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A smooth ruled surface in 4‐space has only parabolic points or inflection points of the real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along which the parallel projection exhibits A$\mathcal {A}$‐singularities of type butterfly or worse. In particular, such parabolic points can be classified as butterfly hyperbolic, parabolic, or elliptic points depending on the value of the discriminant of a binary differential equation (BDE). Also, whenever such discriminant is positive, we ensure that the integral curves of these directions form a pair of foliations on the ruled surface. Moreover, the set of points that nullify the discriminant is a regular curve transverse to the regular curve formed by inflection points of the real type. Finally, using a particular projective transformation, we obtain a simple parametrization of the ruled surface such that the moduli of its 5‐jet identify a butterfly hyperbolic/parabolic/elliptic point, as well as we get the stable configurations of the solutions of BDE in the discriminant curve. [ABSTRACT FROM AUTHOR]

We prove that the modular component , constructed in the Main Theorem in Fania and Flamini (Adv Math 436:109409, 2024. https://doi.org/10.1016/j.aim.2023.109409), of Ulrich vector bundles of rank r and given Chern classes, on suitable threefold scrolls X e over Hirzebruch surfaces F e ≥ 0 , which arise as tautological embeddings of projectivization of very-ample vector bundles on F e , is generically smooth, irreducible and unirational. A stronger result holds for the suitable associated moduli space of vector bundles of rank r and given Chern classes on F e , Ulrich w.r.t. the very ample polarization which turns out to be generically smooth, irreducible and unirational. [ABSTRACT FROM AUTHOR]

We study deformed Hermitian Yang–Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application, we provide many new examples of dHYM connections coupled to a variable background Kähler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the Kähler–Yang–Mills system of Álvarez-Cónsul, Garcia-Fernandez, and García-Prada, and in this limit, our solutions converge smoothly to those constructed by Keller and Tønnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realization as B-branes, the small radius limit, and canonical representatives of complexified Kähler classes. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to research the change in the magnetic field using the curvature theory of ruled surfaces. For this, we examine magnetic curves of the curve whose existence is guaranteed from the derivative formulas of the rotation frame. The Killing vector fields and Lorentz forces of these magnetic curves are calculated. Additionally, correlations between curvatures are obtained. Flux ruled surfaces are obtained and the conditions for developable are achieved. Finally, examples of the magnetic flux developable ruled surfaces are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus γ and degree e in P e - γ . Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for γ ≥ 3 and e ≥ 4 γ + 5 , there exists a non-reduced component H of the Hilbert scheme of smooth curves of genus 3 e + 3 γ and degree 3 e + 1 in P e - γ + 1 . We show that dim T [ X ] H = dim H + 1 = (e - γ + 1) 2 + 7 e + 5 for a general point [ X ] ∈ H . [ABSTRACT FROM AUTHOR]

Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space E 3 . We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis. [ABSTRACT FROM AUTHOR]

Conformable fractional calculus is a relatively new branch of mathematics that seeks to extend traditional calculus to include non-integer order derivatives and integrals. This new form of calculus allows for a more precise description of physical phenomena, such as those related to fractal geometry and chaos theory. It also offers new tools for solving problems in physics, engineering, finance, and other areas. By using conformable fractional calculus, researchers can gain insight into the behavior of systems that would otherwise be difficult or impossible to analyze. In this article, we will discuss the fundamentals of conformable fractional calculus and its applications in various fields. For this purpose ruled surfaces are studied with conformable fractional calculus. The ruled surface is rearranged concerning the conformable surface definition and geometric properties are investigated. [ABSTRACT FROM AUTHOR]

We show that a length-minimizing disk inherits the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs inherit the upper curvature bound from the ambient space. [ABSTRACT FROM AUTHOR]

In this paper, we define three types of 2‐ruled hypersurfaces in the Minkowski 4‐space 피14. We obtain Gaussian and mean curvatures of the 2‐ruled hypersurfaces of type‐1 and type‐2 and some characterizations about its minimality. We also deal with the first Laplace–Beltrami operators of these types of 2‐ruled hypersurfaces in 피14. Moreover, the importance of this paper is the definition of these surfaces by using the octonions in 피14. Thus, this is a new idea and makes the paper original. We give an example of 2‐ruled hypersurface constructed by octonion, and we visualize the projections of the images with MAPLE program. Furthermore, the optical fiber can be defined as a one‐dimensional object embedded in the four‐dimensional Minkowski space 피14. Thus, as a discussion, we investigate the geometric evolution of a linearly polarized light wave along an optical fiber by means of the 2‐ruled hypersurfaces in a four‐dimensional Minkowski space. [ABSTRACT FROM AUTHOR]

The spherical indicatrix of a ruled surface is the image curve of its rulings on the sphere S 2 . In this paper, we present Lorentz forces and magnetic curves produced by the Frenet frame { T e , N e , B e } of the spherical indicatrix of a ruled surface in a magnetic field. We calculate magnetic vector fields of magnetic curves for T e , N e , B e . Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatrix. We obtain developability conditions for these surfaces. Finally, we give some examples to show magnetic flux surfaces. [ABSTRACT FROM AUTHOR]

In this study, we take the striction line which only exists on a ruled surface as an optical fiber. We examine Berry phase model of an optical fiber with a geodesic Frenet frame. We find the direction of the polarized light concerning the plane in which the electric field Ê is located. Also, the electromagnetic trajectories of optical fiber are obtained. Finally, we give the electromagnetic flux ruled surfaces formed by the Killing vector fields of the EM-trajectories of the optical fiber and numerical examples. [ABSTRACT FROM AUTHOR]

Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space E3. We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis.

In this paper, we realize a rapid, smooth iterative morphing algorithm between two immersed and closed spherical curves of arbitrary turning numbers. By compensating in turning numbers, we linearly interpolate between geodesic curvatures and the side lengths of their accurate spherical polygons approximation. Our algorithm is essentially based on a geometrical closure condition for a flow of spherical polygons involving the variation of both sides and exterior angles. A curvature smoothing flow using discrete geodesic curvature is adopted for smooth curve generation. To demonstrate the good properties of the method, a morphing of spherical curves and ruled surfaces is illustrated. [ABSTRACT FROM AUTHOR]

We investigate the existence of Ulrich vector bundles on suitable 3-fold scrolls X e over Hirzebruch surfaces F e , for any e ⩾ 0 , which arise as tautological embeddings of projectivization of very-ample vector bundles on F e that are uniform in the sense of Brosius and Aprodu–Brinzanescu, cf. [10] and [3] respectively. We explicitly describe components of moduli spaces of rank r ⩾ 1 vector bundles which are Ulrich with respect to the tautological polarization on X e and whose general point is a slope-stable, indecomposable vector bundle. We moreover determine the dimension of such components, proving also that they are generically smooth. As a direct consequence of these facts, we also compute the Ulrich complexity of any such X e and give an effective proof of the fact that these X e 's turn out to be geometrically Ulrich wild. At last, the machinery developed for 3–fold scrolls X e allows us to deduce Ulrichness results on rank r ⩾ 1 vector bundles on F e , for any e ⩾ 0 , with respect to a naturally associated (very ample) polarization. [ABSTRACT FROM AUTHOR]