Conformal immersions of Riemannian products in low codimension

Abstract

We prove that conformal immersion of a Riemannian product $$M_0^{n_0}\times M_1^{n_1}$$ as a hypersurface in a Euclidean space must be an extrinsic product of immersions, under the assumption that $$n_0, n_1 \ge 2$$ and that $$M^{n_0}_0\times M^{n_1}_1$$ is not conformally flat. We also state a similar theorem for an arbitrary number of factors, more precisely, a conformal immersion $$f:M^{n_0}_0 \times \cdots \times M^{n_k}_k \rightarrow {{\mathbb {R}}}^{n+k}$$ must be an extrinsic product of immersions if one of the factors admits a plane with vanishing curvature and the remaining factors are not flat.