Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space (Q1736085)

It has been known by \textit{T. Klotz} and \textit{R. Osserman} [Comment. Math. Helv. 41, 313--318 (1966; Zbl 0163.16002)] that totally umbilical spheres and circular cylinders are the only complete surfaces immersed in 3-dimensional Euclidean space \(\mathbb{R}^3\) with nonzero constant mean curvature and its Gaussian curvature does not change the sign. The higher-dimensional case was investigated by \textit{P. Hartman} [Trans. Am. Math. Soc. 245, 363--374 (1978; Zbl 0412.53027)]. In this paper, the authors generalize the above theorem to the case of hypersurfaces with constant higher-order mean curvatures. Theorem: Let \(n\geq 3\) and \(2\leq r < n\) and let \(M\) be a complete hypersurface immersed into the Euclidean space \(\mathbb{R}^{n+1}\) with constant \(r\)-th mean curvature \(H_{r}\neq 0\) and two distinct principal curvatures, one of them being simple. If its Gauss-Kronecker curvature \(K\) does not change sign, then \(K=0\) and \(M\) is isometric to a cylinder \(\mathbb{R} \times \mathbb{S}^{n+1}(\rho) \subset \mathbb{R}^{n+1}\) with radius \(\rho >0\). As a result, they obtain the same as obtained in [the first author and \textit{S. C. García-Martínez}, Geom. Dedicata 156, 31--47 (2012; Zbl 1232.53046)] for the case \(r=2\).