The Codimension of Submanifolds with Negative Extrinsic Curvature

We prove that a substantial isometric immersion into a space form f:Mn→Qcn+p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:M^n\rightarrow \mathbb {Q}_c^{n+p}$$\end{document} with negative extrinsic curvature and flat normal bundle whose first normal bundle has the lowest possible rank possesses substantial codimension p=n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=n-1$$\end{document}. This fact is already known in the rather special case when also Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^n$$\end{document} has constant sectional curvature.