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New relations between $G_2$-geometries in dimensions 5 and 7

There are two well-known parabolic split $G_2$-geometries in dimension five, $(2,3,5)$-distributions and $G_2$-contact structures. Here we link these two geometries with yet another $G_2$-related contact structure, which lives on a seven-manifold. We present a natural geometric construction of a Lie contact structure on a seven-dimensional bundle over a five-manifold endowed with a $(2,3,5)$-distribution. For a class of distributions the induced Lie contact structure is constructed explicitly and we determine its symmetries. We further study the relation between the canonical normal Cartan connections associated with the two structures. In particular, we show that the Cartan holonomy of the induced Lie contact structure reduces to $G_2$. Moreover, the curved orbit decomposition associated with a $\mathrm{G}_2$-reduced Lie contact structure on a seven-manifold is discussed. It is shown that in a neighbourhood of each point on the open curved orbit the structure descends to a $(2,3,5)$-distribution on a local leaf space, provided an additional curvature condition is satisfied. The closed orbit carries an induced $G_2$-contact structure.