الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
Mathematics fields are the philosophy of science. There are several branches of mathematics; some of them are topology, differential equations, numerical analysis and geometric topology. Many studies on the manifolds were studied by Robertson, M. EL- Ghoul and EL-Kholy. We introduce the effect of some types of geometric transformations as folding, deformation and retraction on some manifolds. The effect of these geometric transformations, which transfers the geometric figure, which is not manifold to be a manifold. We introduced the effect of retraction and the folding on some geometric properties of some geometric figures as curvature, dimension and the rotation indices of some curves. This thesis consists of an introduction and three chapters, in the introduction we presents a brief review of some fundamental concepts, and notations and it is the main basic important topological and geometric concepts that help us to follow up the work which are used in the later chapters. In chapter 1 we introduced some types of deformations and some conditional deformations on some manifolds as constant curvature deformation and a type of deformation, which its limits form holes. Also we introduced some types of foldings on some manifolds. The relation between deformation and folding is presented, the effect of deformations and foldings on some geometric properties of some manifolds are discussed . In chapter 2 we introduced some types of retractions and we discussed the effect of these retractions on some geometric figures in Rn , which makes the geometric figure that is not manifold to be a manifold. The limit of retractions is presented, the types of retractions, which fail to change the geometric figure to be a manifold are deduced. The effect of retraction on some singular knots as sheeted Solomon’s seal knot is presented. Moreover The effect of retraction on some geometric properties of some geometric figures as dimension and the rotation indices of some curves is presented. In chapter 3 We introduced the effect of some types of foldings on some geometric figures in Rn , which makes the geometric figure that is not manifold to be a manifold. The limit of foldings is presented, the types of foldings, which fail to change the geometric figure to be a manifold are deduced. The effect of folding on some geometric properties of some geometric figures as dimension and the rotation indices of some curves are presented.