A continuous fibration of
by oriented lines is given by a unit vector field
for
which all of the integral curves are oriented lines. A line fibration is called skew if
no two fibers are parallel. Skew fibrations have been the focus of recent
study, in part due to their close relationships with great circle fibrations of
and with tight
contact structures on
.
Both geometric and topological classifications of the space of skew fibrations have
appeared; these classifications rely on certain rigid geometric properties exhibited by
skew fibrations. Here we study these properties for line fibrations which are not
necessarily skew, and we offer some partial answers to the question: in what sense do
nonskew fibrations look and behave like skew fibrations? We develop and utilize a
technique, called the parallel plane pushoff, for studying nonskew fibrations. In
addition, we summarize the known relationship between line fibrations and contact
structures, and we extend these results to give a complete correspondence. Finally, we
develop a technique for generating nonskew fibrations and offer a number of
examples. All results (except where otherwise stated) hold in the topological
category.
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