Sobolev metrics on spaces of discrete regular curves

Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and considers the induced Riemannian metric on a finite dimensional approximation space. Surprisingly, the theoretical properties of the corresponding finite dimensional Riemannian manifolds have not yet been studied in detail, which is the content of the present article. Our main theorem concerns metric and geodesic completeness and mirrors the results of the infinite dimensional setting as obtained by Bruveris, Michor and Mumford.