A naive guide to web geometry (Q2782003)

The paper under review does not have new results. Its goal is to show some relationships and applications of web geometry. In particular, the author discussesNEWLINENEWLINENEWLINE1) web geometry connections with Lie's theorem on surfaces of double translation and its generalizations (Lie, Poincaré, Darboux, Griffiths, Henkin);NEWLINENEWLINENEWLINE2) the singularity of the curvature forms of various singular 3 web structures in the plane defined by implicit differential equations of the type \(f (x, y, y') = 0\) (Nakai, Mignardi);NEWLINENEWLINENEWLINE3) rank problems for webs of codimension one and exceptional Bol's 5-web and its generalizations (Bol, Damiano, Gelfand--MacPherson, Goldberg, Hénaut);NEWLINENEWLINENEWLINE4) links between web geometry, symplectic geometry and integrable systems (Chern, Brouzet--Molino--Turiel, Fernandes), and in particular, studying of Veronese's webs (Gelfand--Zakharevich, Turiel, Rigal);NEWLINENEWLINENEWLINE5) linear families of one-forms and webs (Cerveau, Cerveau--Neto, Ghys, Nakai, Dufour); 6) application of web geometry to nonlinear geometric optics (Joly, Joly--Métivier--Rauch). NEWLINENEWLINENEWLINEThe author also indicates some open problems: finding a complete classification of double translation structures and investigation of singularities of such structures; and investigation of the space of abelian equations.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00024].