Non-degenerate Surfaces Research Articles

A kind of rotational surfaces with a light-like axis in conformally flat pseudo-spaces of dimensional three

In this work, we define the rotational surface with a light-like axis in conformally flat pseudo-spaces $\left(\mathbb{E}_3^1\right)_\lambda$, where $\lambda$ is a radial-type conformal factor. We relate the principal curvatures of a non-degenerate surface that belongs to conformally equivalent spaces $\left(\mathbb{E}_3^1\right)_\lambda$ and $\mathbb{R}_1^3$, based on the radial conformal factor. Thus, we establish a relationship between the radial conformal factor and the profile curve of the rotational flat surface in $\left(\mathbb{E}_3^1\right)_\lambda$, but also for that of the rotational surface with zero extrinsic curvature.

Nondegenerate Integer Quantum Hall Effect from Topological Surface States in Ag2Te Nanoplates.

The quantum Hall effect is one of the exclusive properties displayed by Dirac Fermions in topological insulators, which propagates along the chiral edge state and gives rise to quantized electron transport. However, the quantum Hall effect formed by the nondegenerate Dirac surface states has been elusive so far. Here, we demonstrate the nondegenerate integer quantum Hall effect from the topological surface states in three-dimensional (3D) topological insulator β-Ag2Te nanostructures. Surface-state dominant conductance renders quantum Hall conductance plateaus with a step of e2/h, along with typical thermopower behaviors of two-dimensional (2D) massless Dirac electrons. The 2D nature of the topological surface states is proven by the electrical and thermal transport responses under tilted magnetic fields. Moreover, the degeneracy of the surface states is removed by structure inversion asymmetry (SIA). The evidenced SIA-induced nondegenerate integer quantum Hall effect in low-symmetry β-Ag2Te has implications for both fundamental study and the realization of topological magneto-electric effects.

On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaces

The complete classification of the orbits on subspaces under the action of the projective stabilizer of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of PGL(2,q2) (and PSL(2,q2)) arising from the Hermitian Veronese curve in PG(3,q2), a maximal rational curve embedded on a smooth Hermitian surface with some fascinating properties. The study of its orbits leads to a new construction of quasi-Hermitian surfaces: sets of points with the same combinatorial and geometric properties as a non-degenerate Hermitian surface.

Non-degenerate Rotational Surfaces of Coordinate Finite II-Type

Non-degenerate Rotational Surfaces of Coordinate Finite II-Type

Helicoidal surfaces in some conformally flat pseudo-spaces of dimensional three

We introduce the complete pseudo-Riemannian manifold [Formula: see text] with a conformally flat pseudo-metric satisfying Einstein’s equation. First, we get a theorem that associates the curvatures of a non-degenerate surface belonging to conformally equivalent spaces, say [Formula: see text] and [Formula: see text]. Next, we evaluate the existence of non-degenerate helicoidal surfaces in some conformally flat pseudo-spaces with pseudo-metrics corresponding to particular conformal factors (for example, a certain rotational symmetry with translational symmetry). We determine these conformal factors according to the causal characters of the axis of rotation. Right after, we get a two-parameter family of non-degenerate helicoidal surfaces with prescribed extrinsic curvature or mean curvature given by smooth functions, corresponding to both spacelike and timelike axes of rotation. As for the lightlike axis, we discuss some particular cases.

Geometry of hyperbolic Cauchy–Riemann singularities and KAM-like theory for holomorphic involutions

This article is concerned with the geometry of germs of real analytic surfaces in $$({\mathbb {C}}^2,0)$$ having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in $$({\mathbb {C}}^2,0)$$ having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions $$\{\tau _1,\tau _2\}$$ at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to $$\{z_1z_2=const\}$$ (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets.

Methodology for predicting the washability of apatite ores (Kirovsky mine, Kola Peninsula)

A statistical approach has been proposed that allows strictly defining the structure of a rock (ore), isolating fields of various structures in barycentric diagrams, and comparing them with the processing results. An original method is proposed for predicting the washability defined as the separation of apatite grains from polymineral aggregations at the crushing and flotation stages. It develops and formalizes the current approach that combines the mineral composition of ores, their structures and textures into a multifactor semiquantitative model. The method is based on the calculation of the probabilities (frequencies) of all types of intergranular contacts recorded in a symmetric matrix under a microscope in petrographic thin sections. Its canonical diagonal shape allows identifying the rock (ore) structure type. The algebraic theory of quadratic forms is used to demonstrate that the structure identified using this method corresponds to one of the types of n-dimensional nondegenerate quadratic surfaces, where n is the number of minerals in the rock (ore). The structure is invariant for a certain fluctuation in the probabilities of intergranular contacts. The calculated probabilities uniquely identify the imaging point in the corresponding barycentric diagram. The Hardy–Weinberg boundary is important as it corresponds to the ideal uniform mixing of minerals and the balance of various intergranular contacts for any modal composition of rocks (ores). It allows distinguishing fields of different structures in barycentric diagrams. The method proposed offers a means to identify ores with different washability characteristics at early exploration stages.

