Curvilinear three-webs (differentially-topological theory) (Q2866888)

A \textit{\(d\)-web of dimension \(k\)} on an \(n\)-dimensional real manifold \(M\) is a collection of \(d\) foliations by \(k\)-dimensional submanifolds of an open subset \(U\subset M\). The study of webs has its origins in the work of S. Lie and H. Poincaré on surfaces of translation. Still, to quote \textit{S.-S. Chern} in [Bull. Am. Math. Soc., New Ser. 6, 1--8 (1982; Zbl 0483.53013)] the theory of \(d\)-webs had its debut in 1926--27 on the beaches of Italy, when W. Blaschke and G. Thomsen realised that in \(\mathbb{R}^2\), a configuration of three foliations by curves has local invariants. Since then, a considerable effort has been invested in attempts to classify webs of various dimensions up to local diffeomorphisms.NEWLINENEWLINEThe manifold \(M\) is often taken to be the Euclidean space, although many interesting webs arise in Grassmannians. For example, a \(k\)-dimensional algebraic subvariety of degree \(d\) in the projective space \(\mathbb{P}^m\) determines a \(d\)-web of codimension \(k\) on the (projective) Grassmannian \(G(m-k,m)\). Indeed, since a point in the Grassmannian is an \((m-k)\)-plane, it intersects the variety in \(d\) points, and one can consider the \((m-k)\)-planes, passing through these points. Chern's article is a beautiful introduction to the subject, its history and problems at the time.NEWLINENEWLINEThe present monograph is devoted to the study of curvilinear three-webs, i.e., the case \(d=3\) and \(n=2\). Historically, curvilinear webs were the first to be studied and used in applications, most notably nomography. However, due to the presence of difficult problems which were only recently resolved, the research in the area had not been very active, at least when compared to the study of higher-dimensional webs. The book of Shelekhov, Lazareva and Utkin claims to fill in this gap and to contain all (known to the authors) meaningful results in the area. Apart from authors' original results, it contains a lot of background material, examples and problems, which makes it suitable for in-class use or individual study.NEWLINENEWLINEThe first chapter is an introduction to 3-webs, including a classification of boundaries and regularity conditions. It also includes a proof of Thomsen's theorem, a discussion of linear and Grassmannian webs and a plethora of beautiful classical results. Chapter 2 is devoted to differential geometry of curvilinear 3-webs, including a derivation of its structure equations, relative and absolute differential invariants. It contains necessary and sufficient conditions for the web to be determined by a linear or Riccati equation. The structure equations for a Grassmannian 3-web are derived. The authors show that a smooth map between Grassmannian three-webs is a projective transformation, which gives a solution to Gronwall's problem. Chapter 3 deals with a problem, formulated by Blaschke in the middle of the previous century: the classification of regular webs, obtained from pencils of circles. A detailed solution, due to Shelekhov and Lazareva, is presented. Using theorems about boundaries of regular webs, they prove that up to ``circle equivalence'' there are 48 types of such webs. Chapter 4 is about special ``Buarau webs'', which are certain pencils of conics. For that, consider a cubic surface in projective 3-space with three lines on it. The planes passing through these lines are conics which form a 3-web. The relative position of the lines is crucial for the regularity properties of the web.NEWLINENEWLINE In the final chapter the authors use Cartan's method of moving frames to study webs on smooth projective surfaces, determined by various choices of a ``cubic absolute''.