Uniformity of geodesic flow in non-integrable 3-manifolds

Almost nothing is known concerning the extension of 3-dimensional Kronecker--Weyl equidistribution theorem on geodesic flow from the unit torus [0, 1) ³ to non-integrable finite polycube translation 3-manifolds. In the special case when a finite polycube translation 3-manifold is the cartesian product of a finite polysquare translation surface with the unit torus [0, 1), we have developed a splitting method with which we can make some progress. This is a somewhat restricted system, in the sense that one of the directions is integrable. We then combine this with a split-covering argument to extend our results to some other finite polycube translation 3-manifolds which satisfy a rather special condition and where none of the 3 directions is integrable.