Cohomogeneity one hypersurfaces of the hyperbolic space
Let \(M^{n}\) be an \(n\)-dimensional Riemannian manifold and \(G\subset \text{Iso}(M^{n})\) a closed and connected subgroup of the group of isometries of \(M^{n}\). The action of \(G\) on \(M\) is said to be of cohomogeneity one if its principal orbits are hypersurfaces. A cohomogeneity one Riemannian manifold \(M^{n}\) is called umbilical if all its principal orbits are umbilical. \textit{F. Podestà} and \textit{A. Spiro} [Ann. Global Anal. Geom. 13, No. 2, 169--184 (1995; Zbl 0827.53007)] have proved that if \( f: M^{n}\rightarrow {\mathbb R}^{n+1} \) is an isometric immersion of an umbilical cohomogeneity one compact Riemannian manifold \(M^{n}\) in the Euclidean space, \(n\geq 4\), then \(f(M^{n})\) is a hypersurface of revolution. In connection with this, \textit{A. C. Asperti, F. Mercuri} and \textit{M. H. Noronha} [Boll. Unione Mat. Ital., VII. Ser., B 11, No. 2, Suppl., 199--215 (1997; Zbl 0882.53006)], have shown that if the \(G\)-principal orbits of the cohomogeneity one compact manifold \(M^{n},\, n\geq 4\), have constant sectional curvature, then they are umbilical and therefore \(f(M^{n})\) is also a hypersurface of revolution of \({\mathbb R}^{n+1}\). \textit{J. A. Seixas} [Hipersuperficies de cohomogeneidade um do espaco euclidiano. Ph. D. Thesis, IMECC-UNICAMP (1996)] obtained similar results for complete cohomogeneity one hypersurfaces of the Euclidean space, under a certain condition on the flat portion of \(M^{n}\). The article under review considers umbilical cohomogeneity one complete hypersurfaces of the hyperbolic space which satisfy a restriction similar to that of Seixas, namely, the Bounded Flatness (BF) condition: A complete hypersurface of the hyperbolic space satisfies the BF-condition if the set of points with type number not greater than one, \(\mathcal{P}\), is not the whole \(M^{n}\), and if the connected components of the projection of \(\mathcal{P}\) on the orbit space are compact. Under these conditions, they prove that \(f(M^{n})\) is either a hypersurface of revolution or an NR-example in \(\mathbf{H}^{n+1}(-1)\). NR-examples are non-rotational hypersurfaces characterized in section 5 of the paper.
scientific article; zbMATH DE number 2015139