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On hypersurfaces with zero r-mean curvature. (English) Zbl 1213.53077

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
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