الفهرس 
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Abstract
Geometric topology and its branches have become hot topics, not only
for almost all fields of mathematics but also for many areas of science such as chemistry, physics, biology and information systems.
The folding of a manifolds is a one of the famous problems in the field
of geometric topology. S.A.Robertson in 1977 [24] is the first one who
introduced this idea, when he crumpled a sheet of paper in his hand and then
crushed it flat against a desk-top and studied the stratification determined by
the folds or the singularties.
More studies on isometric and topological foldings of a manifold are
studied by E. EL-Kholy and S.A.Robertson in [18,25]. Many other types of
foldings studies by E. EL-Kholy and others in [7,19,20]. M. EL-Ghoul and
others introduced many concepts of the folding, unfolding and deformation
retraction of different types of manifolds and fuzzy manifolds III
[8,9,... ,16,23,27]. Various folding problems arising in the physics of
membrane and polymers reviewed by P. DI. Francesco [6].
The limits of folding and unfolding of Riemannian manifolds were
discussed and obtained by M. EL-Ghoul at 1997, 1998 respectively [13,14].
The work of this thesis deals with the Cartesian product of manifolds
(product manifold), and this thesis consists of five chapters.
In chapter I: we gave some definitions and background, which are
needed in the next chapters.
In chapter II: we introduced folding of the Cartesian product of
manifolds, and the relations between folding, unfolding and the deformation
a manifold are discussed. Also theorems which governing these relations
are given. The results of this chapter are accepted in the ”International Journal
of Mathematics and Mathematical Sciences, University of Central Florida,
U.S.A. [8].
Chapter III: deals with the unfolding of the Cartesian product of
manifolds. The relations between the unfolding, retraction and deformations
of the Cartesian product of manifolds are deduced. The limit of the unfolding
of a manifold is obtained. Also theorems governing these relations are given.
Chapter IV: is devoted to introduce the maximum (minimum)
deformation of the Cartesian product of manifolds and their maximum
(minimum) foldingof its components. Also the relations between the
maximum (minimum) deformation of the Cartesian product of manifolds ,the
maximum (minimum) folding and the maximum (minimum) unfolding are
discussed. Theorems governing these relations are acheived.
In chapter V: we imitate the relation between the folding of the
covering space of the product manifold and the folding of the same product
manifolds and their retractions. Also their deformations are discussed in this
chapter and theorems governing these relations are obtained.