Totally geodesic fibrations

Motivation.

Is it possible to cover R³ by continuously varying lines, such that no two are parallel and no two intersect? More precisely, does there exist a fibration of R³ by pairwise skew lines? The answer is yes, and an example can be constructed as follows: fix a line in R³, and surround it with a family of nested, one-sheeted hyperboloids. Each hyperboloid is ruled, resulting in a fibration of R³ by skew lines. This is depicted below.

Two affine subspaces R^p in Rⁿ are called skew if they do not intersect nor contain parallel directions. We refer to a fibration of Rⁿ by pairwise skew, oriented copies of R^p as a (p,n)-fibration. The reason to include the ``skew'' assumption is two-fold. First, without this assumption, the problem is trivial: for any p<n, one can cover Rⁿ by parallel copies of R^p. Second, skew fibrations arise naturally from Hopf fibrations and other great sphere fibrations: the radial projection from a fibered Sⁿ to any tangent hyperplane Rⁿ sends the great sphere fibers to pairwise skew flats in Rⁿ. Thus the study of skew, flat fibrations is partially motivated by the importance of spherical fibrations. In turn, spherical fibrations have been extensively studied due to their relationship with the Blaschke conjecture. See this summary by Ben McKay.

Great circle fibrations of S³ were classified by Herman Gluck and Frank Warner, who showed that the space of all oriented great circle fibration of S³ deformation retracts to the subspace of Hopf fibrations (citation here).

Here I extensively studied (1,3)-fibrations, as well as the relationship between (1,3)-fibrations and great circle fibrations, and I proved that the space of (1,3)-fibrations deformation retracts to the subspace of (projected) oriented Hopf fibrations.

Here and here I studied the relationship between line fibrations and contact structures.

Higher-dimensional skew fibrations.

Ovsienko and Tabachnikov showed here that a (p,n)-skew fibration exists if and only if there exist p linearly independent tangent vector fields on S^n-p-1, thus establishing relationships to Adams' work about vector fields on spheres, as well as to the Hurwitz-Radon function. As a noteworthy special case, a (p,2p+1)-skew fibration exists if and only if S^p is parallelizable, which occurs if and only if p is 1, 3, or 7.