Narrow Results

We give a variant of the Beauville–Narasimhan–Ramanan correspondence for irregular parabolic Higgs bundles on smooth projective curves with fixed semi-simple irregular part and show that it defines a Poisson isomorphism between certain irregular Dolbeault moduli spaces and relative Picard bundles of families of ruled surfaces over the curve.

This paper treats the strict semi-stability of the symmetric powers $S^{k}E$ of a stable vector bundle $E$ of rank $2$ with even degree on a smooth projective curve $C$ of genus $g\ge 2$. The strict semi-stability of $S^{2}E$ is equivalent to the orthogonality of $E$ or the existence of a bisection on the ruled surface ${\mathbb{P}}_C(E)$ whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of $E$ with strictly semi-stable $S^{3}E$. Moreover, it is shown that when $S^{2}E$ is stable, every symmetric power $S^{k}E$ is stable for all but a finite number of $E$ in the moduli of stable vector bundles of rank $2$ with fixed determinant of even degree on $C$.

A ruled surface of revolution with moving axes and angles is a rational tensor product surface generated from a line and a rational space curve by rotating the line (the directrix) around vectors and angles generated by the rational space curve (the director). Only right circular cylinders and right circular cones are ruled surfaces that are also surfaces of revolution, but we show that a rich collection of other ruled surfaces such as hyperboloids of one sheet, 2-fold Whitney umbrellas, and a wide variety other interesting ruled shapes are ruled surfaces of revolution with moving axes and angles. We present a fast way to compute the implicit equation of a ruled surface of revolution with moving axes and angles from two linearly independent vectors that are perpendicular to the directrix of the surface. We also provide an algorithm for determining whether or not a given rational ruled surface is a ruled surface of revolution with moving axes and angles.

In this paper, we consider the minimal genus problem in a ruled 4-manifold M. There are three key ingredients in the studying, the action of diffeomorphism group of M on H²(M,Z), the geometric construction of surfaces representing a cohomology class and the generalized adjunction formula. At first, we discuss the standard form (see Definition 1.1) of a class under the action of diffeomorphism group on H²(M,Z), we prove the uniqueness of the standard form. Then we construct some embedded surfaces representing the standard forms of some positive classes, the generalized adjunction formula is used to show that these surfaces realize the minimal genera.

In this paper, we study structures and properties of Null scrolls. We define the (relative) invariants for Null scrolls by using a kind of standard equation. Using these (relative) invariants of Null scrolls, we give some new characterizations and classifications of Null scrolls and B-scrolls.

In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcı. In this study, we obtain the distribution parameters of the ruled surface and then some results and theorems are presented with special cases. Moreover, some relationships among asymptotic curve and striction line of the base curve of the ruled surface have been found.

In this paper, a one-to-one correspondence is given between the tangent bundle of unit 2-sphere, $T{𝕊}^2$, and the unit dual sphere, ${𝕊}_{𝔻}^2$. According to Study’s map, to each curve on ${𝕊}_{𝔻}^2$ corresponds a ruled surface in Euclidean 3-space, ${ℝ}^3$. Through this correspondence, we have corresponded to each curve on $T{𝕊}^2$ a unique ruled surface in ${ℝ}^3$. Moreover, the relationships between the developability conditions of these ruled surfaces and their striction curves are analyzed. It is shown that the ruled surfaces corresponding to the involute–evolute curve couples on $T{𝕊}^2$ are developable.

Using the curvature theory for the ruled surfaces a technique for robot trajectory planning is presented. This technique ensures the calculation of robot’s next path. The positional variation of the Tool Center Point (TCP), linear velocity, angular velocity are required in the work area of the robot. In some circumstances, it may not be physically achievable and a re-computation of the robot trajectory might be necessary. This technique is suitable for re-computation of the robot trajectory. We obtain different robot trajectories which change depending on the darboux angle function and define trajectory ruled surface family with a common trajectory curve with the rotation trihedron. Also, the motion of robot end effector is illustrated with examples.

This paper reviews the persistent rigid-body motions and examines the geometric conditions of the persistence of some special frame motions on a slant helix. Unlike the Frenet–Serret motion on general helices, the Frenet–Serret motion on slant helices can be persistent. Moreover, even the adapted frame motion on slant helices can be persistent. This paper begins by explaining one-dimensional rigid-body motions and persistent motions. Then, it continues to present persistent frame motions in terms of their instantaneous twists and axode surfaces. Accordingly, the persistence of any frame motions attached to a curve can be characterized by the pitch of an instantaneous twist. This work investigates different frame motions that are persistent, namely frame motions whose instantaneous twist has a constant pitch. In particular, by expressing the connection between the pitch of Frenet–Serret motion and the pitch of adapted frame motion, it demonstrates that both the Frenet–Serret motion and the adapted frame motion are persistent on slant helices.

In this paper, we construct a surface family possessing a Mannheim B-pair of a given curve as an asymptotic curve. Using the Bishop frame of the given Mannheim B curves, we present the surface as a linear combination of this frame and analyze the necessary and sufficient condition for a given curve such that its Mannheim B-pairs are both isoparametric and asymptotic on a parametric surface. The extension to ruled surfaces is also outlined. In addition, necessary and sufficient conditions have been given for the development of this ruled surface family. Finally, we present some interesting examples to show the validity of this study.

In this paper, we define a ruled surface obtained by an infinitesimal bending of a curve in Minkowski $3$-space. We also characterize such ruled surfaces according to causal characters of the curve. Furthermore, we find appropriate conditions of ruled surfaces obtained by infinitesimal bendings of curves to be (lightlike) developable, in particular, cylindrical or conical in Minkowski $3$-space, with examples presented.