Institute for Mathematical Physics Parabolic Geometries and Canonical Cartan Connections Parabolic Geometries and Canonical Cartan Connections

Let G be a (real or complex) semisimple Lie group, whose Lie algebra g is endowed with a so called jkj{grading, i.e. a grading of the form g = g ?k g k , such that no simple factor of G is of type A 1. Let P be the subgroup corresponding to the subalgebra p = g 0 g k. The aim of this paper is to clarify the geometrical meaning of Cartan connections corresponding to the pair (G; P) and to study basic properties of these geometric structures. Let G 0 be the (reductive) subgroup of P corresponding to the subalge-bra g 0. A principal P{bundle E over a smooth manifold M endowed with a (suitably normalized) Cartan connection ! 2 1 (E;g) automatically gives rise to a ltration of the tangent bundle TM of M and to a reduction to the structure group G 0 of the associated graded vector bundle to the ltered vector bundle TM. We prove that in almost all cases the principal P bundle together with the Cartan connection is already uniquely determined by this underlying structure (which can be easily understood geometrically), while in the remaining cases one has to make an additional choice (which again can be easily interpreted geometrically) to determine the bundle and the Cartan connection.