Skew flat fibrations

fibration of ℝ^n by oriented copies of ℝ^p is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and n for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of ℝ^3 by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of S^3 by Gluck and Warner. We show that Salvai’s classification has a topological variation which generalizes to characterize all continuous fibrations of ℝ^n by skew oriented copies of ℝ^p . We show that the space of fibrations of ℝ^3 by skew oriented lines deformation retracts to the subspace of Hopf fibration, and therefore has the homotopy type of a pair of disjoint copies of S^2 . We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of ℂ^n (respectively, ℍ^n ) by skew oriented copies of ℂ^p (respectively, ℍ^p ).