The duality between singular points and inflection points on wave fronts

Local geometry of surfaces in $\mathbf R^4$

The indicatrix or curvature ellipse and the characteristic curve of a surface in $\mathbf R^4$ are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and hyperbolic points, and the inflection points, are also discussed.

The duality of conformally flat manifolds

In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points (i.e. admissible GCF-manifolds), * frontals as hypersurfaces in the lightcone Q^{n+1}_+, * frontals as hypersurfaces in the hyperbolic space H^{n+1}, * sapacelike frontals as hypersurfaces in the de Sitter space S^{n+1}_1. Recently, the duality of conformally flat Riemannian mani…

Natural parameterizations of closed projective plane curves

A natural parametrization of smooth projective plane curves which tolerates the presence of sextactic points is the Forsyth-Laguerre parametrization. On a closed projective plane curve, which necessarily contains sextactic points, this parametrization is, however, in general not periodic. We show that by the introduction of an additional scalar parameter $α\leq \frac12$ one can define a projectively invariant $2π$-periodic global parametrization on every simple closed convex sufficiently smooth projective plane curve without inflection points. For non-quadratic curves this parametrization, whi…

广义Frenet标架与奇异空间曲线的标架序列

Generalized Frenet frames and frame sequences of singular space curves

针对经典Frenet标架在奇异点和曲率零点失效的问题，基于球面前沿理论和Legendre对偶，引入奇异空间曲线的广义Frenet标架及标架序列，统一了Frenet、Bishop标架和球面前沿的渐屈线-渐伸线对应，并导出递推公式。

Bloch-Ros原理及其在曲面理论中的应用

Bloch-Ros principle and its application to surface theory

本文基于Zalcman和Ros的工作提出Bloch-Ros原理，揭示了正规族理论、值分布理论与极小曲面理论间的三位一体现象，并将其推广至多种曲面类，给出了判定亚纯函数高斯曲率估计的有效准则。