ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 6, pp. 56–68.
Allerton Press, Inc., 2018.
Original Russian Text
A.M. Shelekhov, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 6, pp. 63–77.
Three-Webs Deﬁned by Symmetric Functions
A. M. Shelekhov
Tver State University
ul. Zhelyabova 33, Tver, 170000 Russia
Received April 6, 2017
Abstract—We study local di ﬀerential-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.
Keywords: curvilinear k-web, symmetric k-web, k-web equations, Thomsen conﬁguration,
rectilinear k-web, algebraic k-web, curvature of a three-web.
Introduction. Recall that a k-web W (k) on the plane is a collection of k families of smooth curves
in general position. In some recent publications, the k-webs in question are called ordered. We however
will follow the terminology introduced by W. Blaschke, the founder of the diﬀerential-topological theory
of webs. By the domain of a k-web we mean the maximal domain in which the families form transverse
foliations, i.e., leaves of a web are pairwise transverse at each point of its domain.
The paper is devoted to the study of local aspects of the theory of symmetric k-webs, i.e., k-webs
whose equations for a certain choice of parameters of the families are not changed under any permutation
of arguments. Following W. Blaschke , we consider webs up to local diﬀeomorphisms, i.e., up to the
widest equivalence relation. Local diﬀeomorphisms preserve transversality of lines of a k-web, closure
or nonclosure of suﬃciently small conﬁgurations formed by lines of a web.
The most important symmetric webs are rectilinear webs of special form. Recall that a k-web formed
by k families of straight lines (not necessary parallel) is called rectilinear and denoted by LW (k).The
most diﬃcult problems of the theory of curvilinear webs which have more than centennial history are
connected with rectilinear three-webs.
Fist of all this is so-called “anamorphosis problem”, possibility to represent a function in two
variables by nomogram. The complete solution to this problem in terms of diﬀerential invariants of a web
is given in  and is contained in . Another problem also came from nomography and is connected
with the proof of Gronwall conjecture (1912) . Its positive solution was found in . In , it is
given in the following formulation: if two rectilinear three-webs are equivalent, then they are projectively
An important subclass of rectilinear k-webs is formed by algebraic webs deﬁned by homogeneous
algebraic equations of degree k connecting tangential coordinates of current line. In Item 12, we
show that if such an equation deﬁnes a curve of genus 1 (a curve birationally equivalent to a straight
line), then the corresponding rectilinear k-web is symmetric. In the paper, we generalize this property,
namely, we investigate symmetric k-webs or webs SW(k) whose functions, for a choice of parameters of
families, are symmetric functions. The geometric characteristic of symmetric three-webs is as follows.
Such webs carry three three-parameter families of closed Thomsen conﬁgurations (Theorems 1 and 3).
Equations of a rectilinear three-web are symmetric if and only if all its families belong to one family
(Theorem 4). The condition for existence of a symmetric rectilinear three-web (3-web SLW)isgiven