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

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.


53A60 Differential geometry of webs
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