Conformal deformations of submanifolds in codimension two (Q1975889)

It is shown that a certain conformal rigidity holds for conformal immersions \(M^n\to\mathbb{R}^{n+2}\). More precisely, under the assumption that the dimension of any umbilical subspace does not exceed \(n-5\), two conformal immersions of the same manifold are either conformally congruent (that is, equivalent under a conformal mapping of the ambient space) or both can be isometrically realized as hypersurfaces in two Cartan hypersurfaces \(N^{n+1} \to\mathbb{R}^{n+1}\) which in turn are conformal to one another but nowhere conformally congruent. This improves a conformal rigidity theorem by \textit{M. do Carmo} and the first author [Am. J. Math. 109, 963-985 (1987; Zbl 0631.53043)].