Skew and sphere fibrations

TL;DR

This paper investigates the topology and geometry of nondegenerate fibrations of Euclidean space, establishing classification results and their relation to great sphere fibrations, with implications for understanding sphere bundle structures.

Contribution

It introduces the concept of nondegenerate fibrations, studies their properties, and connects them to great sphere fibrations, providing new classification insights and extending local fibrations to global ones.

Findings

Every germ of a nondegenerate fibration extends to a global fibration.

Nondegenerate line fibrations relate to contact structures in odd-dimensional Euclidean space.

Classification results for nondegenerate fibrations in certain dimensions.

Abstract

A great sphere fibration is a sphere bundle with total space $S^n$ and fibers which are great $k$-spheres. Given a smooth great sphere fibration, the central projection to any tangent hyperplane yields a \emph{nondegenerate} fibration of $\mathbb{R}^n$ by pairwise skew, affine copies of $\mathbb{R}^k$ (though not all nondegenerate fibrations can arise in this way). Here we study the topology and geometry of nondegenerate fibrations, we show that every nondegenerate fibration satisfies a notion of Continuity at Infinity, and we prove several classification results. These results allow us to determine, in certain dimensions, precisely which nondegenerate fibrations correspond to great sphere fibrations via the central projection. We use this correspondence to reprove a number of recent results about sphere fibrations in the simpler, more explicit setting of nondegenerate fibrations. For example, we show that every germ of a nondegenerate fibration extends to a global fibration, and we study the relationship between nondegenerate line fibrations and contact structures in odd-dimensional Euclidean space. We conclude with a number of partial results, in hopes that the continued study of nondegenerate fibrations, together with their correspondence with sphere fibrations, will yield new insights towards the unsolved classification problems for sphere fibrations.