  In this paper, for an immersion $f$ of an $n$-dimensional Riemannian manifold
$M$ into $(n+d)$-Euclidean space we give a sufficient condition on $f$ so that,
in case $d\leq 5$, any immersion $g$ of $M$ into $(n+d+1)$-Euclidean space that
induces on $M$ a metric that is conformal to the metric induced by $f$ is
locally obtained, in a dense subset of $M$, by a composition of $f$ and a
conformal immersion from an open subset of $(n+d)$-Euclidean space into an open
subset of $(n+d+1)$-Euclidean space. Our result extends a theorem for
hypersurfaces due to M. Dajczer and E. Vergasta. The restriction on the
codimension is related to a basic lemma in the theory of rigidity obtained by
M. do Carmo and M. Dajczer.
