Remarks on non-maximal integral elements of the Cartan plane in jet spaces

There is a natural filtration on the space of degree-kk homogeneous polynomials in nn independent variables with coefficients in the algebra of smooth functions on the Grassmannian Gr(n,s), determined by the tautological bundle. In this paper we show that the space of ss-dimensional integral elements of a Cartan plane on Jk−1(E,n)Jk−1(E,n), with dimE=n+mdimE=n+m, has an affine bundle structure modeled by the so-obtained bundles over Gr(n,s), and we study a natural distribution associated with it. As an example, we show that a third-order nonlinear PDE of Monge–Ampère type is not contact-equivalent to a quasi-linear one.