Geodesic paths of circles in the plane

Abstract

Let ℰ be the Frechet space of all positively oriented embeddings of the circle in ℝ2 and let ℰ/∼ be the quotient of ℰ modulo orientation preserving diffeomorphisms of the circle. Let π:ℰ→ℰ/∼ be the canonical projection and let \(\mathcal{C}\) denote the space of all constant speed circles. We study geodesics in \(\mathcal{C}\) and \(\pi (\mathcal{C})\) endowed with the Riemannian metrics induced from the canonical weak Riemannian metrics on ℰ and ℰ/∼, respectively. We also study the holonomy of closed paths in \(\pi (\mathcal{C})\).