The paper proves Γ-convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.
problem Proving convergence of discrete tangent-point energies to continuous energies and ropelength.
method Using biarc curves and interpolation, the paper proves Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies and ropelength functional. result Discrete almost minimizing biarc curves converge to ropelength minimizers and minimizers of continuous tangent-point energies.
A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphis…
Characterizes special curves on surface tangent bundles.
problem Understanding curves on surface tangent bundles.
method Characterization of Legendre and slant curves.
result Characterizations for N-Legendre and N-slant curves.
The Samuelson condition is not satisfied by tangent lines of quadratic curves.
problem Area condition for Lagrangian 2-web
method Show that the Samuelson condition is not satisfied
result The Samuelson condition is not satisfied by tangent lines of quadratic curves.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…
The paper studies singularities on parallels of tangent developable surfaces of frontal curves.
problem Understanding singularities on parallels of tangent developable surfaces.
method Generalization of tangent developable surfaces and parallel deformations for frontal curves in arbitrary dimensions.
result Classification of generic singularities on parallels of tangent developable surfaces for frontal curves in 3 or 4 dimensional Euclidean spaces.
Paper defines curvature equivalence for Legendre curves in a plane.
problem No specific problem stated; focuses on Legendre curves.
method Introduced curvature equivalence relation for Legendre curves.
result Local and global classifications of Legendre curves under curvature equivalence.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…
Study of normal and tangent maps to frontals.
problem Understanding geometric and dynamical properties of frontals.
method Geometrical and dynamical analysis of normal and tangent maps.
result Parallels of the tangent map to a frontal curve are right equivalent to the tangent map of a frontal curve under certain conditions.
The study examines Bertrand Legendre curves in the unit tangent bundle over Euclidean plane.
problem Investigating properties of Legendre curves and their associated curves.
method Analyzing Bertrand Legendre curves and their associated curves, including parallel, evolute, and involute curves.
result Existence conditions and inverse operation for Bertrand Legendre curves are provided.
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and differential systems on flag manifolds, via several techniques in differentiable …
Paper proves unique tangent flow at infinity for entropy-limited curve shortening.
problem Proving uniqueness of tangent flows for finite-entropy curve shortening.
method Rescaled backward convergence to a line, entropy analysis, and geometric properties.
result Ancient smooth curve shortening flow has a unique tangent flow at infinity.
In this paper, we define some new associated curves as integral curves of a vector field generated by Frenet vectors of tangent indicatrix of a curve in Euclidean 3-space. We give some relationships between curvatures of these curves. By using these associated curves, we give some methods to construct helices and slant…
Steinhaus conjectured that every closed oriented C1-curve has a pair of anti-parallel tangents. Porter disproved the conjecture by showing that there exist curves with no anti-parallel tangents. Colin Adams rised the question of whether there exists a nontrivial knot in R3 which has no parallel or antiparallel t…
A simple closed curve γ in the real projective plane P2 is called anti-convex if for each point p on the curve, there exists a line which is transversal to the curve and meets the curve only at p. We shall prove the relation i(γ)−2δ(γ)=3 for anti-convex curves, where i(γ) is the number of independent (true…
Analytic curves have infinite codimension of singular germs.
problem Understanding the codimension of singular tangent curves in analytic distributions.
method Formalizing asymptotic statements about finite jets of tangent curves and applying the h-principle.
result The subspace of singular germs has infinite codimension within smooth curves.
We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …
The paper explores Bertrand and framed curves in 3D space.
problem Characterizing Bertrand and framed curves in Euclidean 3-space.
method Analyzing curves where tangent, normal, or binormal lines match another curve's lines.
result Conditions for the existence of Bertrand and framed curves are clarified.
In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its …
The study proves a discrete version of Segre's theorem for polygonal curves.
problem Proving a discrete analog of a four-vertex theorem for spherical curves.
method Using the concept of discrete tangent indicatrix of a polygon.
result A polygon with at least four vertices and a non-self-intersecting discrete tangent indicatrix has at least four flattenings.
It is well-known that if a curve is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold then the projected curve has all its geodesic curvatures constant. In this paper we consider the case of tangent (sphere) bundle over the real, complex and quaternionic space …
A 2-web in the plane is given by two everywhere transverse 1-foliations. In this paper we introduce the study of singular 2-webs, given by any two foliations, which may be tangent in some points. We show that such two foliations are tangent along a curve, which will be called the polar curve of the 2-web, and we study …
We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along th…
We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, …
Geodesics of the same type on curved surfaces are randomly distributed.
problem Distribution of geodesics of the same type on negatively curved surfaces.
method Asymptotic equidistribution with respect to a measure on the unit tangent bundle.
result Geodesics of the same type are asymptotically equidistributed with respect to a measure mS. Study on curves around a Whitney umbrella focusing on geodesic and normal curvatures.
problem Analyzing geometric properties of curves around a specific surface.
method Examined geodesic and normal curvatures, ruled surfaces, and normal developable surfaces.
result Obtained functions representing geometry on a Whitney umbrella.
