This paper has been withdrawn by the authors due to the fact that the webs considered in the paper are ``Veronese-like webs'' which are different from Veronese webs.
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48 results for “Veronese webs”
The paper explores connections between Veronese webs and integrable equations, revealing new symmetries and structures.
problem Understanding the relationships between Veronese webs and integrable equations.
method Established correspondence between Veronese three-dimensional webs and hyper-CR structures, used dispersionless Lax pairs to deform integrable equations, computed contact symmetries and Backlund transformations.
result Found new integrable equations and structures related to Veronese webs, linking finite-dimensional systems to dispersionless integrable PDEs.
Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese websmath-ph
Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web…
Veronese webs appear as the natural way of passing to the quotient of curves in the projective space. In thi paper, we give the link between classical multidimensionnal webs and veronse webs by mean of interpolation.
This work is an introduction to the local geometric theory of Veronese webs developed in the last twenty years. Among the different possible approach, here one has chosen the point of view of differential forms. Moreover, in order to make its reading easier, this text is self-contained in which directly regards Verones…
It is shown how the well-known class of bihamiltonian structures in general position can be extended to a wider class. A generalization of the corresponding notion of a Veronese web for this wider class is presented (in the general position case Veronese webs form complete systems of local invariants for bihamiltonian …
We exploit the correspondence between the three-dimensional Lorentzian Einstein-Weyl geometries of the hyper-CR type, and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless Hirota equation. We also demonstrate how to construct hyper-CR Einstein--Weyl struc…
We prove that there is a correspondence between projective structures defined by torsion-free connections with skew-symmetric Ricci tensor and Veronese webs on a plane. The correspondence is used to characterise the projective structures in terms of second order ODEs.
We construct point invariants of ordinary differential equations that generalise the Cartan invariants of equations of order two and three. The vanishing of the invariants is equivalent to the existence of a totally geodesic paraconformal structure which consist of a paraconformal structure, an adapted -conne…
Given a 1-parameter family of 1-forms $\g(t)= \g_0+t\g_1+...+t^n\g_n$, consider the condition $d\g(t)\wedge\g(t)=0$ (of integrability for the annihilated by $\g(t)$ distribution ). We prove that in order that this condition is satisfied for any it is sufficient that it is satisfied for different value…
The paper connects web theory to heavenly PDEs and Einstein metrics.
problem Understanding the correspondence between self-dual metrics and vector fields.
method Using Nijenhuis operators and web theory to construct new heavenly PDEs.
result New integrable heavenly PDEs derived from Nijenhuis operators.
A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese…
Veronese minimizes normal curvatures to sphere.
problem Bounding normal curvatures of submanifolds.
method Veronese embeddings of projective planes.
result Optimal bound on normal curvatures guarantees sphere.
The study broadens the concept of cyclic polytopes to Veronese polytopes.
problem Extending the framework of cyclic polytopes to a broader class of polytopes.
method Described facial structure and combinatorial characterisation of facets via σ-parity alternating sequences.
result Established a bijective correspondence between combinatorial types of Veronese polytopes and partitions of finite sets.
Study calculates reach and curvature of a specific geometric variety.
problem Computing geometric properties of a specific variety.
method Computed reach, extremal curvature, and volume of a tubular neighborhood.
result Computed geometric properties of the Segre-Veronese variety.
Research shows asymptotic non-defectivity of Segre-Veronese varieties.
problem Defectivity of Segre-Veronese varieties.
method Analyzing the asymptotic behavior of Segre-Veronese varieties using line bundles and embedding.
result Asymptotically, Segre-Veronese varieties are not h-defective for a certain range of h.
Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.
problem Investigate non-geodesic connections in warped Segre-Veronese manifolds.
method Investigate a one-parameter family of warped geometries, presenting closed expressions for maps and distance.
result Segre-Veronese manifolds are not geodesically connected in Euclidean geometry but can be for some warping parameters.
Sharp Veronese rigidity theorem for submanifolds of unit ball.
problem Veronese rigidity of submanifolds under harmonic structure.
method Intrinsic harmonic structure assumptions, Bochner-Gauss mechanism, shape operators.
result Sharp lower bound on maximal normal curvature for specific submanifolds.
Study algebraic invariants from lightning self-attention models.
problem Understanding polynomial coefficients of self-attention mechanisms.
method Identify algebraic invariants using polynomial coefficients and coordinate geometry.
result Found linear and nonlinear families of algebraic invariants.
Structure of chambers cut out by Veronese arrangements of hyperplanes in the real projective spacesmath.GT
Research examines arrangements of hyperplanes in real projective spaces, focusing on specific cases.
problem Analyzing the structure of hyperplane arrangements in real projective spaces.
method Investigates arrangements of hyperplanes in the -dimensional real projective space, with a focus on and or .
result Provides insights into the structure of chambers cut out by these specific hyperplane arrangements.
Holomorphic deformations of weighted projective spaces yield Finsler spheres with closed geodesics.
problem Finding Finsler 2-spheres with constant curvature and closed geodesics.
method Establishing a correspondence between Finsler structures and Weyl connections on orbifolds.
result Holomorphic deformations of Veronese embeddings provide examples of Finsler 2-spheres with constant curvature and closed geodesics.
Estimates the probability of a random symmetric tensor being close to rank-one.
problem Estimating the probability of a random symmetric tensor being close to rank-one.
method Using Weyl's tube formula and techniques from Random Matrix theory, we study metric invariants of the real Veronese variety.
result Explicit formula for the reach and curvature coefficients of the real Veronese variety with respect to the Bombieri-Weyl metric.
