On fibrations with flat fibers

TL;DR

This paper investigates conditions under which affine spaces can be decomposed into skew affine subspaces, linking the problem to classical results in topology and quadratic forms.

Contribution

It characterizes pairs (p,n) for affine space fibrations by skew subspaces, connecting geometric configurations with topological and algebraic theories.

Findings

Characterizes pairs (p,n) for affine space fibrations

Links geometric fibrations to Adams' theorem and Hurwitz-Radon theory

Provides conditions for the existence of such fibrations

Abstract

We describe pairs (p,n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms.