Abelian equations and rank problems for planar webs

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 theorem: A planar 4-web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem “to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3.” Finally, we find invariant characterization of planar 5-webs of maximum rank and prove than in general such webs are not linearizable. The text was submitted by the authors in English.     

On linearization of planar three-webs and Blaschke's conjecture

We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. To cite this article: V.V. Goldberg, V.V. Lychagin, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

4-Webs in the Plane and Their Linearizability

We investigate the linearizability problem for different classes of 4-webs in the plane. In particular, we apply the linearizability conditions, recently found by Akivis, Goldberg and Lychagin, to confirm that a 4-web MW (Mayrhofer's web) with equal curvature forms of its 3-subwebs and a nonconstant basic invariant is always linearizable (this result was first obtained by Mayrhofer in 1928; it also follows from the papers of Nakai). Using the same conditions, we further prove that such a 4-web with a constant basic invariant (Nakai's web) is linearizable if and only if it is parallelizable. Next we study four classes of the so-called almost parallelizable 4-webs APW _a ,a=1,2,3,4 (for them the curvature K=0 and the basic invariant is constant on the leaves of the web foliation X _a ), and prove that a 4-web APW _a is linearizable if and only if it coincides with a 4-web MW _a of the corresponding special class of 4-webs MW. The existence theorems are proved for all the classes of 4-webs considered in the paper.     

Three-webs with covariantly constant curvature and torsion tensors

In this paper, we study a special class of multidimensional 3-webs with covariantly constant curvature and torsion tensors. In the first part, we prove that 3-webs of the class belong to G-webs, i.e., there is a subfamily of adapted frames whose components of curvature and torsion tensors are constant. The structure of the homogeneous space G/H carrying the 3-web is described. Structure equations of the G-group are found. In the second part, we find structure equations of the W ^??-web and finite equations of some special web classes.     

On the problem of linearizability of a 3-web

In this paper we study the linearizability problem for 3-webs on a two-dimensional manifold. With an explicit computation we examine a 3-web whose linearizability was claimed in [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654] and was contested later in [V.V. Goldberg, V.V. Lychagin, On the Blaschke conjecture for 3-webs, J. Geom. Anal. 16 (1) (2006) 69–115] and [V.V. Goldberg, V.V. Lychagin, On linearization of planar three-webs and Blaschke’s conjecture, C. R. Acad. Sci. Paris, Ser. I. 341 (3) (2005)]. On the basis of the theories of [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654], we give an effective method for computing the linearizability criterion, and we prove that this particular web is linearizable by finding explicitly the affine deformation tensor and the corresponding flat linear connection.     

Geodesic Webs on a Two-Dimensional Manifold and Euler Equations

We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the web 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 to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d>4, provided that additional d−4 second-order invariants vanish.     

4-Webs on hypersurfaces of 4-axial space

V. B. Lazareva investigated 3-webs formed by shadow lines on a surface embedded in 3-dimensional projective space and assumed that the lighting sources are situated on 3 straight lines. The results were used, in particular, for the solution of the Blaschke problem of classification of regular 3-webs formed by pencils of circles in a plane. In the present paper, we consider a 4-web W formed by shadow surfaces on a hypersurface V embedded in 4-dimensional projective space assuming that the lighting sources are situated on 4 straight lines. We call the projective 4-space with 4 fixed straight lines a 4-axial space. Structure equations of 4-axial space and of the surface V , asymptotic tensor of V , torsions and curvatures of 4-web W, and connection form of invariant affine connection associated with 4-web W are found.     

On the Gronwall Conjecture

The Gronwall conjecture states that a planar 3-web which admits more than one distinct linearization is locally equivalent to an algebraic web. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with web curvature that vanishes to order three at a point. The differential relation on the third-order jet of web curvature provides an explicit criterion for unique linearization.     

Linearizability of d-webs, d ≥ 4, on two-dimensional manifolds

We find d − 2 relative differential invariants for a d-web, d ≥ 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 conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g ₄, and d − 4 PDEs of the second order with respect to f and g ₄,..., g _d . For d = 4, this result confirms Blaschke’s conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give the Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage.     

On the geometry of the (2n+1)-webs in 2n-dimensional affinely connected spaces A2n

An approach for the investigation of the geometry of (2n+1) -webs in 2n-dimensional affinely connected spaces A_2n by means of the prolonged differentiation is given in [1] Using this approach in the present paper some properties of (2n+1)-webs in A_2n are studied. Necessary and sufficient conditions under which the generalized affinor of Hess of a (2n+1)-web is covariantly constant are found. The space A_2n, containing a (2n+1)-web with a constant covariant generalized affinor of Hess, is determined. Conditions for geodesicity of the lines of a given (2n+1)-web in A_2n in terms of its tensor are found, also.The present investigation is partially supported by the Ministry of Science and Higher Education of People's Republic of Bulgaria under grant 1021.     

