Hypersurfaces with null higher order mean curvature (Q611924)

The constancy condition of higher elementary symmetric functions of the principal curvatures (= higher mean curvatures \(H_r\)) is an interesting problem in the global theory of hypersurfaces. For the compact case and a non-vanishing constant see [\textit{R. Walter}, Math. Ann. 270, 125--145 (1985; Zbl 0536.53054)]. In the present paper, the authors deal with the case of vanishing higher mean curvature of complete and orientable hypersurfaces of dimension \(n\) in space forms. They make the assumption that an open part does not consist of totally geodesic hypersurfaces. Another assumption is that \(H_{r+1}\) vanishes identically and \(H_r\) never vanishes. One of the results states that, under these assumptions, the manifold is foliated by complete totally geodesic submanifolds of dimension \(n-r\).