ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 6, pp. 80–84.
Allerton Press, Inc., 2018.
Original Russian Text
I.S. Strel’tsova, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 6, pp. 92–97.
Projective Invariants of Planar 2-Webs
I. S. Strel’tsova
(Submitted by V.V. Shurygin)
Astrakhan State University
ul. Tatishcheva 20a, Astrakhan, 414056 Russia
Received December 4, 2017
Abstract—In this paper, we compute the rational projective diﬀerential invariants of linear planar 2-
webs. The invariants are used to ﬁnd conditions under which two linear planar 2-webs are equivalent
with respect to the group of projective transformations.
Keywords: projective diﬀerential invariants, invariant derivations.
Introduction. The problem of equivalence is one of the basic ones in the theory of planar webs .
Here, the choice of structure group is essential. The most useful are groups of projective and conformal
 transformations of the plane, as well as the group of all diﬀeomorphisms of the plane.
In this paper, we consider linear 2-webs, i.e., those generated by two transversal foliations of the
plane consisting of straight lines. It is clear that such webs are locally equivalent with respect to the
group of all diﬀeomorphisms of the plane.
Another classic group in the theory of webs is the group of projective transformations of the plane
(R) which acts straightforwardly in the class of linear webs. In this paper, we give projective
diﬀerential invariants of linear 2-webs which we use later for the projective classiﬁcation of such webs.
In the ﬁrst place, this problem is reduced to the seek of projective diﬀerential invariants for the pair
of diﬀerent solutions of the Euler equation. We show (see Theorem 3) that every rational projective
diﬀerential invariant of a linear 2-web is generated by two diﬀerential invariants of the second and third
, respectively, and by their invariant derivations.
We call a linear 2-web regular, if the values of basis invariants J
on this web are functionally
independent. For regular linear 2-webs, we suggest a constructive way to solve the problem of their
1. Linear webs and the Euler equation. A linear foliation on the plane, whose leaves are lines, can
be given by equations L(t):
a(t)x + b(t)y +1=0, (1)
where t is a parameter deﬁning a leaf.
Deﬁneafunctionu(x, y) ontheplanewhosevalueatthepoint(x, y) equals the value of parameter t
for the line passing through this point, i.e., (x, y) ∈ L(u(x, y)).
We diﬀerentiate twice Eq. (1), where t = u(x, y), and eliminate functions a, b and their derivatives, to
obtain the following diﬀerential equation (the ﬂex equation ):
Thus we obtain the following representation of linear foliations.
Proposition 1. Any linear foliation in a domain D ⊂ R
is generated by contours of the function
u(x, y) satisfying in this domain the ﬂex Eq. (2) with exterior derivative du =0.