Overview
This work explores the geometric implications of the analytic continuation of the Frenet curvature 1-form for an analytic plane curve. It focuses on the derivation of a distinguished conformal coordinate net in the neighborhood of such a curve, generated directly by its curvature.
Research Context
Fermi coordinates offer a local coordinate system in the vicinity of a plane curve. However, these coordinates are not conformal and do not directly originate from the curvature form. The central question addressed is whether a real analytic plane curve uniquely defines a conformal coordinate net that is generated by its curvature.
Approach
The study considers the Frenet curvature 1-form of an analytic curve, denoted as $\omega_F=-k\,ds$. The approach involves examining the holomorphic continuation of this form, expressed as $\Omega=K(q)\,dq$. This holomorphic differential is a central element in the reconstruction of the conformal coordinate net.
Findings
- The holomorphic continuation $\Omega=K(q)\,dq$ of the Frenet curvature 1-form $\omega_F=-k\,ds$ naturally determines a conformal coordinate net.
- This determination occurs through the reconstruction formula $z_q=\exp\left(i\int\Omega\right)$.
- The holomorphic differential $\Omega$ is identified as a conformal connection differential.
- This conformal connection differential is associated with the coordinate net determined by the holomorphic continuation.
- The restriction of $\Omega$ to the initial curve coincides with the original curvature form.
- For a pair of dual conformal nets, the derivative of the conformal transition map can be represented geometrically as $z_q=-\overline{\left(\frac{\widetilde\omega}{\omega}\right)}$. Here, $\omega$ and $\widetilde{\omega}$ represent the complex curvature coefficients of the two nets.
- While individual complex curvature coefficients $\omega$ and $\widetilde{\omega}$ are not generally holomorphic, their normalized conjugate dual ratio is holomorphic.
- An analytic plane curve thus determines a distinguished conformal coordinate net in its neighborhood via the holomorphic continuation of its curvature form.