FOLDING OF 2-MANIFOLDS

Abstract

The project of this the'sis concerns a field of mathematics called "Geometric Topology", which essentially studies various structures and properties of manifolds and complexes.

Ismoetric foldings between Riemannian manifolds rriay be characterized as maps that send piecewise geodesic segments to piecewise geodesic segments of the same length.

The field of isometric foldings began with Robetison's, work [20] who studied the stratification detennined by the folds or the singularities, and relates this• str•ucture to classical ideas of Hopff degree, and volume. Then the themy of isometric foldings, has been pushed by both Robertson and El-K.110ly [21] to include covering spaces and many other different aspects. Again the idea of . topological folding is modeled by both of them on that of isometric folding, but in the absence of metr'ical structure [7].

A simplicial folding defined on simplicial complexes first defmed by

El-K11oly and El-Ghoul [6].

The notion of cellular folding of a cell complex is invented by El-K110ly and Al-Khursani and various properties of this type of folding are also studied by them and others [4, 5, 8].

The cellular folding j : M------+ N of a surface M onto another N zs one of the most interesting subjects to study , since the set of singularities of j in this case is a graph lr embedded in M . If If is a regular graph, then the cellular