We introduce the notions of the pseudospherical normal Darboux images for the curve on a lightlike surface in Minkowski 3-space and study these Darboux images by using technics of the singularity theory. Furthermore, we give a relation between these Darboux images and Darboux frame from the viewpoint of Legendrian dualities.

We define framed curves (or frontals) on Euclidean 2-sphere, give a moving frame of the framed curve and define a pair of smooth functions as the geodesic curvature of a regular curve. It is quite useful for analysing curves with singular points. In general, we can not define evolutes at singular points of curves on Euclidean 2-sphere, but we can define evolutes of fronts under some conditions. Moreover, some properties of such evolutes at singular points are given.

The main goal of this paper is to characterize evolutes at singular points of curves in hyperbolic plane by analysing evolutes of null torus fronts. We have done some work associated with curves with singular points in Euclidean 2-sphere [H. Yu, D. Pei, X. Cui, J. Nonlinear Sci. Appl., 8 (2015), 678-686]. As a series of this work, we further discuss the relevance between singular points and geodesic vertices of curves and give different characterizations of evolutes in the three pseudo-spheres. c 2015 All rights reserved.Keywords: Evolute, null torus front, null torus framed curve, hyperbolic plane. 2010 MSC: 51B20, 53B50, 53A35. PreliminariesAs a subject closely related to nonlinear sciences, singularity theory [1,2,3,4,7] has been extensively applied in studying classifications of singularities of submanifolds in Euclidean spaces and semi-Euclidean spaces [11,12]. However, little information has been got at singular points from the view point of differential geometry. In this paper we characterize the behaviors at singular points of curves in hyperbolic plane.If a curve has singular points, we can not construct its moving frame. However, we can define a moving frame of a frontal for a framed curve in the unit tangent bundle. Along with the moving frame, we get a pair of smooth functions as the geodesic curvature of a regular curve. It is quite useful to analyse curves with singular points. Because we can get information at singular points through analysing framed curves. We have researched curves with singular points in Euclidean 2-sphere in [13]. In general, one can not define evolutes at singular points of curves on Euclidean 2-sphere, but we define evolutes of fronts under some conditions.

The main goal of this paper is to study the singularities of null hypersurfaces of pseudonull curves. To do this we construct a null frame and a Lorentz distance-squared function of the pseudonull curve. The relations are shown between singularities of the null hypersurfaces and those, of the Lorentz distance-squared function. And we reveal the geometric meanings of the singularities of such hypersurfaces. In addition, we also introduce some properties of the nullcone Gaussian surface of the pseudonull curve.

Abstract:In this paper, we define the supersolvable order of hyperplanes in a supersolvable arrangement, and obtain a class of inductively free arrangements according to this order. Our main results improve the conclusion that every supersolvable arrangement is inductively free. In addition, we assert that the inductively free arrangement with the required induction table is supersolvable.