manuscripta math. 154, 185–193 (2017) © Springer-Verlag Berlin Heidelberg 2016
Sergey I. Agafonov
Note on generic singularities of planar ﬂat 3-webs
Received: 9 January 2016 / Accepted: 8 December 2016
Published online: 21 December 2016
Abstract. We propose a deﬁnition of genericity for singular ﬂat planar 3-webs formed by
integral curves of implicit ODEs and give a classiﬁcation of generic singularities of such
A planar d-web is formed by d foliations in the plane. At each point of the plane, we
have d leaves passing through the point, one from each foliation of the web. A point
is called regular if any two of these d leaves are transverse. Consider the pseu-
dogroup of local diffeomorphisms of the plane and the corresponding equivalence
relation on the set of d-web germs. Any two planar 2-web germs are equivalent
whenever the base points are regular. This is not true for 3-web germs (see ).
There is a local invariant, which has, in fact, a topological nature. The differential-
geometric counterpart of this invariant is the so-called Blaschke curvature.
Deﬁnition 1. 3-web is ﬂat (or hexagonal) if its germ at any regular point is equiv-
alent to the web formed by three families of parallel lines.
A 3-web is ﬂat if and only if the Blaschke curvature vanishes identically (see, for
instance ). This curvature is a scalar 2-form, therefore any general classiﬁcation
of 3-webs with respect to local diffeomorphisms will necessarily have functional
moduli. Namely, such a classiﬁcation will inevitably involve arbitrary functions of
By the above deﬁnition, any two ﬂat 3-web germs are equivalent provided that
the base points are regular. Hence the “personality” of a ﬂat 3-web is encoded in
Web structure is ubiquitous in mathematics and its applications, the Blaschke
curvature often being the obstacle to obtaining a “reasonable” classiﬁcation. There-
fore ﬂat 3-webs play a distinguished role.
For example, hexagonal 3-webs have a 3-dimensional local symmetry algebra
at regular points, while a generic 3-web does not admit any inﬁnitesimal symmetry
S. I. Agafonov (
): Departamento de Matemática, UNESP-Universidade Estadual Paulista,
São José do Rio Preto, Brazil. e-mail: firstname.lastname@example.org
Mathematics Subject Classiﬁcation: Primary 53A60; Secondary 34M35