LOCAL CLASSIFICATION OF ALMOST FLAT (1,1) -TENSOR FIELDS

Consider a (1, 1) tenser field J, on a real or complex manifold whose Nijenhuis torsion vanishes. Let rho = t(m) + h(m-1) t(m-1) +...+ h(0) be its characteristic polynomial and set E = Ker dh(0) boolean AND...b oolean AND Ker dh(m-1). Then JE subset of E, and J/E defines a G-struc ture on each integral submanifold N of E [i.e. such that T-p N = E (p) for all p epsilon N]. IS the first-order structure function of all of these G-structures vanishes, then there exists a dense open set such that we can find coordinates, around each of its its points, on which J is written with affine coefficients. Moreover we give the complete l ocal classification of J on this open set.