Linearizability of d-webs, d?≥?4, on two-dimensional manifolds

We find d – 2 relative differential invariants for a d-web, d 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 g₄(x, y),..., g_d(x, y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g₄, and d – 4 PDEs of the second order with respect to f and g₄,..., g_d. For d = 4, this result confirms Blaschkes conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give the Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage.     

Curvature of curvilinear 4-webs and pencils of one forms: Variation on a theorem of Poincaré, Mayrhofer and Reidemeister

A curvilinear d-web W = (F ₁ , . . . , F _d ) is a configuration of d curvilinear foliations F _i on a surface. When d = 3, Bott connections of the normal bundles of F _i extend naturally to equal affine connection, which is called Chern connection. For 3 < d, this is the case if and only if the modulus of tangents to the leaves of F _i at a point is constant. A d-web is associative if the modulus is constant and weakly associative if Chern connections of all 3-subwebs have equal curvature form. We give a geometric interpretation of the curvature form in terms of fake billiard in §2, and prove that a weakly associative d-web is associative if Chern connections of triples of the members are non flat, and then the foliations are defined by members of a pencil (projective linear family of dim 1) of 1-forms. This result completes the classification of weakly associative 4-webs initiated by Poincaré, Mayrhofer and Reidemeister for the flat case. In §4, we generalize the result for n + 2-webs of n-spaces. Received: September 23, 1996     

Three-Webs Defined by Symmetric Functions

We study local differential-geometrical properties of curvilinear k-webs defined by symmetric functions (webs SW(k)). This class of k-webs contains in particular algebraic rectilinear k-webs defined by algebraic curves of genus 0. On a web SW(3), there are three three-parameter families of closed Thomsen configurations. We find equations of a rectilinear web SW(k) in terms of adapted coordinates and prove that the curvature of a symmetric three-web is a skew-symmetric function with respect to adapted coordinates. In conclusion, we formulate some open problems.     

{\mathfrak {sl}}_3-Web bases,intermediate crystal bases and categorification

We give an explicit graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra \(K_S\) . In order to do this, we identify Kuperberg’s basis for the \({\mathfrak {sl}}_3\) -web space \(W_S\) with a version of Leclerc–Toffin’s intermediate crystal basis and we identify Brundan, Kleshchev and Wang’s degree of tableaux with the weight of flows on webs and the \(q\) -degree of foams. We use these observations to give a “foamy” version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra. We restrict ourselves to the \({\mathfrak {sl}}_3\) case over \(\mathbb {C}\) here, but our approach should, up to the combinatorics of \({\mathfrak {sl}}_N\) -webs, work for all \(N>1\) or over \(\mathbb {Z}\) .     

On the problem of classification of regular 4-webs formed by pencils of spheres

Shelehov’s theorem on bondary curves of a regular curvilinear 3-web is generalized to the case of an arbitrary regular codimension 1 (n + 1)-web; an example is given of a regular 4-web formed by pencils of spheres in the three-dimensional conformal space (W. Blaschke’s problem); it is proved that a spherical 4-web of the basic type cannot be regular.     

On the geometry of point correspondences of three conformal spaces

Point correspondences of three conformal spaces are studied on the base of G. F. Laptev??s invariant methods. We establish the basic equations and geometrical objects of the point correspondences in question. We construct invariant normalizations of the spaces, single out the basic tensors of the correspondences, establish a connection of the correspondences with the theory of multidimensional 3-webs, and find the torsion and the curvature tensors of a point correspondence. For a series of particular cases, we prove existence theorems.     

Linearizability of the polynomial differential systems with a resonant singular point

Integrability and linearizability of polynomial differential systems are studied. The computation of generalized period constants is a way to find necessary conditions for linearizable systems for any rational resonance ratio. A method to compute generalized period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As the application, we discuss linearizable conditions for several Lotka-Volterra systems, and where this is the first time that the linearizability is considered for 3:−4 and 3:−5 resonances.     

Terminal control problem for affine systems

We suggest a method for solving the terminal control problem for multidimensional affine systems that are not linearizable by feedback. We prove a sufficient condition for the existence of a solution and present a numerical procedure for its construction.     

Nonholonomic (n + 1)-webs

We consider a nonholonomic (n + 1)-web NW on an n-dimensional manifold M, i.e., n + 1 codimension 1 distributions on M. We prove that a web NW on M is equivalent to a G-structure with structure group λE, the group of scalar matrices. We find the structure equations of a web NW and the integrability conditions of the distributions of a web NW. It is shown that on a manifold with nonholonomic (n + 1)-web an affine connection Γ arises naturally for which the distributions of the web are totally geodesic. We consider the case when the connection Γ has zero curvature and, in particular, when a web NW is defined by invariant distributions on a Lie group. In the case when all distributions of a web NW on a Lie group are integrable, we find the equations of this group in terms of local coordinates.