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

Abstract

There is a natural filtration on the space of degree- k homogeneous polynomials in n 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 s -dimensional integral elements of a Cartan plane on J k − 1 ( E , n ) , with dim E = 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.