2 Preliminaries

Denote by \(\mathbb{S}^1=\left\{x_1^2+x_2^2=1\right\}\subset \mathbb R^2\) the unit circle in \(\mathbb R^2\). Unless otherwise stated, we always assume that \(\mathbb{S}^1\) inherits the standard orientation from \(\mathbb R^2\) and consider \(\mathrm{d}\theta\) to be the associated volume form and \(\partial_\theta\) the dual vector field. Abusing notation slightly, let \(\theta:\mathbb{S}^1\to [0,2\pi )\) be the compatible chart with \(\theta(\mathbf{e}_1)=0\). Let \(\pi: \mathbb R\to \mathbb{S}^1\) be the covering map so that \(\pi^*\mathrm{d}\theta=\mathrm{d}x\) and \(\pi(0)=\mathbf{e}_1\) – here \(x\) is the usual coordinate on \(\mathbb R\). Denote by \(I: \mathbb{S}^1\to \mathbb{S}^1\) the involution given by \(I(p)=-p\). Hence, \(\theta(I(p))=\theta(p)+\pi \mod 2\pi\).

A degree-one curve, \(\sigma\), is uniquely determined by the data \((\mathbf{T}_\sigma, \mathbf{x}_\sigma, L(\sigma))\). Denote by \(\mathcal{D}^{k+1,\alpha}\) the set of degree-one curves of class \(C^{k+1,\alpha}\) and by \(\mathcal{D}^{k+1,\alpha}_+\subset \mathcal{D}^{k+1,\alpha}\) the set of strictly convex degree-one curves of of class \(C^{k+1,\alpha}\). The length element associated to \(\sigma\) is \(\mathrm{d}s=\frac{L(\sigma)}{2\pi} \mathrm{d}x=\frac{L(\sigma)}{2\pi}\pi^* \mathrm{d}\theta=\pi^*\widetilde{\mathrm{d}s}\). If \(\sigma\in \mathcal{D}^2\), then the geodesic curvature, \(\kappa_\sigma\), of \(\sigma\) satisfies \(\kappa_\sigma=\pi^*\tilde{\kappa}_\sigma\) where \(\tilde{\kappa}_\sigma\in C^{k-1,\alpha}(\mathbb{S}^1)\) satisfies \[\int_{\mathbb{S}^1} \tilde{\kappa}_\sigma \; \widetilde{\mathrm{d}s}= 2\pi.\] Conversely, given such a \(\kappa_\sigma\) there is a degree-one curve with geodesic curvature \(\kappa_\sigma\). Abusing notation slightly, we will not distinguish between \(\mathrm{d}s\) and \(\widetilde{\mathrm{d}s}\) and between \(\kappa_\sigma\) and \(\tilde{\kappa}_\sigma\). Clearly, \(\sigma \in\mathcal{D}^2_+\) if and only if \(\kappa_\sigma>0\).

The standard parameterization of \(\mathbb{S}^1\) is given by the data \((\mathbf{T}_0, \mathbf{e}_1, 2\pi)\) where \[\mathbf{T}_0(p)=-\sin(\theta(p))\mathbf{e}_1+\cos(\theta(p))\mathbf{e}_2.\] Let \(\mathrm{ Diff}_+^{k,\alpha}(\mathbb{S}^1)\) denote the orientation preserving diffeomorphisms of \(\mathbb{S}^1\) of class \(C^{k,\alpha}\) – that is bijective maps of class \(C^{k,\alpha}\) with inverse of class \(C^{k,\alpha}\). Endow this space with the usual \(C^{k,\alpha}\) topology. For \(\sigma\in \mathcal{D}_+^{k+1,\alpha}\), we call the map \[\phi_{\sigma}=\mathbf{T}_{0}^{-1}\circ \mathbf{T}_{\sigma}\] the induced diffeomorphism of \(\sigma\). Clearly, the induced diffeomorphism of the standard parameterization of \(\mathbb{S}^1\) is the identity map.

For \(f \in C^k(\mathbb{S}^1)\), let \(f'=\partial_{\theta}f\), \(f^{\prime\prime}=(f^{\prime})^{\prime}\) and likewise for higher order derivatives. Observe that, for \(\phi \in \mathrm{Diff}^1_+(\mathbb{S}^1)\), we have \(\phi'>0\) where \(\phi'\in C^0(\mathbb{S}^1)\) satisfies \(\phi^*\mathrm{d} \theta=\phi'\mathrm{d}\theta\). A simple computation shows that if \(\sigma\in \mathcal{D}_+^2\), then \(\phi'_\sigma= {\kappa_\sigma} \frac{L(\sigma)}{2\pi}\).