Geodesic homotopies

In this paper we wish to endow the manifold M of smooth curves in ℝ_n (either closed curves or open curves with fixed endpoints) with a Riemannian structure that allows us to treat homotopies between two curves C₀ and C₁ as trajectories with computable lengths. If we regard a curve as all possible reparameterizations of a C¹ mapping of the unit interval into ℝ_n, then the tangent space at a point on M corresponds to all possible non-tangential flows of the underlying curve. We note that a Riemannian metric corresponds to a choice of inner-product on this tangent space. Once this is defined, we can compute distances between curves and consider the natural problem of finding geodesics which will yield minimal length homotopies between C₀ and C₁. We begin by noting that a seemingly natural inner-product corresponding to a geometric version of ℓ₂ does not yield useful geodesics and will instead introduce a sequence of conformally modfied inner-products which has interesting limit properties.