Differential invariants of generic parabolic Monge–Ampère equations

Abstract

Some new results on geometry of classical parabolic Monge-Ampère equations (PMA) are presented. PMAs are either integrable, or nonintegrable according to integrability of its characteristic distribution. All integrable PMAs are locally equivalent to the equation u xx = 0. We study nonintegrable PMAs by associating with each of them a 1-dimensional distribution on the corresponding first order jet manifold, called the directing distribution. According to some property of this distribution, nonintegrable PMAs are subdivided into three classes, one generic and two special ones. Generic PMAs are completely characterized by their directing distributions, and we study canonical models of the latters, projective curve bundles (PCB). A PCB is a 1-dimensional subbundle of the projectivized cotangent bundle of a 4-dimensional manifold. Differential invariants of projective curves composing such a bundle are used to construct a series of contact differential invariants for corresponding PMAs. These give a solution of the equivalence problem for generic PMAs with respect to contact transformations. The introduced invariants measure in an exact manner nonlinearity of PMAs. In this article we study geometry of classical parabolic Monge-Ampere equations (PMAs) on the basis of a new approach sketched in [16]. It is based on the observation that a PMA E ⊂ J 2 (π), π being a 1-dimensional fiber bundle over a bidimensional manifold, is completely characterized by a 2-dimensional Lagrangian distribution D E on J 1 (π) , called the characteristic distribution of E, and vice versa. Such distributions and, accordingly, the corresponding PMAs, are naturally subdivided into four classes of integrable, generic and two special types of equations (see [16] and sec.4). In this classification, integrable PMAs are those whose characteristic distributions are integrable. Since all integrable Lagrangian distributions are locally contact equivalent, all integrable PMAs are locally contact equivalent to one of them, say, to the equation u xx = 0. On the contrary, nonintegrable Lagrangian distributions are very diversified and our main goal here is to describe their multiplicity, i.e., equivalence classes of PMAs with respect to contact transformations.With this purpose we associate with a Lagrangian distribution a projective curve bundle (shortly, PCB) over a 4-dimensional manifold N. A PCB over N is a 1dimensional smooth subbundle of the "projectivized" cotangent bundle P T * (N) of N.