Isometric foldings

An isometric folding from a Riemannian manifold M to another N is a map which sends piecewise geodesic segments on M to piecewise geodesic segments on N of the same length. The set (ImM, N) of all isometric foldings from M to N has, as a subset of all continuous functions from M to N, a natural C0 topology. The main purpose of this Thesis is to obtain descriptions of the topological spaces of all isometric self-foldings of the Riemannian 2-sphere S2, of the Euclidean plane Re2 and of the hyperbolic plane cal H2, given in Chapters 3, 4 and 5 respectively. In Chapter 1 we review the general theory of isometric foldings of Riemannian manifolds. In Chapter 2 we study a subspace of (Im M, N) formed by all isometric foldings of finite type, that is by all isometric foldings f whose set of singularities, (points where f fails to be differentiable), partitions M into a finite number of regions on which f is an isometric immersion. Finally in Chapter 6 we give a brief outline of the main results and some open problems are proposed.