A Characterization of Horizontally Totally Geodesic Hypersurfaces in Heisenberg Groups

In the sub-Riemannian Heisenberg group Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}<^>n$$\end{document}, with n >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 2$$\end{document}, we prove that hyperplanes are the only entire hypersurfaces with vanishing horizontal symmetric second fundamental form. This result is applied in a subsequent work to establish a Bernstein-type theorem in H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}<^>2$$\end{document}, marking a significant step towards understanding the sub-Riemannian Bernstein problem.