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 …
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The abstract discusses maximal rank 4-webs formed by confocal conics and their properties.
problem Characterizing confocal conics using 4-webs of maximal rank.
method Examining three different 4-webs formed by confocal conics and proving they are of maximal rank.
result Confocal conics are characterized from the web theory viewpoint.
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…
We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web …
Geodesic flows on surfaces have specific fractional-linear integrals related to constant cross-ratios.
problem Characterizing geodesic flows on surfaces with fractional-linear integrals.
method Proving the dimension of fractional-linear integrals and giving a geometric criterion.
result The dimension of fractional-linear integrals is either 3 or 5, corresponding to constant curvature.
Geodesic flows with specific integrals are linked to special 4-webs.
problem Characterizing geodesic flows with commuting quadratic integrals.
method Characterization through geodesic 4-webs and geometric hypothesis.
result Metrics of Stäckel type for geodesic flows with specific integrals.
New model for rational tropical points using -webs and measures.
problem Understanding rational tropical points of Fock-Goncharov moduli space.
method Introducing rational bounded -laminations and defining tropical coordinate systems.
result Established a bijection between rational tropical points and -webs.
Study of -webs on surfaces, proving cluster algebra structure.
problem Understanding -webs and their cluster algebra properties.
method Introduced skein algebra and cluster structure, proved positivity.
result Proved is a quantum cluster algebra.
We consider the stable ruled surface over an elliptic curve. There is a unique foliation on transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.
Le rang des tissus de Nakaimath.DG
Proves Nakai webs have rank 0 or 1, provides examples.
problem Classifying the rank of Nakai's webs.
method Analyzes cross-ratios and foliations to prove rank.
result Nakai webs have rank 0 or 1.
The study generalizes Blaschke curvature for higher-dimensional webs and identifies infinite classes of isomorphism.
problem Understanding the structure and classification of -webs by curves in -dimensional manifolds.
method Definition and analysis of generalized Blaschke curvature, construction of isomorphism classes, and identification of invariants.
result There are infinitely many classes of isomorphism for 4-webs by curves of rank one in 3-dimensional space.
We find d - 2 relative differential invariants for a d-web, d \geq 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f (x,y) and g_4 (x,y),...,g_d (x,y), then necessary and sufficient c…
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…
The paper characterizes measures preserving independence through planar web geometry.
problem Characterizing measures with preserved independence.
method Planar web geometry and inhomogeneous Abelian functional equations.
result The independence-preserving property is preserved by coordinatewise reparametrizations and forms a natural invariant.