الفهرس 
Perfect Folding of a Surface to a PolygonOnly 14 pages are availabe for public view
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Abstract
The project of this thesis concerns a field of mathematics called Geometric Topology, which essentially studies various structures and properties of manifolds This thesis study perfect foldings of surfaces and graphs. A folding of a compact connected surface M onto a polygon Pn with n vertices is a mapping 𝑓 : 𝑀 →𝑃𝑛 whose singular point set is a graph Γ𝑓 embedded in M such that its image f(Γ𝑓) coincides with the boundary of 𝑃𝑛 and that the vertices of Γ𝑓 are sent to the vertices of 𝑃𝑛 . A regular folding is a folding whose associated graph Γ𝑓 is a regular graph. A perfect folding is a regular folding with an extra condition. The study of foldings of a manifold into another manifold began with S. A. Robertson’s work on isometric folding of a Riemannian manifold M into another N which send any piecewise geodesic path in M to a piecewise geodesic path with the same length in N ,[1].