Holonomy reductions of Cartan geometries and curved orbit decompositions

Abstract

We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modeling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a “curved orbit decomposition.” The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal, and CR geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third-order differential equation that governs the existence of a compatible Einstein metric, and in conformal geometry we discuss almost-Einstein scales. Finally, we discuss analogues of the two latter equations in CR geometry, which leads to invariant equations that govern the existence of a compatible Kähler–Einstein metric.