Abstract
This paper attempts to introduce and study new connection between fuzzy retractions, fuzzy foldings and fuzzy deformation retracts of open fuzzy spheres in fuzzy Lobachevsky space and fuzzy open ball in fuzzy Euclidean space. Types of fuzzy foldings and fuzzy deformation retracts of fuzzy open sphere are discussed. Types of minimal fuzzy retractions of fuzzy open sphere are presented. The fuzzy foldings of fuzzy open sphere is deduced, also the limit of this folding is obtained. The connection between fuzzy foldings and fuzzy central projection map is achieved; also the connection between fuzzy retractions and fuzzy exponential map is discussed. Some applications are presented. Keywords: Fuzzy Retractions, Fuzzy Foldings, Fuzzy Deformation Retracts, Central Projection Map, Fuzzy Open Sphere in Fuzzy Lobachevsky Space.
Highlights
Introduction and BackgroundLobachevsky space represents one of the most intriguing and emblematic discoveries in the history of geometry
The aim of the present paper is to describe the above phenomena geometrically, concerned with the study of the new types of fuzzy retractions, fuzzy deformation retracts and fuzzy folding of fuzzy open sphere Sn − (g) in fuzzy Lobachevsky space as presented by El-Ahmady [1,2,3,4,5,6,7,8,9,10,11,12]
The relation between the fuzzy exponential map and fuzzy folding of a fuzzy sphere S n → T p (S n) ⊂ L n+1 discussed from the following commutative diagram
Summary
Lobachevsky space represents one of the most intriguing and emblematic discoveries in the history of geometry If it were introduced for a purely geometrical purpose, they came into prominence in many branches of mathematics and physics. Ja→s γS, nwihsearpeieJc=ew[0is,1e]f.uIzfz¡~y does not preserve lengths, ¡~ is a topological folding of fuzzy sphere in fuzzy Lobachevsky space [8, 9,13,14,15,16,17,18,19] It h:e∪TuMhp pei e rish →oyfpu∪ezrzMm yiasnufiocflohdltidnhsga∃t MI o(fjMd o)∪w=M nM iM~⊆ansSu dncahnityshMa~taiμbie=floolμndjgifnotogr every corresponding points i.e. μ(ai) = μ(aj) [6] (Figure 1). The relation between the fuzzy exponential map and fuzzy folding of a fuzzy sphere S n ⊂ L n+1 discussed from the following commutative diagram
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