New relations between G2 geometries in dimensions 5 and 7

Abstract

There are two well-known parabolic split [Formula: see text] geometries in dimension 5, [Formula: see text] distributions and [Formula: see text] contact structures. Here we link these two geometries with yet another [Formula: see text] related contact structure, which lives on a [Formula: see text]-manifold. More precisely, we present a natural geometric construction that associates to a [Formula: see text] distribution a [Formula: see text]-dimensional bundle endowed with a canonical Lie contact structure. We further study the relation between the canonical normal Cartan connections associated with the two structures and we show that the Cartan holonomy of the induced Lie contact structure reduces to [Formula: see text]. This motivates the study of the curved orbit decomposition associated with a [Formula: see text] reduced Lie contact structure on a [Formula: see text]-manifold. It is shown that, provided an additional curvature condition is satisfied, in a neighborhood of each point in the open curved orbit the structure descends to a [Formula: see text] distribution on a local leaf space. The closed orbit carries an induced [Formula: see text] contact structure.