Skew flat fibrations

Abstract

A fibration of ${\mathbb{R}}^n$ by oriented copies of ${\mathbb{R}}^p$ 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 ${\mathbb{R}}^3$ 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 ${\mathbb{R}}^n$ by skew oriented copies of ${\mathbb{R}}^p$. We show that the space of fibrations of ${\mathbb{R}}^3$ by skew oriented lines deformation retracts to the subspace of Hopf fibrations, 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 ${\mathbb{C}}^n$ (${\mathbb{H}}^n$) by skew oriented copies of ${\mathbb{C}}^p$ (${\mathbb{H}}^p$).

Mathematische Zeitschrift, 2015, http://link.springer.com/article/10.1007/s00209-015-1538-0

References in corpus (1)

• On fibrations with flat fibers

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