Pedal curves of hyperbolic frontals and their singularities

Singularities of parallels to tangent developable surfaces

It is known that the class of developable surfaces which have zero Gaussian curvature in three dimensional Euclidean space is preserved by the parallel transformations. A tangent developable surface is defined as a ruled developable surface by tangent lines to a space curve and it has singularities at least along the space curve, called the directrix or the the edge of regression. Also the class of tangent developable surfaces are invariant under the parallel deformations. In this paper the notions of tangent developable surfaces and their parallels are naturally generalized for frontal curves…

带有奇点的子流形的总绝对曲率

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。

欧氏空间中前曲线螺旋面的奇点

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。

Surfaces of revolution of frontals in the Euclidean space

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.

Wigner焦散与欧几里得平面中前向曲线的中心对称集的奇点

Singularities of the Wigner Caustic and the Centre Symmetry Set of Frontal Curves in the Euclidean Plane

将Wigner焦散与中心对称集推广至前向曲线，给出尖点判据、降阶关系及尖点计数不等式推广，并分析奇异平行对的共振传递结果。

Pseudospherical surfaces with singularities

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…

Affine connections and singularities of tangent surfaces to space curves

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, or, more generally a directed curve or a frontal curve which has well-defined tangent directions along the curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the c…

带奇点的闭曲线的全绝对曲率

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。

A generalization of Zakalyukin's lemma, and symmetries of surface singularities

Zakalyukin's lemma asserts that the coincidence of the images of two wave front germs implies the right equivalence of corresponding map germs under a certain genericity assumption. The purpose of this paper is to give an improvement of this lemma for frontals. Moreover, we give several applications for singularities on surfaces.

Singularities of tangent surfaces to generic space curves

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umeh…

The axial curvature for corank 1 singular surfaces

For singular corank 1 surfaces in $\mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes the singular curvature for frontal type singularities. We then study contact properties of the surface with respect to the plane orthogonal to the axial vector and show how they are related to the axial curvature. Finally, for certain fold type singularities, we relate the axial curvature with the Gaussian curvature of an appropriate blow up.

Flat Surfaces with singularities in Euclidean 3-space

It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $ν$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,ν)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this pape…