Abstract
Let ℰ be the Fréchet 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 \(\pi (\mathcal{C})\).
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References
Binz, E.: Two natural metrics and their derivatives on a manifold of embeddings. Monatsh. Math. 89, 275–288 (1980)
Kriegl, A., Michor, P.: The Convenient Setting for Global Analysis. Surveys and Monographs, vol. 53. AMS, Providence (1997)
Michor, P., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8, 1–48 (2006)
Michor, P., Ratiu, T.: On the geometry of the Virasoro-Bott group. J. Lie Theory 8, 293–309 (1998)
Salvai, M.: Force free conformal motions of the sphere. Differ. Geom. Appl. 16, 285–292 (2002)
Salvai, M.: Another motivation for the hyperbolic space: Segments moving on the line. Math. Intell. 29, 6–7 (2007)
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Partially supported by FONCyT, CONICET and SECyT (UNC).
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Salvai, M. Geodesic paths of circles in the plane. Rev Mat Complut 24, 211–218 (2011). https://doi.org/10.1007/s13163-010-0036-5
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DOI: https://doi.org/10.1007/s13163-010-0036-5