Kentaro Saji, Keisuke Teramoto
We investigate behavior of principal curvatures and principal vectors near a non-degenerate singular point of the first kind of frontals. As an application, we extend the notion of Ribaucour transformations to frontals with singular points.
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We study singularities and geometric properties of surfaces given by the singular loci of normal congruence of frontals with pure-frontal singular points. These surfaces consist of the normal ruled surface and focal surfaces of the initial frontal. For the normal ruled surface, we give characterizations of singularities in terms of geometric invariants of the initial frontal defined on the set of singular points. For focal surfaces, we show relation between certain singularities of them and geometric property of the given frontal. Moreover, we consider behavior of Gaussian curvature of focal s…
We characterize the extendibility of the normal curvature on frontals and we give a representation formula of this type of frontals. Also we give representation formulas for wavefronts on all types of singularities and others sub classes of these. Some applications to asymptotic curves and lines of curvature on frontals are made.
Equiaffine structure on frontals
本文将等仿射结构与 Blaschke 向量场推广到前沿面，给出其存在性的充要条件及例子，并建立了前沿面版本的仿射微分几何基本定理。
For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves. Moreover, we give properties of surfaces of revolution with singularities and cones.
On vertices of frontals in the Euclidean plane
本文研究带奇点平面曲线（frontals/fronts）的顶点，利用渐屈线给出一般顶点定义，并建立闭前曲线满足四顶点定理的条件，辅以正反例子。
This paper introduces pedal curves of spacelike frontals in the hyperbolic 2-space. We mainly investigate the singularities of these hyperbolic pedal curves of spacelike frontals for non-singular and singular dual curve germs. We then show that for non-singular dual curve germs, the singularities of a pedal curve are dependent on the singularities of the first hyperbolic Legendrian curvature germ and the location of the pedal point, while for singular dual curve germs, they depend upon the singularities of both hyperbolic Legendrian curvature germs and also the location of the pedal point. We …
A characterization of the Legendre involution on the class of generic frontals
该研究在温和假设下刻画了一般前向类上的勒让德对合，证明不动点集与勒让德对合完全一致的对合就是勒让德对合，并同时给出其自然复化结论。
Helicoidal surfaces of frontals in Euclidian space as deformations of surfaces of revolution, with singularities
本文利用 Legendre 曲线与标架曲面理论研究带奇异性的前曲线螺旋面，给出其成为 frontal/front 的条件及曲率公式，并证明平行曲面与焦曲面的螺旋性及奇点稳定性。
Helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space
本文定义了非光状前向在洛伦兹-闵可夫斯基3空间中的两种螺旋面，并研究其成为光锥框架底面的条件。通过构造适当的微分同胚变换并利用(i,j)-尖点和(i,j)-尖点边缘准则，建立了两种类型螺旋面在奇异区域上的识别定理。
Singularities of helicoidal surfaces of frontals in the Euclidean space
本文研究欧氏空间中由含奇点前曲线生成的螺旋面，将其刻画为广义标架基曲面，并利用 Legendre 曲线曲率给出其不变量、曲率及奇点判据。
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…
The total absolute curvature of submanifolds with singularities
本文研究了带有奇点的子流形（称为frontals）在欧几里得空间中的总绝对曲率，证明了其总绝对曲率大于等于贝蒼恒数之和，并在总绝对曲率为2且所有奇点为第一类时，图像与仿射子空间的闭合凸域一致。
A geometric construction is provided that associates to a given flat front in $\mathbb{H}^3$ a pair of minimal surfaces in $\mathbb{R}^3$ which are related by a Ribaucour transformation. This construction is generalized associating to a given frontal in $\mathbb{H}^3$ , a pair of frontals in $\mathbb{R}^3$ that are envelopes of a smooth congruence of spheres. The theory of Ribaucour transformations for minimal surfaces is reformulated in terms of a complex Riccati ordinary differential equation for a holomorphic function. This enables one to simplify and extend the classical theory, that in pr…
The total absolute curvature of closed curves with singularities
研究将 Fenchel 定理推广至 $\mathbb{R}^n$ 中带奇点的闭 frontal，给出全绝对曲率下界及等号情形，并刻画尖点数量与简单性的关系。
We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all non-degenerate singularities of both types, as well as many degenerate singularities. We also give a method for solving the singular geometric Cauchy problem: construct a pseudospherical frontal containing a given regular space curve as a non-degenerate singular curve. The solutio…
We give criteria for which a principal curvature becomes a bounded $C^\infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.
We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of singularities and geometrical properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature of the focal surface.
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