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We prove local well-posedness of the geodesic equations both on the Banach manifold $\\mathcal{I}^{q}(S^1,\\mathbb{R}^{d})$ and on the Fr\\'{e}chet-manifold $\\operatorname{Imm}(S^1,\\mathbb{R}^d)$ provided the order of the metric is greater or equal to one. 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