  We characterize the existence of an Ulrich vector bundle on a variety $X
\subset P^N$ in terms of the existence of a subvariety satisfying some precise
conditions. Then we use this fact to prove that a complete intersection of
dimension $n \ge 4$, which if $n=4$ is very general and not of type $(2,2)$,
does not carry any Ulrich bundles of rank $r \le 3$ unless $n=4, r=2$ and $X$
is a quadric.