The paper classifies helix curves on a pseudo-Riemannian surface.
problem Classifying helix curves on pseudo-Riemannian surfaces.
method Analyzing geodesic flow vector fields and pseudo-Riemannian metrics.
result All helix curves are circular helixes with constant curvature and torsion.
We study metrics on the shape space of curves that induce a prescribed splitting of the tangent bundle. More specifically, we consider reparametrization invariant metrics G on the space Imm(S1,R2) of parametrized regular curves. For many metrics the tangent space $T_c\operatorname{Imm}(S^1,…
The envelope of straight lines affine normal to a plane curve C is its affine evolute; the envelope of the affine lines tangent to C is the original curve, together with the entire affine tangent line at each inflexion of C. In this paper, we consider plane curves without inflexions. We use some techniques of singulari…
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
The paper finds lower bounds for volumes of complex geometric structures.
problem Estimating the volume of complex geometric structures.
method Reduction to a counting problem in the unit tangent bundle, solved using exponential multiple mixing for the geodesic flow.
result First known lower bound for the volume of these manifolds in terms of curve length.
In this paper, we investigate the tangent indicatrix of the curve C with constant curvature. Tangent indicatrix of the curve C is characterized with det(C^(3),C^(4),C^(5))=0 in Minkowski 3-space E13. Moreover, we study null slant helices using the determinant approach and give the following characterization: A curve C …
Characterizes minimizing curves in Riemannian manifolds.
problem Finding optimal paths in curved spaces.
method Characterization of prox-regular sets and tangent cones.
result Necessary condition for minimizing curves in prox-regular sets.
Study shows a universal local obstruction to the Samuelson condition for tangent Lagrangian 2-webs.
problem Obstruction to the Samuelson condition for tangent Lagrangian 2-webs.
method Local analysis of tangent lines and their intersection maps.
result A universal local phenomenon produces a nonzero mixed derivative, obstructing the Samuelson condition.
Veering branched surfaces help construct geodesic flows on curved surfaces.
problem Constructing geodesic flows on negatively curved surfaces.
method Introduce veering branched surfaces and surgeries, then use them to construct veering triangulations that correspond to geodesic flows.
result Explicit constructions of veering branched surfaces corresponding to geodesic flows on negatively curved surfaces.
New method shortens and straightens curves, proving convergence and well-posedness.
problem Shortening and straightening of curves.
method Conceptual shift in curve shortening to tangent aligning, variational study of geometric flows.
result Proves convergence to a straight line and global well-posedness for various geometric flows.
Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of t…
Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.
problem Understanding the geometry of smooth surfaces in 3D space.
method Analyzes the vertex curve, related to differential geometry and symmetry sets of isophote curves.
result Establishes connections between the vertex curve and other geometric curves like parabolic and flecnodal curves.
Curve Shortening Flow preserves circularity for convex projections.
problem Understanding the behavior of curves under Curve Shortening Flow.
method Contradiction argument and analysis of tangent flows.
result Smooth curves with convex projections become asymptotically circular under Curve Shortening Flow.
New construction shows VMRTs of unbendable curves can be Legendrian.
problem Characterize VMRTs of unbendable rational curves under contact structures.
method Used geometry of contact lines and symplectic geometry of distributions.
result VMRTs of Legendrian submanifolds can be realized.
Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n-manifold. In particular, we show that if the curve satisfies certain n…
We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve…
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…
Study of rational curves in complex manifolds with specific normal bundles.
problem Characterizing rational curves in complex manifolds with given normal bundles.
method Analyzing differential and projective geometric properties of rational curves and their tangents.
result Classification of rational curves into Goursat and Cartan types based on their geometric properties.
Let (M12,g) be a Minkowski surface and (T1M12,g1) its unit tangent bundle endowed with the pseudo-Riemannian induced Sasaki metric. We extend in this paper the study of the N-Legendre and N-slant curves which the inner product of normal vector and Reeb vector is zero and nonzero constan…
Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagran…
We provide a new angle and obtain new results on a class of metrics on length-normalized curves in d dimensions, represented by their unit tangents expressed as a function of arc-length, which are functions from the unit interval to the (d−1)-dimensional unit sphere. These metrics are derived from the combined acti…