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…
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Proofs curvature trace for planar webs.
problem Proving curvature trace for planar webs.
method Complete proof of curvature trace formula for planar d-webs.
result Trace of curvature equals sum of Blaschke curvatures of sub 3-webs.
We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d > n) of hypersurfa…
Programs Maple for curvature computation of planar webs defined by polynomial differential equations.
problem Computing curvature of planar webs defined by polynomial differential equations.
method Developed a Maple program to compute curvature for planar d-webs defined by polynomial differential equations of degree d.
result Curvature matrix of calibrated ordinary d-webs is concentrated on the last lines of the matrix.
Study infinitesimal symmetries in planar differential equations.
problem Qualitative study of differential equations and their integral curves.
method Use meromorphic connections methods to analyze infinitesimal symmetries.
result Infinitesimal symmetries of planar -webs are understood.
We prove that a d-web near a point in n-space, where n is greater than 2 and d is greater than 2n-1, is equivalent to an algebraic web, if it has maximal rank or, more generally, if it has (2d - 3n + 1) abelian relations the 1-jets of which are linearly independent. In case n=3, this is a theorem of Bol. The general ca…
We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its foliations. For a class of Legendrian -webs defined by simple second order ODE's, w…
The curvature of web curves is studied in 3D manifolds.
problem Understanding the maximal rank of web structures in 3D.
method Defined a tautological connection on a bundle whose curvature vanishes for maximal rank.
result Exceptional 6-web and its subwebs in 3D have maximal rank.
MOY calculus in type Dmath.QA
Defines a new link invariant for type D webs.
problem Invariants for framed unoriented links in type D.
method Positive state sum for type D webs, derived from quantum so(2N) modules.
result Relates to Reshetikhin-Turaev's invariant.
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 …
The study establishes a criterion for the holomorphy of curvature in smooth webs and applies it to dual webs of homogeneous foliations.
problem Establishing conditions for the holomorphy of curvature in smooth webs and their duals.
method Developed an effective criterion for the holomorphy of curvature in smooth -webs and applied it to dual webs of homogeneous foliations.
result Characterized the holomorphy of the curvature of dual webs of homogeneous foliations on .
Study of flat Legendre transforms on complex projective plane.
problem Characterizing homogeneous foliations with flat dual web.
method Effective criteria for flatness of dual -web, explicit examples, classification results.
result Up to automorphism of , there are 11 homogeneous foliations of degree 3 with flat dual web.
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…
Holomorphic curvature of planar webs along invariant curves
problem Holomorphic curvature of planar webs
method Show that curvature is holomorphic along invariant curves
result Curvature is holomorphic along invariant curves if and only if curvature of subwebs is holomorphic
Effective method to compute web rank in complex manifolds.
problem Computing the rank of holomorphic webs in complex manifolds.
method Effective method using polynomial degrees and vanishing curvature.
result Computed ranks refine the web's rank and can be stopped when curvature vanishes.
Study differential operators and their solutions on manifolds, proving upper bounds and curvature.
problem Understanding the dimension of solution spaces for differential equations on manifolds.
method Analyzing ordinary and calibrated differential operators, constructing vector bundles and connections.
result Upper bounds and curvature obstructions for solution spaces, proving concentration theorems.
The paper generalizes abelian relations for webs of arbitrary codimension.
problem Understanding abelian relations in webs of arbitrary codimension.
method Defining p-ordinary and strongly p-ordinary webs, calculating ranks and defining tautological connections.
result For p-ordinary webs, the p-rank is finite and can be bounded.