Local Conformal Rigidity in Codimension $\leq$ 5

Abstract

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.