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

Abstract

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.Let W d be a d-web given by d one-parameter foliations of curves on a two-dimensional manifold M 2 . The web W d is linearizable (rectifiable) if it is equivalent to a linear d-web, i.e., to a d-web formed by d one-parameter foliations of straight lines on a projective plane.The problem of finding a criterion of linearizability of webs was posed by Blaschke in the 1920s (see, for example, his book [2], §17 and §42) who claimed that it is hopeless to find such a criterion. Blaschke in [2] formulated the problems of finding conditions for the linearizability of 3-webs ( § 17) and 4-webs ( § 42) given on M 2 . Comparing the numbers of absolute invariants for a general 3-web W 3 (respectively, a general 4-web W 4 ) and a linear 3-web (respectively, a linear 4-web), Blaschke made the conjectures that conditions of linearizability for a 3-web W 3 should consist of four relations for the ninth order web invariants (four PDEs of ninth order) and those for a 4-web W 4 should consist of two relations for the fourth order web invariants (two PDEs of fourth order).A criterion for linearizability is very important in web geometry and in its

On the Blaschke conjecture for 3-webs

“…the above equation becomes (1). It is easy to see that any two of such normalized triplets ω 1 , ω 2 , ω 3 and ω s 1 , ω s 2 , ω s 3 determine the same 3-web W 3 if and only if…”

“…We shall apply this construction to 3-webs on a two-dimensional manifold M. Let π = τ * : T * (M ) → M be the cotangent bundle, and let W 3 be a 3-web defined by the differential 1-forms {ω 1 , ω 2 , ω 3 } normalized by (1).…”

“…In [1] a complete solution of the linearizability problem for d-webs, d ≥ 5, was also presented. In [11] the linearizability conditions found in [1] were applied to check whether some known classes of 4-webs are linearizable.…”

“…In [1] a complete solution of the linearizability problem for d-webs, d ≥ 5, was also presented. In [11] the linearizability conditions found in [1] were applied to check whether some known classes of 4-webs are linearizable.In the present paper we continue to use the Akivis approach (see [1]) for establishing criteria of linearizability of 3-webs. In this approach, the linearizability problem is reduced to the solvability of the system of nonlinear partial 1…”

Quasilinear systems with linearizable characteristic webs

“…An n-web is said to be linearizable (rectifiable) if it is locally diffeomorphic to n families of lines, not necessarily parallel. We emphasize that the condition of linearizability is far more subtle than that of parallelizability, see [5,6,3,15,13,2,12, 1] and references therein for a discussion of the linearizability problem.…”