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 Kähler-Einstein metric.

v2: major revision; 30 pages v3: final version to appear in Duke Math. J

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• New relations between $G_2$-geometries in dimensions 5 and 7
• Invariant prolongation of the Killing tensor equation
• Asymptotic symmetries of projectively compact order one Einstein manifolds
• First BGG operators on homogeneous conformal geometries
• First BGG operators via homogeneous examples
• Differential geometry of ${\mathsf{SO}}^{\ast }(2n)$-type structures -- Integrability
• The ambient obstruction tensor and conformal holonomy
• Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds
• Ti and Spi, Carrollian extended boundaries at timelike and spatial infinity
• Parabolic Compactification of Homogeneous Spaces
• Parabolic quasi-contact cone structures with an infinitesimal symmetry
• Perturbations of Fefferman spaces over CR three-manifolds