This the first of a set of three papers about the Compression
Theorem: if Mm is embedded in Qq×R
with a normal vector field and if q-m≥1, then the given vector field
can be straightened (ie, made parallel to the given R
direction) by an isotopy of M and normal field in Q×R.
The theorem can be deduced from Gromov's theorem on directed embeddings
and is implicit in the preceeding discussion. Here we give a direct
proof that leads to an explicit description of the finishing
In the second paper in the series we give a proof in the spirit of
Gromov's proof and in the third part we give applications.