A Semple-type approach to a problem of Goursat: The multi-flag case

We consider the problem of classifying the orbits within a tower of fibrations with fibers diffeomorphic to projective planes and we generalize the tower of fiber bundles due to J. Semple. This tower, which was rediscovered by Montgomery and Zhitomirskii in the context of subriemannian geometry, admits a natural action of the diffeomorphism group of affine 3-space, and these orbits correspond to classes of Goursat multi-flags. We demonstrate that it is possible to classify many of these orbits by elementary means by appealing to some basic tools in projective geometry, and the combinatorics of spatial curves.