Holonomy reductions of Cartan geometries and curved orbit decompositions

Details Holonomy reductions of Cartan geometries and curved orbit decompositions Holonomy reductions of Cartan geometries and curved orbit decompositions

2011-03-23

arxiv.org/abs/1103.4497v2

Mathematics

Differential Geometry

v2: major revision; 30 pages

Scientific paper

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.

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