Double-bowl state in photonic Dirac nodal line semimetal

The past decade has seen a proliferation of topological materials for both insulators and semimetals in electronic systems and classical waves. Topological semimetals exhibit topologically protected band degeneracies, such as nodal points and nodal lines. Dirac nodal line semimetals (DNLS), which own four-fold line degeneracy, have drawn particular attention. DNLSs have been studied in electronic systems but there is no photonic DNLS. Here in this work, we provide a new mechanism, which is unique for photonic systems to investigate a stringent photonic DNLS. When truncated, the photonic DNLS exhibits double-bowl states (DBS), which comprise two sets of perpendicularly polarized surface states. In sharp contrast to nondegenerate surface states in other photonic systems, here the two sets of surface states are almost degenerate over the whole-spectrum range. The DBS and the bulk Dirac nodal ring (DNR) dispersion along the relevant directions, are experimentally resolved.

On the Group of Holomorphic Automorphisms of a Model Surface

In this paper, it is proved that the group $$\mathrm{Aut}\, Q$$ of all holomorphic automorphisms of a holomorphically homogeneous nondegenerate model surface $$Q$$ is a subgroup of the group of birational isomorphisms of the ambient complex space (the Cremona group) of uniformly bounded degree. The degree is estimated in terms of the dimension of the ambient space (Theorem 4). It is shown that no condition of the theorem can be weakened. In the paper, the question of the connectivity of $$\mathrm{Aut}\, Q$$ is also considered (Theorem 7). This paper is directly adjacent to the previous paper of the author [7].

Competency approach to the study of the second-order surface interface as the geometric location of points in space

The necessity of the formation of students' functional literacy as a competency approach to the training of future Mathematics teachers is substantiated on the example of studying of one of the topics of analytical geometry. It has been established that a prerequisite for the development of any competency prescribed in the standards of secondary education is the initial existence of a sufficiently new concept of functional literacy for a student of a certain level. The basic literacy comes down to the ability to read, write and express of one's thoughts correctly. Let us consider the issue of functional literacy from the point of view of the pedagogic specialty. Acquaintance with the well-known textbooks of analytic geometry allows us to say that 2nd order algebraic surfaces in Euclidean space are determined in most cases algebraically by means of equations. A constructive approach is also of use – surfaces are obtained by rotating 2nd degree curves around their symmetry axes and by deformation of the resulting surfaces by compression. The metric approach, as it used for 2nd order curves, is restricted only by the formulation of problems to find the certain locus of points in space. The exception is the article Dmitriy Perepyolkin which was published in 1936. In this paper the locus of points in space with the following characteristic property is studied – the ratio of the distance to a given point to the distance to a given straight line is constant. The strait line is assumed not to contain the point. The study is held out in pure geometrical manner – it is done using the method of sections and known loci of points on the surface. In the present article we study the locus of points in space defined by metric relation to a certain set of pairs of points, lines and planes. It is shown that any non-degenerate 2nd order surface can be considered as a certain locus of points of space and this interpretation is not unique.

Общий анализ формы линии пересечения двух однотипных поверхностей второго порядка

The presented paper is devoted to classification questions of fourth-order spatial curves, obtained as a result of intersection of non-degenerate second-order surfaces (quadrics) from the point of view of the forms of the original quadrics generating this curve. At the beginning of the paper is performed a brief historical overview of appearance of well-known and widely used curves ranging from ancient times and ending with the current state in the theory of curves and surfaces. Then a general analysis of the influence of the shape parameters and the relative position of original surfaces on the shape of the resulting curve and some of its parameters (number of components, presence of singular points, curve components flatness or spatiality) is carried out. Curves obtained as a result of intersection of equitype surfaces are described in more detail. The concept of interacting surfaces is introduced, various possible cases of the forms of the quadrics generating the curve are analyzed. A classification of fourth-order curves based on the shape parameters and relative position of second-order surfaces is proposed as an option. Illustrations of the resulting curve shapes with different shape parameters and location of generating quadrics are given. All surfaces and curves are considered in real affine space, taking into account the possibility of constructing them using descriptive geometry methods. Possible further research directions related to the analysis of the curves under discussion are briefly considered. In addition, are expressed hypotheses related to these curves use in the process of studying by students of technical universities the courses in analytical geometry, descriptive geometry, differential geometry and computer graphics. The main attention is paid to forms, therefore a wide variability of the surface shape in the framework of its described equation has been shown, provided by various values of numerical parameters.

A proof of Sørensen’s conjecture on Hermitian surfaces

In this article we prove a conjecture formulated by A. B. Sørensen in 1991 on the maximal number of F q 2 \mathbb {F}_{q^2} -rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree d ≤ q . d \le q.

