We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order \(q\in [0,\infty )\) . We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if \(q>1/2\) . Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if \(q>3/2\) , whereas if \(q<3/2\) then finite-time blowup may occur. The geodesic completeness for \(q>3/2\) is obtained by proving metric completeness of the space of \(H^q\) -immersed curves with the distance induced by the Riemannian metric.
