Properties of the connection associated with planar webs and applications

We give various results and applications using the connection $(E,\nabla)$ associated with a $d$-web. Precisely, we exhibit fundamental invariants of the web related to the differential equation of first order which presents the web. They cast some new lights on the connection and its construction, both conceptually and effectively. We describe 4 and 5-webs from this point of view and show for instance that the connection gives account for the linearizability conditions of the web. Moreover, we get characterization of maximal rank webs such as exceptional 5-webs, and 4-webs \emph{via} a new proof of the Poincar\'e theorem in terms of differential systems. We establish the trace formula related to the determinant bundle $(\det E, \det \nabla)$ and the extracted 3-webs. Furthermore, the theorem of determination of the rank is proved to give an explicit criterion for measuring the rank of a web.