Hypersurfaces with two distinct principal curvatures in a real space form

In this paper, we study hypersurfaces with two distinct principal curvatures in a real space form Mn+1(c). Denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi_{ij}}$$\end{document} the trace free part of the second fundamental form of Mn, and let ρ2 be the square of the length of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi_{ij}}$$\end{document}. If ρ2 is constant, we obtain two rigidity results and give some characterization of the Riemannian products in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M^{n+1}(c): S^k(a) \times S^{n-k}(\sqrt{1-a^2})}$$\end{document} for c = 1, Rk × Sn-k(a) for c = 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^k(\tanh^2 \varrho-1) \times S^{n-k}(\coth^2 \varrho-1)}$$\end{document} for c = −1, where 1 ≤ k ≤ n − 1.