Conformal immersions of Riemannian products in low codimension

We prove that conformal immersion of a Riemannian product M0n0×M1n1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_0^{n_0}\times M_1^{n_1}$$\end{document} as a hypersurface in a Euclidean space must be an extrinsic product of immersions, under the assumption that n0,n1≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_0, n_1 \ge 2$$\end{document} and that M0n0×M1n1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^{n_0}_0\times M^{n_1}_1$$\end{document} is not conformally flat. We also state a similar theorem for an arbitrary number of factors, more precisely, a conformal immersion f:M0n0×⋯×Mknk→Rn+k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:M^{n_0}_0 \times \cdots \times M^{n_k}_k \rightarrow {{\mathbb {R}}}^{n+k}$$\end{document} 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.