Properties of Sobolev-type metrics in the space of curves

We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S¹ → ℝⁿ (n ≥ 2, S ¹ is the circle). We study geometries on the manifold of curves, provided by Sobolev-type Riemannian metrics H^j. These metrics have been shown to regularize gradient flows used in computer vision applications (see [13, 14, 16] and references therein). We provide some basic results on H^j metrics; and, for the cases j = 1, 2, we characterize the completion of the space of smooth curves. We call these completions "H ¹ and H² Sobolev-type Riemannian manifolds of curves". This result is fundamental since it is a first step in proving the existence of geodesics with respect to these metrics. As a byproduct, we prove that the Fréchet distance of curves (see [7]) coincides with the distance induced by the "Finsler L^∞ metric" defined in §2.2 of [18].