Study calculates indices and nullities of focal manifolds in spheres.
problem Calculating indices and nullities of focal manifolds in spheres.
method Analyzes isoparametric hypersurfaces in spheres with three distinct principal curvatures.
result Index equals the ambient space dimension, nullity determined by Killing vector fields.
A submanifold in a real space form attaining equality in the DDVV inequality at every point is called a Wintgen ideal submanifold. They are invariant objects under the Moebius transformations. In this paper, we classify those Wintgen ideal submanifolds of dimension m>3 which are Moebius homogeneous. There are three cla…
In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. We apply our theory to isotropic Willmore two-spheres in and derive a necessary condition for such ( possibly branched) isotropic surfaces to descend to (…
The article studies factorization structures in geometry and their applications to cones and polytopes.
problem Understanding and characterizing factorization structures in geometry.
method Comprehensive study of factorization structures, including structure theory, construction of compatible polytopes and cones, and derivation of generalised Gale's evenness condition.
result Established generalised Vandermonde identities and found examples of Delzant and rational Delzant compatible polytopes.
We describe some buildings related to complex Kac-Moody groups. First we describe the spherical building of SLn(C) (i.e. the projective geometry PG(Cn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field . Then …
Estimates eigenvalues for surfaces in RP^3, proving a strict inequality.
problem Quantitative estimate of eigenvalues for surfaces in RP^3.
method Combination of canonical Veronese embedding, conformal test function method, rigidity argument, and classification of surfaces.
result Strict inequality for eigenvalues when χ(Σ) ≤ 0.
New criteria found for Willmore submanifolds in Lie group orbits.
problem Criteria for Willmore submanifolds in Lie group orbits.
method Criteria for Willmore submanifolds based on orbit type stratification.
result Found Willmore orbits in each stratified subset of orbit type.
Formula calculates MOY webs and link polynomials.
problem No specific problem stated; focuses on evaluation.
method Closed formula for exterior webs and link polynomials.
result Closed formula for evaluating MOY webs and link polynomials.
We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré's theorem: a planar 4-web of maximum rank is lineari…
Sharp pinching theorem for submanifolds in spheres.
problem Characterizing submanifolds in spheres based on curvature bounds.
method Conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.
result Complete submanifolds with specific curvature bounds are either totally geodesic or Clifford tori/Veronese surfaces.
Paper introduces a new web robot traffic generator.
problem Lack of realistic web robot traffic generators.
method Statistical and Bayesian models fitted to robot traffic data.
result Generated traffic mimics real robot traffic characteristics.
Classifies hexagonal circular 3-webs with cubic polar curves.
problem Classifying hexagonal circular 3-webs with algebraic polar curves of degree three.
method Analyzes hexagonal circular 3-webs on unit sphere with polar points on a twisted cubic.
result Completes the classification of hexagonal circular 3-webs with algebraic polar curves of degree three.
We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the solutions of the corresponding associativity equation. We show that such webs are he…
We give various results and applications using the connection associated with a -web. Precisely, we exhibit fundamental invariants of the web related to the differential equation of first order which presents the web. They cast some new lights on the connection and its construction, both conceptually an…
We investigate the linearizability problem for different classes of 4-webs in the plane. In particular, we apply a recently found in [AGL] the linearizability conditions for 4-webs in the plane to confirm that a 4-web MW (Mayrhofer's web) with equal curvature forms of its 3-subwebs and a nonconstant basic invariant is …
It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold of the sphere and such that the normal holonomy group is not transitive (on t…
In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine struc…
Projective invariants described for 3-web geometry, resolving Gronwall's conjecture.
problem Projective invariants of linear 3-webs and resolving Gronwall's conjecture.
method Projective torsion-free Cartan connections, web leaves as geodesics, algorithm for resolving Gronwall's conjecture.
result Resolved Gronwall's conjecture for specific 3-webs.
New instanton homology for webs counts Tait colorings.
problem Counting Tait colorings of complex webs.
method Introducing a local system of coefficients to deform instanton homology.
result Rank of deformed instanton homology equals number of Tait colorings.
Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps, derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese…
Study local invariants of divergence-free webs in geometry.
problem Characterize triviality of divergence-free webs.
method Introduce two local invariants: differential and geometric.
result Triviality of either invariant characterizes trivial divergence-free web-germs.
Investigates webs related to cluster algebras and polylogarithms.
problem Understanding webs associated with cluster algebras and polylogarithms.
method Introducing AMP webs and analyzing their properties, proving results and conjectures.
result Many webs associated with polylogarithms and cluster algebras are AMP webs.
Defines generic singularities in planar flat 3-webs and classifies them.
problem Classifying singularities in planar flat 3-webs.
method Proposes a definition of genericity and classifies singularities.
result Classifies generic singularities of planar flat 3-webs.
Much information available on the web is copied, reused or rephrased. The phenomenon that multiple web sources pick up certain information is often called trend. A central problem in the context of web data mining is to detect those web sources that are first to publish information which will give rise to a trend. We p…
For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a transversal distribution is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and sufficient conditions of its integrability and prove the existence theorem for webs W (3, 2,…
In this paper we study the linearizability problem for 3-webs on a 2-dimensional manifold. With an explicit computation based on the theory developed in the paper "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp. 2643-2654), we examine a 3-web whose linearizability was claimed in the same paper. We …