Web geometry and the equivalence problem of the first order partial differential equations (Q2782010)

Consider an implicit differential equation of the type NEWLINE\[NEWLINE f (x, y, y') = 0.\tag{1} NEWLINE\]NEWLINE If one solves (1), one obtains \(d\) explicit first-order differential equations NEWLINE\[NEWLINE y' = f_1 (x, y), \dots, y' = f_d (x, y).\tag{2} NEWLINE\]NEWLINE The solutions of (2) form a configuration of \(d\) foliations of codimension one at a generic point in \((x,y)\)-space, i.e., a \(d\)-web (a solution web of (2)). It is known that the holomorphic equivalence classes of \(d\)-webs \((d \geq n + 1)\) form subsets of infinite codimension in the jet space of \(d\)-tuples of level functions defined at \(0 \in {\mathbb C}^n\). This means that an analytic classification of the solution webs fails in the ordinary sense. In fact, it is also known [see, for example, \textit{V. I. Arnol'd, S. M. Gusein-Zade} and \textit{A. N. Varchenko}, Singularities of differentiable maps. Vol. I, Birkhäuser Boston, Boston, MA (1985; Zbl 0554.58001)] that analytic equivalence classes of the solution webs defined by PDEs with first integrals have infinite-dimensional moduli, which are parametrized by the space of smooth functions defined on the configuration space \({\mathbb C}^n\) at \(0\). The parameter space is called the function moduli. The author showed earlier [see Topology 26, No. 4, 475-504 (1987; Zbl 0647.57018)] that for such a web structure, topological equivalence is automatically given by a holomorphic diffeomorphism. Thus, even topological classification does not make sense.NEWLINENEWLINENEWLINEIn the paper under review, the author generalizes the above idea to define invariant forms for some cases. He also investigates the relation of singularities of the affine connection (the pole and zero of the connection form and the curvature form) to the singularities of the web structure, develops a general theory of geometric invariants of PDEs and looks for geometric understanding of the structure of propagation of wave fronts, which is focused in the theory of optical caustics.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00024].