Translation surfaces in affine 3-space

In this paper, we study translation surfaces in three dimensional affine space. We characterize the finite type non-degenerate translation surfaces with respect to the first affine and the second affine fundamental forms.

Homogeneous Surfaces in $$\mathbb C^4$$ Associated with a 5-Dimensional Completely Nondegenerate Cubic Model Surface of CR-Type $$(1,3)$$

The action on the space $$\mathbb C^4$$ of the 7-dimensional Lie group of infinitesimal holomorphic automorphisms of a completely nondegenerate cubic model surface $$Q$$ of CR-type $$(1,3)$$ is considered. All orbits of the given action are found and their biholomorphic classification is given. One of the orbits coincides with the surface $$Q$$ (the 5-dimensional orbit), two orbits are 6-dimensional, and the remaining part of the space $$\mathbb C^4$$ , the complement to the above orbits, foliates into 7-dimensional real orbits. It is also proved that the algebra of infinitesimal holomorphic automorphisms of all orbits, except for the twelve holomorphically degenerate orbits, coincides with the algebra of the surface $$Q$$ .

Boundary of the Milnor fiber of a Newton non-degenerate surface singularity

We give an algorithm for the description as a graph manifold of the boundary of the Milnor fiber of a Newton non-degenerate surface singularity embedded in a germ of toric three-dimensional variety, starting from its local Newton polyhedron. This algorithm is based on a previous work by the author describing a general method for the computation of the boundary of the Milnor fiber of any smoothing of a reduced non-isolated singularity of complex surface.

Maximum Number of Points on Intersection of a Cubic Surface and a Non-Degenerate Hermitian Surface

In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in the case when $d=2$. In this paper, we prove that the conjecture is true for $d=3$ and $q \ge 8$. We further determine the second highest number of rational points on the intersection of a cubic surface and a non-degenerate Hermitian surface. Finally, we classify all the cubic surfaces that admit the highest and second highest number of points in common with a non-degenerate Hermitian surface. This classifications disproves one of the conjectures proposed by Edoukou, Ling and Xing.

On a binary system of Prendiville: The cubic case

We prove sharp decoupling inequalities for a class of two dimensional non-degenerate surfaces in R5, introduced by Prendiville [13]. As a consequence, we obtain sharp bounds on the number of integer solutions of the Diophantine systems associated with these surfaces.

Tuning the topological surface states in Sb2Te3/CrSb heterostructures by interface spacing

Motivated by the recent experimental progress in TI/CrSb heterogeneous film, the magnetic and electronic structures of the Sb2Te3/CrSb heterostructure are investigated through first-principle calculations and a low-energy effective model. Atoms near the interface are magnetized due to the magnetic proximity effect from CrSb. Accordingly, a surface state gap of about 2.4 meV is induced in the band structure. Interestingly, the induced gap can be further tuned from the interface spacing between Sb2Te3 and CrSb, where the energy difference between the two nondegenerate surface Dirac cones plays a role. Our results reveal that the anomalous Hall effect observed in TI/CrSb heterostructure originates from the surface state gap from magnetic introducing. Our study sheds new light on the understanding of the surface state gap in TI/FM (AFM) heterostructures.

A New Approach to Minimal and Maximal Hypersurfaces in Product Spaces

In this paper we introduce a new method for the study of non-degenerate hypersurfaces immersed into product spaces of the form $$M^n\times \mathbb {R}$$ , with $$M^n$$ a Riemannian manifold, which are naturally endowed with two metrics: the standard Riemannian metric $$\left\langle , \right\rangle _M+dt^2$$ , and the Lorentzian metric $$\left\langle , \right\rangle _M-dt^2$$ . Naturally we can consider two mean curvatures and two Gaussian curvatures associated to the Riemannian and Lorentzian metrics. In this setting, we prove that a hypersurface having zero mean curvature with respect to both metrics must be foliated by hypersurfaces which are minimal submanifolds of the ambient space. As an application we prove that non-degenerate surfaces in a product space with zero mean curvature with respect to both metrics must be open pieces of slices, cylinders over geodesics or helicoids. Furthermore, we characterize flat surfaces as the unique non-degenerate surfaces in the Lorentz–Minkowski space having the same Gaussian curvature with respect to both metrics.

A Study on a Generalized Relaxed Curvature Energy Action

We investigate the variational problem of the generalized relaxed elastic line defined as the problem of finding critical points of the functional obtained by adding the twisting energy to the bending energy functional, on a non-degenerate surface in Minkowski 3-space. There arise two different situations for the curve $\alpha $ given on any non-degenerate surface S in Minkowski 3-space according to the absolute value expression in the curvature and torsion formulas. We study the problem for both cases and as a result we characterize the generalized relaxed elastic line with an Euler-Lagrange equation and 3 boundary conditions in both cases. Finally, we search special solutions for the differential equation system obtained with regard to the geodesic curvature, geodesic torsion and normal curvature of the curve.