Analytic Continuation of the Frenet Curvature Form of an Analytic Plane Curve

Abstract

Fermi coordinates provide a natural local coordinate system near a plane curve. Nevertheless, they are not conformal and are not generated directly by the curvature form. The present work asks whether a real analytic plane curve determines a distinguished conformal coordinate net generated by its curvature. Let \[ \omega_F=-k\,ds \] be the Frenet curvature 1-form of an analytic curve. We show that its holomorphic continuation \[ \Omega=K(q)\,dq \] naturally determines a conformal coordinate net through the reconstruction formula \[ z_q=\exp\left(i\int\Omega\right). \] The holomorphic differential \(\Omega\) is identified with a conformal connection differential associated with the coordinate net and whose restriction to the initial curve coincides with the original curvature form. For a pair of dual conformal nets, the derivative of the conformal transition map admits the geometric representation \[ z_q=-\overline{\left(\frac{\widetilde\omega}{\omega}\right)}, \] where \omega and \widetilde{\omega} denote the complex curvature coefficients of the two nets. Although neither curvature coefficient is holomorphic in general, their normalized conjugate dual ratio is holomorphic. Thus an analytic plane curve determines, through the holomorphic continuation of its curvature form, a distinguished conformal coordinate net in its neighborhood.