On Hypersurfaces with Zero r-Mean Curvature

Abstract.

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 a hypersurface 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 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) × O(n − s) × {1} or O(n − 1) × I ₂ on 2-dimensional surfaces.