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 modelling 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. In CR-geometry we discuss an invariant equation that governs the existence of a compatible K\"{a}hler-Einstein metric.Comment: v2: major revision; 30 pages v3: final version to appear in Duke Math.

Projective compactness and conformal boundaries

“…In this case, the section L(τ ) of S 2 T * is parallel for the the tractor connection, thus defining a reduction of projective holonomy to a pseudo-orthogonal group. Via the general theory of holonomy reductions developed in [9], one obtains an induced conformal structure on the boundary, which by Proposition 10 coincides with the one discussed in this article. The general theory further implies that one can obtain the conformal standard tractor bundle by restricting the projective standard tractor bundle to the boundary, endowing it with the bundle metric L(τ ).…”

“…The argument which was used to prove this in Proposition 3.2 of [9] actually can be applied in a significantly more general situation, as we will show next.…”

Projective compactifications and Einstein metrics

“…In this case, L(τ ) defines a non-degenerate bundle metric (necessarily of indefinite signature) on the standard tractor bundle over M . This gives rise to a reduction of projective holonomy to an orthogonal group as studied in Section 3.3 of [8] and in Section 3.1 of [9], with the closed curved orbit given by the boundary ∂M and the open curved orbit given by the interior M. In particular, as shown in these references, the boundary ∂M inherits a conformal structure.…”

“…Projective holonomy reductions to orthogonal groups have been studied in detail in Section 3.2 of [8] and in Section 3.1 of [9] and we use the results obtained there. If we start with a Riemannian metric g, then the reduction will be to the orthogonal group SO(n + 1) ⊂ SL(n + 1, R) and this amounts to a positive Einstein Riemannian metric in the projective class.…”

“…Now of course, the boundary sphere S n can be viewed as the ray projectivization of ker(I) \ {0} and H −1 describes a parallel section of the symmetric square of the cotractor bundle for the resulting flat projective structure. Now the orbit decomposition is described in [8] and [9]. It exactly corresponds to the restriction of H −1 to a ray being positive definite, negative definite, and zero, respectively.…”