ar X iv : m at h / 06 06 00 7 v 1 [ m at h . G T ] 1 J un 2 00 6 Curves of Finite Total Curvature

We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry. To explore these ideas, we consider theorems of Fáry/Milnor, Schur, Chakerian and Wienholtz. Here we introduce the ideas of discrete differential geometry in the simplest possible setting: the geometry and curvature of curves, and the way these notions relate for polygonal and smooth curves. The viewpoint has been partly inspired by work in geometric knot theory, which studies geometric properties of space curves in relation to their knot type, and looks for optimal shapes for given knots. We start out by considering total curvature; following Milnor [Mil50] we give a unified treatment where polygonal and smooth curves are both contained in the larger class of FTC (finite total curvature) curves. We explore the connection between FTC curves and BV functions. Then we examine the theorems of Fáry/Milnor, Schur and Chakerian in terms of FTC curves. We sketch a proof of a result by Wienholtz in integral geometry. We end by looking at ways to define curvature density for polygonal curves. A companion article [DS06] examines more carefully the topology of FTC curves, showing that any two sufficiently nearby FTC graphs are isotopic. Our whole approach in this survey should be compared to that of Alexandrov and Reshetnyak [AR89], who develop much of their theory for curves having one-sided tangents everywhere, a class somewhat more general than FTC. 1. Length and total variation We want to consider the geometry of curves. Of course curves – unlike higher-dimensional manifolds – have no local intrinsic geometry. So we mean the extrinsic geometry of how the curve sits in some ambient space M . Usually M will be in euclidean d-space E, but the study of space curves naturally leads also to the study of curves on spheres. Thus we also allow M to be a compact smooth Riemannian manifold; for convenience we