I g : J → M 4 is a piecewise fuzzy geodesic and of the same length as g, where J = [0,1]. If I does not preserve lengths, I is a topological folding of fuzzy Space Time M 4 [1, 4, 5, 6, 8, 9, 22].

Main Results

Applications

Conclusion