Local Conformal Rigidity in Codimension ≤ 5

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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≤ 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.