## 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.

### MSC:

 53A60 Differential geometry of webs
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