arXiv research

The paper describes the geometric properties of line congruences' singularities.

Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with re…

New framework for analyzing line fields on surfaces, proving stability under specific conditions.

The paper classifies singularities of line congruences in 4D space.

Our aim is to generalize the result that two generic complex line arrangements are equivalent. In fact for a line arrangement A we associate its defining polynomial, the product of a_ix+b_iy+c_i, so that A = (f=0). We prove that the defining polynomials of two generic line arrangements are, up to a small deformation, t…

Paper maps Hamiltonians and line elements in manifolds.

The Enneper surface and helix surfaces are unique in their geometric properties.

Study stability thresholds of big line bundles, proving bounds and generalizing results.

The paper studies how points and lines can move while preserving incidences.

Generalizes cohomology ring result for combinatorial line arrangements.

This article is devoted to the study of cyclides osculating general surfaces. We show that generically, at any point of a surface, one has a one-parameter family of cyclides tangent to a surface curve of order three and among them just one is tangent to this curve of order four. This one will be called the osculating c…

Study three discrete envelope types of polygon bisection lines.

New discretizations of principal curvature lines discovered.

We show that a generic real projective n-dimensional hypersurface of degree 2n-1 contains "many" real lines, namely, not less than (2n-1)!!, which is approximately the square root of the number of complex lines. This estimate is based on the interpretation of a suitable signed count of the lines as the Euler number of …

New BDEs reveal singular surfaces from line congruences.

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines $E$. We prove that its general element is a smooth surface containing $E$ and no other line. Afterwards we prove that …

Study on straight-line flows for generative modeling with theoretical obstructions.

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements a…

The paper explores reflection principles for lightlike line segments on maximal surfaces.

There is a natural duality between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$ that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…

We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we…

Study weak geodesic lines in Kähler metric space, disproving a conjecture.

For a pair of points in a smooth closed convex planar curve $γ$, its mid-line is the line containing its mid-point and the intersection point of the corresponding pair of tangent lines. It is well known that the envelope of the mid-lines ($EML$) is formed by the union of three affine invariants sets: Affine Envelope Sy…

Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.

The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines …

Isothermic nets created from special maps for smooth surfaces.

The study generalizes a specific geometric correspondence to higher dimensions.

Study complex lines in symplectic geometry, generalizing previous results.

The paper explores affine geometry of line congruences using singularity theory.

Six quaternionic lines with optimal angles found in 2D quaternion space.

Paper generalizes Hamiltonian mechanics using line bundles.

Invariant Kähler metrics on line bundles are derived from the Calabi ansatz.

In this paper we show that to each planar line arrangement defined over the real numbers, for which no two lines are parallel, one can write down a corresponding relation on Dehn twists that can be read off from the combinatorics and relative locations of intersections. This leads to an alternate proof of Wajnryb's gen…

The paper explores the geometry of level lines of quasiperiodic functions with many periods.

Study the monodromy and center-focus problems for rational maps defined by products of generic lines.

Study on complex line fields on almost-complex manifolds, proving existence conditions.

We prove that every right-angled Artin group embeds into the $C^{\infty}$ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the $C^{\infty}$ diffeomorphism group of the real line.

Let X be a complex-projective contact manifold whose second Betti-number is one. It has long been conjectured that X should then be rational-homogeneous, or equivalently, that there exists an embedding of X into a projective space whose image contains lines. Using methods introduced in math.AG/0206193, we show that X i…

Wilson lines generate positive Laurent polynomials in decorated triangulations.

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

Identifying a potentially large number of simultaneous line outages in power transmission networks in real time is a computationally hard problem. This is because the number of hypotheses grows exponentially with the network size. A new "Learning-to-Infer" method is developed for efficient inference of every line statu…

A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress…

We prove that a finite type curve is an $ξ$-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type $ξ$-asymptotic line. These results obtained here are generalizations, for plane fields, of the results of V. Arnold [4].

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular,…

The study examines conditions that prevent null geodesic lines in spacetimes, impacting cosmological geometry.

Let X be a complex Fano-manifolds with second Betti-number 1 which carries a contact structure. It follows from previous work that such a manifold can always be covered by lines. Thus, it seems natural to consider the geometry of lines in greater detail. In this brief note we show that if x in X is a general point, the…