$G_2$-geometry in contact geometry of second order

Abstract

In [13],[14],[15], we formulate the contact equivalence of systems of second order partial differential equations for a scalar function as the Contact Geometry of Second Order or as the geometry of PD-manifolds of second order, generalizing works [3], [4] of E.Cartan. Especially, in [13], generalizing the famous G2-models in [3], we observed, for each exceptional simple Lie algebra Xl, we could find the overdetermined system (Al) and the single equation of Goursat type (Bl), whose symmetry algebras are isomporphic with Xl and formulated this fact as the G2-geometry. The main purpose of the present paper is to construct the (local) models for overdetermined systems (Al) explicitly for each exceptional simple Lie algebra and also for the classical type analogy for BD type. We will also give parametric descriptions of the single equation of Goursat type (Bl). Our constructions are based on the explicit calculation, in terms of Chevalley basis, of the structure of the Goursat gradation of each exceptional simple Lie algebra and each simple Lie algebra of BD type.