A general method to construct invariant PDEs on homogeneous manifolds

Abstract

Let [Formula: see text] be an [Formula: see text]-dimensional homogeneous manifold and [Formula: see text] be the manifold of [Formula: see text]-jets of hypersurfaces of [Formula: see text]. The Lie group [Formula: see text] acts naturally on each [Formula: see text]. A [Formula: see text]-invariant partial differential equation of order [Formula: see text] for hypersurfaces of [Formula: see text] (i.e., with [Formula: see text] independent variables and [Formula: see text] dependent one) is defined as a [Formula: see text]-invariant hypersurface [Formula: see text]. We describe a general method for constructing such invariant partial differential equations for [Formula: see text]. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup [Formula: see text] of the [Formula: see text]-prolonged action of [Formula: see text]. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space [Formula: see text] and in the conformal space [Formula: see text]. Our method works under some mild assumptions on the action of [Formula: see text], namely: A1) the group [Formula: see text] must have an open orbit in [Formula: see text], and A2) the stabilizer [Formula: see text] of the fiber [Formula: see text] must factorize via the group of translations of the fiber itself.