The geometry of generalized flat ribbons

This thesis is concerned with the Riemannian geometry of developable surfaces in Euclidean space, their generalization and application. One of our main interests is to model surfaces by means of intrinsically flat ribbons. Naturally, we consider tangent (first-order) approximations, where each ribbon has the same distribution of tangent planes, i.e., the same normal field, as the surface along a given curve. Each ribbon is thus isometric to a planar ribbon constructed along the so-called Cartan development of the original surface curve. We show that the planar and the approximating (bent) ribbons are dual, rolling-related, constructions. In particular, the geodesic torsion and the normal curvature of the surface curve completely determine the rotational part of the rolling as well as the ruling angle function of the ribbons. On this ground, we present a rolling-based method for approximating surfaces via collections of flat ribbons, which we call ribbonizations. They are in some sense akin to the triangulations typically used in finite element methods and in computeraided geometric design. In higher dimensions, we study the local problem of approximating a hypersurface by means of a flat hyper-ribbon along a prescribed curve. We show that the well-known two-dimensional condition for the existence and uniqueness of the approximating ribbon naturally extends to this more general setting. In higher codimension, we limit our analysis to the family of flat and ruled (or developable) submanifolds. More precisely, we solve the following problem: Given a smooth distribution D of m-dimensional planes along a smooth regular curve γ in Rm+n, find all m-dimensional developable submanifold of Rm+n that pass through γ and whose tangent bundle along γ is precisely D. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution.