On planar web geometry through Abelian relations and connections. (English) Zbl 1069.53020

Hénaut, Alain

Ann. Math. (2) 159, No. 1, 425-445 (2004).

Using a canonical normalization of a planar \(d\)-web given by a differential equation \(F(x,y,y')=0\) the author gives a method to find analytical invariants of this \(d\)-web. A connection \((\varepsilon, \bigtriangledown)\) associated with a planar \(d\)-web is constructed. The connection is a generalization of the Blaschke 3-web connection and its \(\mathbb{C}\)-vector space of horizontal sections is isomorphic to the \(\mathbb{C}\)-vector space of Abelian relations of \(d\)-webs. The connection \((\varepsilon, \bigtriangledown)\) is integrable if and only if \(d\)-web is of maximal rank. Using only the methods introduced in his paper the author proves the main theorem for linear \(d\)-webs.

Reviewer: A. M. Shelekhov (Tver’)

MSC:

53A60

Differential geometry of webs

Keywords:

\(d\)-web; linear web; Abelian relation of web; rank of web; connection of web