Ordinary and calibrated differential operators, connections and application to curvilinear webs

Abstract

Abelian relations of a curvilinear web are the solutions of the partial differential equation defined by a certain differential operator that we prove to be always "ordinary" and "calibrated". Thus we begin by explaning why the space of solutions of the partial differential equation defined by such an ordinary and calibrated homogeneous linear differential operator (of arbitrary order) is isomorphic to the space of the sections of a certain vector bundle with vanishing covariant derivative for a certain tautological connection. We then apply this result to the case of webs by curves, recovering the upper-bound for the rank of such a web given in [D. Damiano, PhD Thesis, Brown University, 1986] та [D. Damiano, Amer. J. Math. (1983) 105:6, 1325-1345], and defining finally the "curvature" of the web which vanishes iff the web has maximal rank.