Geometric aspects of rank-3 vector bundles over surfaces and 2-plane distributions on 5-manifolds

We study geometric aspects of horizontal 2-plane distributions on the complement of the zero section in the 5-dimensional total space of a rank-3 vector bundle equipped with connection over a surface. We show that any surface in 3-dimensional projective space can be associated to such a geometric structure in 5-dimensions, and establish a dictionary between the projective differential geometry of the surface and the growth vector of the 2-plane distribution.