Ferreira, Walterson P.; Tenenblat, Keti On hypersurfaces with zero r-mean curvature. (English) Zbl 1213.53077 Result. Math. 52, No. 3-4, 261-280 (2008). Summary: We consider hypersurfaces of simply connected space forms, with zero r-mean curvature, associated to a totally geodesic hypersurface by Ribaucour transformations. We characterize such hypersurfaces in terms of solutions of a nonlinear partial differential equation. In particular, we obtain the differential equations whose solutions produce hypersurfaces of the Euclidean space \(\mathbb{R}^{n+1}\), with zero r-mean curvature. We characterize the hypersurfaces corresponding to special solutions of these differential equations. Such solutions provide cylinders, explicit hypersurfaces with zero \((n - 1)\)-mean curvature, rotational hypersurfaces, with zero r-mean curvature, and also hypersurfaces generated by the action of the groups \(O(s) \times O(n - s) \times \{1\}\) or \(O(n - 1) \times I _{2}\) on 2-dimensional surfaces. MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces Keywords:zero r-mean curvature; Ribaucour transformations; totally geodesic hypersurface PDFBibTeX XMLCite \textit{W. P. Ferreira} and \textit{K. Tenenblat}, Result. Math. 52, No. 3--4, 261--280 (2008; Zbl 1213.53077) Full Text: DOI