.REGULAR FOLDING OF SURFACES

Abstract

In this thesis, we investigate the so-called regular foldings of surfaces. The study of regular folding of surfaces began with the work of H. R. Farrn, E. EL-Kholy and S. A. Robertson [13]. The field of cellular foldings began with S. A. Robertson's work on isometric foldings of Riemannian manifold Minto another Riemannian manifold N which send any piecewise geodesic path in M to a pieCewise geodesic path with the same length in N [25]. As a topological version of this notion of isometric foldings, the notion of cellular foldings are invented by E. El-Kholy and H. A. Al-Khurasani [10]. A regular folding of a surfaces onto a polygons is a cellular folding with some sy=etry. In this introductory chapter, we outline this thesis. In Section 1.1, we give some basic definitions needed in later sections of this chapter. In Section 1.2, we explain the main themes of this thesis. In Sections 1.3 -1.7, we give the su=ary of Chapters 2,3,4,5 and 6, respectively.

1.1. Definitions.

In this section, for the convenience of later discussions, we give the defintion of regular foldings of a surface to a polygon. We begin with the more general notions of cell complexes and cellular maps