Isometric and topological folding of manifolds

Abstract

Local isometries between Riemannian manifolds may be characterised as maps that send geodesic segments to geodesic segments of the same length. Isometric foldings are likewise characterised by such a property, with the difference that we use piecewise geodesic segments instead of geodesic segments. The theory of isometric foldings studies the stratification determined by the folds or singularities, and relates this structure to classical ideas of Hopf degree, volume and covering spaces. 

The idea of topological folding is modelled on that of isometric folding, but in the absence of metrical structure the definition is necessarily inductive. Again a stratification by folds is obtained, and a body of theorems concerning neat foldings has been established. These theorems have a strongly algebraic flavour, and are related to certain aspects of graphs on surfaces and of covering space theory in general.

The first three chapters deal with the theory for manifolds of any dimension. In the final chapter, the special case of surfaces is examined is greater detail.

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