AI 中文总结
研究解析平面曲线能否由曲率确定特殊保形坐标网，通过弗伦内曲率1 - 形式的全纯延拓，利用重构公式等确定保形坐标网，给出对偶网映射导数表示，证明曲线可借此确定邻域内特殊保形坐标网。
AI 中文摘要
费米坐标在平面曲线附近提供了自然的局部坐标系。然而，它们不保形且并非直接由曲率形式生成。本文探讨实解析平面曲线是否能确定由其曲率生成的特殊保形坐标网。设\(\omega_F = -k\,ds\)为解析曲线的弗伦内曲率1 - 形式，其全纯延拓\(\Omega = K(q)\,dq\)通过重构公式\(z_q=\exp\left(i\int\Omega\right)\)自然地确定保形坐标网。全纯微分\(\Omega\)与坐标网相关的保形联络微分等同，其在初始曲线上的限制与原始曲率形式一致。对于一对对偶保形网，保形过渡映射的导数有几何表示\(z_q = -\overline{\left(\frac{\widetilde\omega}{\omega}\right)}\)。虽然一般曲率系数不全纯，但它们的归一化共轭对偶比全纯。因此，解析平面曲线通过其曲率形式的全纯延拓在其邻域确定了特殊保形坐标网。
英文摘要
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 \[ ω_F=-k\,ds \] be the Frenet curvature 1-form of an analytic curve. We show that its holomorphic continuation \[ Ω=K(q)\,dq \] naturally determines a conformal coordinate net through the reconstruction formula \[ z_q=\exp\left(i\intΩ\right). \] The holomorphic differential \(Ω\) 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 \[ z_q=-\overline{\left(\frac{\widetildeω}ω\right)}, \] where ωand \widetildeω 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.
Comments9 pages. Corrected author name
