Abstract We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties known from their smooth counterparts. Special emphasis is paid to discrete flows built from Bäcklund transformations in the respective space forms. The invariants of the flows form a hierarchy of curves and we show that discrete elastic and constrained elastic curves can be characterized as elements of this hierarchy. This work also includes an introductory chapter on discrete curve theory in space forms, where we find discrete Frenet-type formulas and describe an associated family related to a fundamental theorem.

Abstract In algebraic geometry there is a well-known categorical equivalence between the category of normal proper integral curves over a field k and the category of finitely generated field extensions of k of transcendence degree 1. In this paper we generalize this equivalence to the category of normal quasi-compact universally closed separated integral k-schemes of dimension 1 and the category of field extensions of k of transcendence degree 1. Our key technique are morphisms of finite expansion which can be considered as relaxation of morphisms of finite type. Since the schemes in the generalized category have many properties similar to normal proper integral curves, we call them normal integral universally closed curves over k.